--- /srv/rebuilderd/tmp/rebuilderdFLHBNm/inputs/macaulay2-common_1.25.06+ds-8_all.deb +++ /srv/rebuilderd/tmp/rebuilderdFLHBNm/out/macaulay2-common_1.25.06+ds-8_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2025-08-25 11:02:26.000000 debian-binary │ --rw-r--r-- 0 0 0 515716 2025-08-25 11:02:26.000000 control.tar.xz │ --rw-r--r-- 0 0 0 30156752 2025-08-25 11:02:26.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 515548 2025-08-25 11:02:26.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 30156028 2025-08-25 11:02:26.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.25.06+ds-8 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 295483 │ │ │ +Installed-Size: 295486 │ │ │ Depends: fonts-glyphicons-halflings (>= 1.009~3.4.1+dfsg), fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 1.0.0), libjs-bootstrap (>= 3.4.1+dfsg), libjs-d3 (>= 3.5.17), libjs-jquery (>= 3.6.1+dfsg+~3.5.14), libjs-katex (>= 0.16.10+~cs6.1.0), libjs-nouislider (>= 15.8.1+ds), libjs-three (>= 111+dfsg1), node-clipboard (>= 2.0.11+ds+~cs9.6.11) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -2857,25 +2857,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 27932 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10640 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41444 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___A__Infinity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 917 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 918 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4348 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_a__Infinity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 56403 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_burke__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3427 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_display__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3016 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_extract__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1714 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_golod__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_is__Golod__A__Inf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2183 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_picture.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14643 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_a__Infinity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 67508 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_burke__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_display__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10392 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_extract__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9277 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_golod__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5933 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_has__Minimal__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6465 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_is__Golod__A__Inf.html │ │ │ @@ -2969,29 +2969,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7615 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4178 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1945 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1748 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1946 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1747 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1888 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1889 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 9248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11329 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11328 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6620 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7258 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9579 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9580 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9741 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9319 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12483 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13802 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8276 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4437 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/AlgebraicSplines/ │ │ │ @@ -3178,15 +3178,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3474 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2077 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_cohomology__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1620 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1294 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3200 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2382 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2381 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_sym__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1879 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_universal__Extension.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3763 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/___Exterior.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4089 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/___Regularity.html │ │ │ @@ -3194,15 +3194,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6517 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_bgg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12129 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_cohomology__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5266 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9044 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10036 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13003 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_projective__Product.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14096 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14095 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_sym__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7650 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_tate__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7928 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/_universal__Extension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12306 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9532 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5311 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BIBasis/ │ │ │ @@ -3227,18 +3227,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 76682 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4226 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2909 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 419 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5570 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5584 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4242 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2912 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289851 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -3426,15 +3426,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3722 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/_twist__Inv__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3690 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/_twist__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 68866 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 58264 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25666 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BernsteinSato/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/dump/ │ │ │ --rw-r--r-- 0 root (0) root (0) 177282 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump │ │ │ +-rw-r--r-- 0 root (0) root (0) 177286 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___B_sq__Constants.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini.out │ │ │ -rw-r--r-- 0 root (0) root (0) 659 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini_spinput_spconfiguration.out │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Is__Projective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/example-output/___Number__To__B_sq__String.out │ │ │ @@ -3472,22 +3472,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3466 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Main__Data__Directory.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5822 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Number__To__B_sq__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3727 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Path__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4558 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Top__Directory.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4574 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Use__Regeneration.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4599 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/___Variable_spgroups.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Component__Member__Test.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15617 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15618 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10011 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Pos__Dim__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9215 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Refine__Sols.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8788 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10591 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Track__Homotopy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9140 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12162 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12760 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12163 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12761 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Incidence__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8549 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Main__Data__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6062 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Parameter__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8481 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_import__Solutions__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12225 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Input__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12047 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Section.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11912 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/_make__B_sq__Slice.html │ │ │ @@ -3502,15 +3502,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10474 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Bertini/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 141158 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6933 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8038 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6928 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6926 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2267 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1576 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Equality_spchecks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1927 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Labels.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2080 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/_action_lp__Complex_cm__List_cm__List_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2099 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/_action_lp__Module_cm__List_cm__List_rp.out │ │ │ @@ -3532,15 +3532,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 62 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8643 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15227 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14939 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18503 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18501 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8439 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9891 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6339 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6812 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4335 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_spoperations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6855 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Equality_spchecks.html │ │ │ @@ -3841,15 +3841,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 124847 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 930 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___H__H^__Z__Z_sp__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 980 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___H__H_sp__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 876 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___H__H_us__Z__Z_sp__Cell__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 484 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___Ring__Map_sp_st_st_sp__Cell__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 479 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___Ring__Map_sp_st_st_sp__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 792 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary__Cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 583 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary__Map_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 206 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex__R__Pn.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex__Sphere.out │ │ │ -rw-r--r-- 0 root (0) root (0) 707 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex__Torus.out │ │ │ @@ -3882,15 +3882,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6685 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___Cell.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11611 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8891 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___H__H^__Z__Z_sp__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8497 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___H__H_sp__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9160 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___H__H_us__Z__Z_sp__Cell__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7185 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___Ring__Map_sp_st_st_sp__Cell__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7180 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___Ring__Map_sp_st_st_sp__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8176 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary__Cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8532 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary__Map_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6528 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex__R__Pn.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6430 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex__Sphere.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6436 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex__Torus.html │ │ │ @@ -3933,19 +3933,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1447 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 721 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact_lp__Chain__Complex_cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 578 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1961 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1962 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3453 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3451 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/CodingTheory/example-output/_enumerate__Vectors.out │ │ │ @@ -4199,15 +4199,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 186 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_hyp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 330 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_information__Rate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_length_lp__Linear__Code_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12822 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_linear__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1920 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_locally__Recoverable__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 947 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_messages.out │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/CodingTheory/example-output/_size_lp__Linear__Code_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2134 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_syndrome__Decode.out │ │ │ -rw-r--r-- 0 root (0) root (0) 970 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_tanner__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2381 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_to__String_lp__Linear__Code_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2652 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_toric__Code.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2644 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_toric__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_universe__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_v__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 199 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_vasconcelos__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 889 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_vector__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 127 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_weight.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1143 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_zero__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1826 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_zero__Sum__Code.out │ │ │ @@ -4246,15 +4246,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5792 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Polynomial__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8436 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Sets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7632 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Strat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5834 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/___Vanishing__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5400 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_alphabet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6193 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_ambient__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6746 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_bitflip__Decode.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24657 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24635 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6057 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_choose__Strat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5752 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_codewords.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13086 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cyclic__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8839 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_cyclic__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6465 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_dim_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5970 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_dual__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5798 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_enumerate__Vectors.html │ │ │ @@ -4270,15 +4270,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6329 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_hyp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6176 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_information__Rate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6005 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_length_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40926 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_linear__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9034 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_locally__Recoverable__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7000 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_messages.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7115 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_minimum__Weight.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22745 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22751 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7143 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_parity__Check__To__Generator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11485 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_quasi__Cyclic__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6855 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_rand__L__D__P__C.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5780 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_rand__No__Repeats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8811 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_random__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5776 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_reduced__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6452 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_reed__Muller__Code.html │ │ │ @@ -4287,38 +4287,38 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6052 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_ring_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12862 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_shorten.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5685 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_shortest__Path.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6693 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_size_lp__Linear__Code_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8667 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_syndrome__Decode.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6589 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_tanner__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7490 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_to__String_lp__Linear__Code_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9262 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_toric__Code.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9254 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_toric__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6718 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_universe__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5848 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_v__Number.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6575 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_vasconcelos__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6374 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_vector__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5317 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_weight.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6300 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_zero__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7352 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/_zero__Sum__Code.html │ │ │ -rw-r--r-- 0 root (0) root (0) 36467 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26490 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15240 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13418 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 13261 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13260 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1045 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 982 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg_lp__Normal__Toric__Variety_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3840 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/___Silent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10103 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8434 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25594 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25593 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5794 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3603 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 107816 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -4401,15 +4401,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4599 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4596 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2102 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1754 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 761 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1322 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_complexity.out │ │ │ @@ -4433,24 +4433,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1851 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 857 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1336 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10478 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1284 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1314 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 379 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ --rw-r--r-- 0 root (0) root (0) 448 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 381 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 449 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5986 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4935 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10327 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5284 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Hom__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Layered.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4742 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -4495,18 +4495,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10902 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35373 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15036 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 706402 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -4842,29 +4842,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 21288 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16146 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 36652 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 36656 2025-08-25 11:02:26.000000 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2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6197 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7545 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7270 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ @@ -5348,106 +5348,106 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 14169 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10678 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5712 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/CotangentSchubert/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239212 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2310 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Codim__Bs__Inv.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19800 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ 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(0) root (0) 9704 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_vb_vb_sp__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17701 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17703 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16999 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 51963 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4930 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/_coefficient__Ring_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7609 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cremona/html/_coefficients_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) 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drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cyclotomic/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 455 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/_cyclotomic__Field.out │ │ │ @@ -5497,46 +5497,46 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 184137 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/ │ │ │ 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│ │ -rw-r--r-- 0 root (0) root (0) 6428 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html │ │ │ @@ -6141,36 +6141,36 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 41804 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33842 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16747 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EdgeIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EigenSolver/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EigenSolver/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 7766 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EigenSolver/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EigenSolver/example-output/ │ │ 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./usr/share/doc/Macaulay2/Elimination/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 98399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/example-output/ │ │ │ @@ -6356,37 +6356,37 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10407 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EngineTests/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8398 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EngineTests/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6990 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EngineTests/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10834 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 366 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 367 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_multiple__Cover.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2170 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2171 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5203 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5211 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_multiple__Cover.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11546 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11547 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5529 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3452 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52732 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Equivariant__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Priority__Queue.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Monomial__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring_lp__Ring_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 222 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_delete__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1283 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1277 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 348 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_exponent__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 449 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_inc__Orbit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2025-08-25 11:02:26.000000 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-rw-r--r-- 0 root (0) root (0) 10316 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6607 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_delete__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6472 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8999 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8993 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6126 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_exponent__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7150 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_inc__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6020 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_insert_lp__Priority__Queue_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5369 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_length_lp__Priority__Queue_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_merge__P__Q.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5290 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_min_lp__Priority__Queue_rp.html │ │ │ @@ -6600,70 +6600,70 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10138 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FGLM/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4568 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FGLM/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2981 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FGLM/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 142955 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 29589 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 29586 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Point__Options.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14279 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 14277 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1054 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 337 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Good__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Smallest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_get__Submatrix__Of__Rank.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1790 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1787 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Rank__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 434 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out │ │ │ --rw-r--r-- 0 root (0) root (0) 25045 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 25048 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 273 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_reorder__Polynomial__Ring.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6134 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Det__Strategy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50413 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50410 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5373 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Max__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4415 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Min__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4447 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Modulus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Point__Options.html │ │ │ --rw-r--r-- 0 root (0) root (0) 27490 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 27488 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14641 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9190 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Good__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5703 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Random__Submatrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Largest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Smallest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10115 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_get__Submatrix__Of__Rank.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11278 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11275 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Dim__At__Most.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9506 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5364 2025-08-25 11:02:26.000000 ./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) 10203 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7912 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ --rw-r--r-- 0 root (0) root (0) 43512 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 43515 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6458 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24783 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26704 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7533 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1850 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1848 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_affine__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_co1__Fitting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 848 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gauss__Col.out │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gotzmann__Test.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_next__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 197 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_quot__Scheme.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 10958 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10956 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_affine__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7448 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_co1__Fitting.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7377 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gauss__Col.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gotzmann__Test.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7246 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_next__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5849 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_quot__Scheme.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9538 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/index.html │ │ │ @@ -6698,15 +6698,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Real__Type_sp__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 99 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__String__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 133 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__String__Type_sp__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Struct__Type_sp__Visible__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Foreign__Object.out │ │ │ -rw-r--r-- 0 root (0) root (0) 226 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out │ │ │ --rw-r--r-- 0 root (0) root (0) 525 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 524 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 92 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Void__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 321 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 181 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out │ │ │ -rw-r--r-- 0 root (0) root (0) 262 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Array__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out │ │ │ @@ -6742,15 +6742,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6208 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Struct__Type.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7402 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Struct__Type_sp__Visible__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12459 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5940 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Foreign__Object.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5323 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5527 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6378 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6377 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5597 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Void__Type.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6731 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12352 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_fast_sp__Fourier_sptransform_spexample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6555 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Array__Type.html │ │ │ @@ -6886,15 +6886,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 105269 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4034 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2459 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2457 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5582 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -6906,15 +6906,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10580 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4665 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4769 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4735 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5945 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14756 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26000 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24542 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24540 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13295 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12519 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9887 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19600 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20028 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7092 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -6966,15 +6966,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8076 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8074 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7017,15 +7017,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8393 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19825 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7021 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7191 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7746 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6964 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5716 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5537 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26820 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8087,42 +8087,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 71259 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Graphs/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 57446 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Graphs/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27398 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Graphs/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56327 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 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./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11279 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5699 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18009 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 705 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -8142,28 +8142,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18704 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/___Point_sp_eq_eq_sp__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/___Point_sp_st_sp__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 585 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/Jets/html/_jets__Info.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4884 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7171 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -8811,59 +8811,59 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2786 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2262 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3423 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1001 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 269 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2261 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3426 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1005 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 267 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6875 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6874 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1031 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1075 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2222 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2220 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4913 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4489 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9537 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13299 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9792 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10619 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6850 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6737 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9791 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10622 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6854 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6735 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18189 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7392 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9428 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14901 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14900 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5106 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8478 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8111 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6177 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8020 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7118 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7204 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8344 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8322 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8320 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5851 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7523 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10854 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6384 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31911 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22352 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8781 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -9023,15 +9023,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70564 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 595 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 594 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -9048,15 +9048,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5005 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5659 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5749 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7711 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7710 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9462 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4957 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6907 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6587 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6730 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6764 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7032 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -9218,30 +9218,30 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 60488 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/___Linear__Truncations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ --rw-r--r-- 0 root (0) root (0) 700 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 701 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1046 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Linear__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 904 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Quasi__Linear.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1150 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) 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2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Quasi__Linear.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8199 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8381 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10456 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.html │ │ │ @@ -9454,15 +9454,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -9512,15 +9512,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 767 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1114 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 484 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_vb_sp__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 912 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 911 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2609 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 313 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_pl_pl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_sl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 299 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_st_st_sp__Ring.out │ │ │ @@ -9707,15 +9707,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 618 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -9730,23 +9730,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 592 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1803 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1523 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1521 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2015 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -9762,15 +9762,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1519 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1541 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ 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│ @@ -9857,15 +9857,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21210 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21213 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10006,15 +10006,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2025-08-25 11:02:26.000000 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0 root (0) root (0) 315 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -10045,15 +10045,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Mutable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Open.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Output__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Prime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Primitive.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1881 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1880 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ready_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Real.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Skew__Commutative.out │ │ │ @@ -10146,30 +10146,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-08-25 11:02:26.000000 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root (0) 551 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1805 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -10244,31 +10244,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_override.out │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13708 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2025-08-25 11:02:26.000000 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│ @@ -10821,15 +10821,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4540 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Max__Reduction__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4532 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Maximal__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 53050 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Method__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5456 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Method__Function__Binary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12315 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Method__Function__Single.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21533 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Method__Function__With__Options.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4036 2025-08-25 11:02:26.000000 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drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5724 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/___Q__Qi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Eliminate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11550 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10568 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Lead__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10503 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__R__U__R.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13943 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14214 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7806 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18043 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4504 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Msolve/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47286 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump │ │ │ @@ -13372,28 +13372,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 18059 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12849 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7429 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2241 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5318 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4235 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4491 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4750 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5012 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4647 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11856 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11855 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6602 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9224 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10098 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8008 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14051 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13133 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -13401,28 +13401,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35826 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 413 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 260 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 412 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 261 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6386 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6706 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8885 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6040 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5704 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6039 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5705 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6831 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9585 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6921 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15265 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9576 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4371 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/toc.html │ │ │ @@ -13475,15 +13475,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5192 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 379736 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1954 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp!.out │ │ │ --rw-r--r-- 0 root (0) root (0) 20228 2025-08-25 11:02:26.000000 ./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) 20227 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5148 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_bs_sp__Multiprojective__Variety.out │ │ │ @@ -13491,15 +13491,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1341 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1275 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2025-08-25 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./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -13522,31 +13522,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1007 2025-08-25 11:02:26.000000 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│ │ -rw-r--r-- 0 root (0) root (0) 155 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2025-08-25 11:02:26.000000 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6518 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -14311,15 +14311,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8147 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608995 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3515 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3508 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14383,20 +14383,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 937 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1561 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1560 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1000 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1001 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1963 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1965 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1337 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_polytope_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2219 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_projective_spspace.out │ │ │ @@ -14421,15 +14421,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 12853 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12846 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11949 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14079 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9111 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -14498,20 +14498,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10561 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13649 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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) 11118 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9104 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10406 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10405 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10956 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10434 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10435 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19155 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13815 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13552 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13554 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12084 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10736 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10563 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10843 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_polytope_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11081 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_projective_spspace.html │ │ │ @@ -14819,37 +14819,37 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40489 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24404 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6254 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1177 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1178 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 578 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Numerical__Interpolation__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1600 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Pseudo__Witness__Set.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1643 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1626 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_is__On__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1341 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1332 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Nullity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12322 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Source__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18900 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set.html │ │ │ @@ -14934,15 +14934,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 25053 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16568 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9573 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 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./usr/share/doc/Macaulay2/NumericalSemigroups/html/_buchweitz__Semigroups.html │ │ │ @@ -15035,16 +15035,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 691 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Degree__Shifts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 285 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Vector__In__Width_sp_pl_sp__Vector__In__Width.out │ │ │ @@ -15058,33 +15058,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Schreyer__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 890 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_image_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1124 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_install__Generators__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9935 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7891 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_standard__Sols.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6388 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_vertex__Spanning__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40232 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18097 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8240 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Oscillators/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PHCpack/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PHCpack/dump/ │ │ │ @@ -16753,15 +16753,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2469 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Hibi_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 485 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Intersection_splattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__L__C__M-lattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2468 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2470 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 306 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_all__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_antichains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_atoms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 253 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_boolean__Lattice.out │ │ │ @@ -16880,15 +16880,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Hibi_spideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7164 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Intersection_splattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6041 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__L__C__M-lattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38795 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5707 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_st.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8668 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8670 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6042 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6593 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_all__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_antichains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7024 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5632 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_atoms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5773 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Posets/html/_augment__Poset.html │ │ │ @@ -17043,30 +17043,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 81139 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2696 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/___Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3736 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated_spprimes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2222 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2223 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1390 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17690 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5762 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_irreducible__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8232 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8732 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8733 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10172 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8209 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11813 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21071 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5735 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9886 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7321 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ @@ -17359,15 +17359,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21833 2025-08-25 11:02:26.000000 ./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) 28399 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -17401,15 +17401,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9061 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9848 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47481 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 54047 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4123 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12440 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8586 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14692 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12230 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9284 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -17545,27 +17545,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2678 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1315 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1522 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5055 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 510 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1149 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3937 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4295 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9992 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7241 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7257 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6973 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14308 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8062 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8060 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11390 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9528 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4800 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -17574,43 +17574,43 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4115 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2798 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50254 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_is__Smooth__Curve.out │ │ │ --rw-r--r-- 0 root (0) root (0) 400 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6061 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/___Details.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6085 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/___Printing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5533 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_is__Smooth__Curve.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8850 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8849 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7683 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Genus14.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7429 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Genus15.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7520 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Via__Plane__Model.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7419 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Via__Space__Model.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11531 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10272 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4915 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 276 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_canonical__Curve__Genus14.out │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2925 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Canonical__Curve__Genus8with8__Points.out │ │ │ --rw-r--r-- 0 root (0) root (0) 484 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 485 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus8__Degree14in__P6.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5382 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_canonical__Curve__Genus14.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5485 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9332 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Canonical__Curve__Genus8with8__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7433 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7434 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6239 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus8__Degree14in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8399 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4166 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ @@ -17622,23 +17622,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 289 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 285 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 421 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8939 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8938 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1615 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10300 2025-08-25 11:02:26.000000 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│ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 93198 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -18099,15 +18099,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_is__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3453 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_jacobian__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1022 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction_lp..._cm__Tries_eq_gt..._rp.out │ │ 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./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6638 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Regularity/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4864 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Regularity/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3191 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Regularity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36804 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/example-output/ │ │ │ @@ -18305,80 +18305,80 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 114689 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1231 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/___Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 454 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Discriminant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2692 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2425 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2424 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6588 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ --rw-r--r-- 0 root (0) root (0) 17437 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17438 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1436 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_conormal__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4385 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1757 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4380 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1756 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7159 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dualize.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7652 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_generic__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3136 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3133 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6758 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1903 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6760 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1904 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 122875 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5793 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 122873 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5791 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1268 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_veronese.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 41 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5949 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Proj.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5166 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Assume__Ordinary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4966 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Duality.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9160 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6173 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4737 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6413 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8671 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10654 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14352 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25096 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25097 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7973 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10977 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12076 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9234 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12071 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9233 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13158 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dualize.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14857 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14856 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8152 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_generic__Polynomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12389 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9185 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9182 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8219 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12570 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10010 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12572 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10011 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14374 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 130501 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16280 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 130499 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16278 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7768 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/_veronese.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19593 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20713 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7336 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 418 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6282 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6316 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5266 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Keep__Statistics__Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5527 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Pre__Run__Script.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5029 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7604 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23865 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7601 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23899 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5662 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2_lp..._cm__Keep__Files_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7559 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9683 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7490 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4878 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -18414,15 +18414,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8415 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SCSCP/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6794 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SCSCP/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3956 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SCSCP/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101515 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2513 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2514 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_arithmetic_spwith_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_compressing_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 623 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_creating_spgates.out │ │ │ -rw-r--r-- 0 root (0) root (0) 142 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_declare__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 635 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_differentiating_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1597 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluate_lp__S__L__Program_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 524 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluating_spgates.out │ │ │ @@ -18457,15 +18457,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4088 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Compiled__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7422 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Interpreted__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4854 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8288 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_measuring_spthe_spsize_spof_spcircuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6033 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_undeclare__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_value__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8900 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_working_spwith_spgate_spmatrices.html │ │ │ --rw-r--r-- 0 root (0) root (0) 36823 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 36824 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20897 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6550 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42778 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3218 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -18700,33 +18700,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 64666 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 45881 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22045 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44888 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2824 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2826 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3437 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5542 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2088 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2087 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6984 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9656 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9254 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3737 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3751 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11974 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11976 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13012 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15236 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3541 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10080 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10079 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14702 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14335 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3882 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9100 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18378 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18702 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3961 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse1.html │ │ │ @@ -18750,24 +18750,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52965 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1860 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1858 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6560 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7263 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16736 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16734 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15598 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11944 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12011 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13853 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11866 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4273 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -18801,15 +18801,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1581 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7722 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -18921,15 +18921,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5605 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6370 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Incidence__Correspondence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4163 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Intersection__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4202 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Isotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8064 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8058 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4976 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6353 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4277 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7875 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11726 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -19559,15 +19559,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_get__Facet__Bases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 967 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_graph__From__Slack__Matrix.out │ │ │ 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15577 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_specific__Slack__Matrix.out │ │ │ @@ -19615,15 +19615,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10463 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5375 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Flag__Indices_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Object_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7130 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Vars_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10128 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6100 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal_lp..._cm__Saturate_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16349 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8305 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8307 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7939 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_set__Ones__Forest.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15037 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Circuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5349 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/SparseResultants/html/_rank_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_reverse__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5628 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_ring_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5589 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7887 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sort__Shape.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65531 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65532 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7079 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21933 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20968 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11034 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178042 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -19966,107 +19966,107 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 203984 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 914 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___G__Mtables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ --rw-r--r-- 0 root (0) root (0) 953 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 954 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1190 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1191 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1172 2025-08-25 11:02:26.000000 ./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) 1173 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1555 2025-08-25 11:02:26.000000 ./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) 297 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 286 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 298 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 287 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 599 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_from__Ordinary__To__Gushel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 716 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ --rw-r--r-- 0 root (0) root (0) 785 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 971 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 786 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 973 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 530 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4324 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 530 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 531 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6372 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6988 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7943 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___G__Mtables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9888 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Hodge__Special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6265 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Intersection__Of__Three__Quadrics__In__P7.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4602 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Singular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9519 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10070 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8773 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Verbose.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_ambient__Fivefold.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9595 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9596 2025-08-25 11:02:26.000000 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│ │ │ --rw-r--r-- 0 root (0) root (0) 6653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6745 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6654 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6746 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6642 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5909 2025-08-25 11:02:26.000000 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9405 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6394 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8483 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7954 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8180 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7955 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8182 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15460 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15461 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6948 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7039 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10491 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6591 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12074 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12073 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11528 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./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 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7730 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9302 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6144 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11933 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10746 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5400 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7019 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7020 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37413 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33496 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14735 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 228935 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/SpectralSequences/example-output/ │ │ │ @@ -20913,25 +20913,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1327 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ --rw-r--r-- 0 root (0) root (0) 487 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1824 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 488 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1822 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1491 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1513 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4326 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4385 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -20964,25 +20964,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9151 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7168 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10611 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12408 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16744 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16745 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8554 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8215 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15811 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8216 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15809 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10346 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9661 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16392 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16391 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5824 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7890 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14994 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14993 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17708 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37577 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34814 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12484 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 135465 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -21180,30 +21180,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37607 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27702 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11513 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24388 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1554 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1429 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 653 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1147 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 572 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1034 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1155 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3201 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2781 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 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│ │ │ +-rw-r--r-- 0 root (0) root (0) 6627 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6976 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7158 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13904 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13484 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21626 2025-08-25 11:02:26.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2025-08-25 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./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8174 2025-08-25 11:02:26.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13901 2025-08-25 11:02:26.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7600 2025-08-25 11:02:26.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7447 2025-08-25 11:02:26.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4016 2025-08-25 11:02:26.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4015 2025-08-25 11:02:26.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3422 2025-08-25 11:02:26.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31435 2025-08-25 11:02:26.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 9585 2025-08-25 11:02:26.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13677 2025-08-25 11:02:26.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 9581 2025-08-25 11:02:26.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13688 2025-08-25 11:02:26.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7188 2025-08-25 11:02:26.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2025-08-25 11:02:26.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10698 2025-08-25 11:02:26.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5773 2025-08-25 11:02:26.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2025-08-25 11:02:26.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4688 2025-08-25 11:02:26.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8877 2025-08-25 11:02:26.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44182 2025-08-25 11:02:26.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19378 2025-08-25 11:02:26.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51414 2025-08-25 11:02:26.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 18359 2025-08-25 11:02:26.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12642 2025-08-25 11:02:26.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10435 2025-08-25 11:02:26.000000 ./usr/share/info/Visualize.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10700 2025-08-25 11:02:26.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33249 2025-08-25 11:02:26.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33229 2025-08-25 11:02:26.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6123 2025-08-25 11:02:26.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8866 2025-08-25 11:02:26.000000 ./usr/share/info/XML.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 49484 2025-08-25 11:02:26.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11345 2025-08-25 10:49:14.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-08-25 11:02:26.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 1.6649s elapsed │ │ │ + -- 1.37168s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 1.93901s elapsed │ │ │ + -- 1.57284s 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)
│ │ │ - -- 1.6649s elapsed
│ │ │ + -- 1.37168s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 1.93901s elapsed
│ │ │ + -- 1.57284s 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) │ │ │ │ - -- 1.6649s elapsed │ │ │ │ + -- 1.37168s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 1.93901s elapsed │ │ │ │ + -- 1.57284s 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 │ │ │ - -- .086914s elapsed │ │ │ + -- .0366431s 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) │ │ │ - -- .614531s elapsed │ │ │ + -- .519998s 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; │ │ │ - -- 6.22936s elapsed │ │ │ + -- 4.7473s 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) │ │ │ - -- .036592s elapsed │ │ │ + -- .0132487s 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) │ │ │ - -- .0579303s elapsed │ │ │ + -- .0606966s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.83516s elapsed │ │ │ + -- 1.47426s 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
│ │ │ - -- .086914s elapsed
│ │ │ + -- .0366431s 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 │ │ │ │ - -- .086914s elapsed │ │ │ │ + -- .0366431s 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)
│ │ │ - -- .614531s elapsed
│ │ │ + -- .519998s 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;
│ │ │ - -- 6.22936s elapsed
│ │ │ + -- 4.7473s 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)
│ │ │ │ - -- .614531s elapsed
│ │ │ │ + -- .519998s 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;
│ │ │ │ - -- 6.22936s elapsed
│ │ │ │ + -- 4.7473s 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)
│ │ │ - -- .036592s elapsed
│ │ │ + -- .0132487s 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)
│ │ │ - -- .0579303s elapsed
│ │ │ + -- .0606966s 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;
│ │ │ - -- 1.83516s elapsed
│ │ │ + -- 1.47426s 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)
│ │ │ │ - -- .036592s elapsed
│ │ │ │ + -- .0132487s 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)
│ │ │ │ - -- .0579303s elapsed
│ │ │ │ + -- .0606966s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.83516s elapsed
│ │ │ │ + -- 1.47426s 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 0.438373s (cpu); 0.358343s (thread); 0s (gc)
│ │ │ + -- used 0.456935s (cpu); 0.383125s (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 0.584346s (cpu); 0.420024s (thread); 0s (gc)
│ │ │ + -- used 0.632135s (cpu); 0.47979s (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 0.438373s (cpu); 0.358343s (thread); 0s (gc)
│ │ │ + -- used 0.456935s (cpu); 0.383125s (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
│ │ │ │ - -- .749652s elapsed
│ │ │ │ + -- .715549s 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
│ │ │ │ - -- 33.8497s elapsed
│ │ │ │ + -- 28.7s 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)
│ │ │ │ - -- 15.078s elapsed
│ │ │ │ + -- 12.6432s 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 0.135933s (cpu); 0.137017s (thread); 0s (gc)
│ │ │ + -- used 0.174238s (cpu); 0.175965s (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 0.584346s (cpu); 0.420024s (thread); 0s (gc)
│ │ │ + -- used 0.632135s (cpu); 0.47979s (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 0.438373s (cpu); 0.358343s (thread); 0s (gc)
│ │ │ │ + -- used 0.456935s (cpu); 0.383125s (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 0.584346s (cpu); 0.420024s (thread); 0s (gc)
│ │ │ │ + -- used 0.632135s (cpu); 0.47979s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:27 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:26 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=9
│ │ │  QmVuY2htYXJr
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Mon Aug 25 12:34:12 UTC 2025
│ │ │ --- Linux sbuild 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-08-02) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Sat Oct  4 04:45:28 UTC 2025
│ │ │ +-- Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.06, compiled with gcc 14.3.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .0997777 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .163973 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 Mon Aug 25 12:34:12 UTC 2025
│ │ │ --- Linux sbuild 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-08-02) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Sat Oct  4 04:45:28 UTC 2025
│ │ │ +-- Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.06, compiled with gcc 14.3.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .0997777 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .163973 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 Mon Aug 25 12:34:12 UTC 2025
│ │ │ │ --- Linux sbuild 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-
│ │ │ │ -08-02) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249
│ │ │ │ +-- beginning computation Sat Oct  4 04:45:28 UTC 2025
│ │ │ │ +-- Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.48-1 (2025-09-20) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.25.06, compiled with gcc 14.3.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .0997777 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .163973 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-
│ │ │ │  1.25.06+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -438,15 +438,15 @@
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│ │ │  #:len=24
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│ │ │  #:len=29
│ │ │  aW1wb3J0SW5jaWRlbmNlTWF0cml4KFN0cmluZyk=
│ │ │  #:len=281
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│ │ ├── ./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/M2-73368-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-126384-0/0", 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/M2-73368-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-126384-0/0", 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/M2-73368-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-126384-0/0", 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/M2-73368-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-126384-0/0", 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/M2-73368-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-126384-0/0", 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/M2-73368-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-126384-0/0", 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
│ │ │ - -- .446192s elapsed
│ │ │ + -- .394805s 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
│ │ │ - -- .596132s elapsed
│ │ │ + -- .424255s 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
│ │ │ - -- .749652s elapsed
│ │ │ + -- .715549s 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
│ │ │ - -- 33.8497s elapsed
│ │ │ + -- 28.7s 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)
│ │ │ - -- 15.078s elapsed
│ │ │ + -- 12.6432s 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
│ │ │ - -- .446192s elapsed
│ │ │ + -- .394805s 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
│ │ │ │ - -- .446192s elapsed
│ │ │ │ + -- .394805s 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
│ │ │ - -- .596132s elapsed
│ │ │ + -- .424255s 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
│ │ │ │ - -- .596132s elapsed
│ │ │ │ + -- .424255s 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
│ │ │ - -- .749652s elapsed
│ │ │ + -- .715549s 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
│ │ │ - -- 33.8497s elapsed
│ │ │ + -- 28.7s 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)
│ │ │ - -- 15.078s elapsed
│ │ │ + -- 12.6432s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.135933s (cpu); 0.137017s (thread); 0s (gc)
│ │ │ + -- used 0.174238s (cpu); 0.175965s (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 0.135933s (cpu); 0.137017s (thread); 0s (gc)
│ │ │ │ + -- used 0.174238s (cpu); 0.175965s (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/___Ring__Map_sp_st_st_sp__Cell__Complex.out
│ │ │ @@ -26,13 +26,13 @@
│ │ │  
│ │ │  o10 : List
│ │ │  
│ │ │  i11 : D = f ** C;
│ │ │  
│ │ │  i12 : cells(1,D)/cellLabel
│ │ │  
│ │ │ -               2
│ │ │ -o12 = {a*b, b*c }
│ │ │ +          2
│ │ │ +o12 = {b*c , a*b}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │  
│ │ │  i7 : C = cellComplex(R,F);
│ │ │  
│ │ │  i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -22,17 +22,17 @@
│ │ │  
│ │ │  o7 = C
│ │ │  
│ │ │  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 }                                       }
│ │ │ -                      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 }
│ │ │ +                        4   5   5   4    3 2   2 3
│ │ │ +o8 = HashTable{0 => {x*y , x , x , x y, x y , x y }                                       }
│ │ │ +                      5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4
│ │ │ +               1 => {x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , x y }
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_face__Poset_lp__Cell__Complex_rp.out
│ │ │ @@ -20,18 +20,18 @@
│ │ │  
│ │ │  i10 : f = newCell({e12,e23,e34,e41});
│ │ │  
│ │ │  i11 : C = cellComplex(R,{f});
│ │ │  
│ │ │  i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 |
│ │ │ -                       | 0 1 0 0 0 1 1 0 1 |
│ │ │ -                       | 0 0 1 0 1 0 1 0 1 |
│ │ │ -                       | 0 0 0 1 1 0 0 1 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ +                       | 0 1 0 0 1 0 1 0 1 |
│ │ │ +                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ +                       | 0 0 0 1 0 1 0 1 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out
│ │ │ @@ -19,16 +19,16 @@
│ │ │                             
│ │ │       -1     0      1      2
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       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}
│ │ │ +       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}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : cells(2,H)/cellLabel
│ │ │  
│ │ │         5 4    4 5
│ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out
│ │ │ @@ -29,13 +29,13 @@
│ │ │          2      2   2
│ │ │  o13 = {a b, b*c , b , a*c}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2 2   2   2     2    2 2       2
│ │ │ -o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │ +        2 2   2 2       2     2    2   2
│ │ │ +o14 = {a b , b c , a*b*c , a*b c, a b*c }
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/___Ring__Map_sp_st_st_sp__Cell__Complex.html
│ │ │ @@ -134,16 +134,16 @@
│ │ │                
    i11 : D = f ** C;
    
│ │ │              
    
│ │ │                
    i12 : cells(1,D)/cellLabel
│ │ │  
│ │ │ -               2
│ │ │ -o12 = {a*b, b*c }
│ │ │ +          2
│ │ │ +o12 = {b*c , a*b}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  
│ │ │ │  o10 = {y*z, x*y}
│ │ │ │  
│ │ │ │  o10 : List
│ │ │ │  i11 : D = f ** C;
│ │ │ │  i12 : cells(1,D)/cellLabel
│ │ │ │  
│ │ │ │ -               2
│ │ │ │ -o12 = {a*b, b*c }
│ │ │ │ +          2
│ │ │ │ +o12 = {b*c , a*b}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_b_e_l_C_e_l_l_C_o_m_p_l_e_x -- relabels a cell complex
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _R_i_n_g_M_a_p_ _*_*_ _C_e_l_l_C_o_m_p_l_e_x -- tensors labels via a ring map
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.html
│ │ │ @@ -112,27 +112,27 @@
│ │ │                
    i7 : C = cellComplex(R,F);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};
│ │ │ │  i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};
│ │ │ │  i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};
│ │ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │ │  i7 : C = cellComplex(R,F);
│ │ │ │  i8 : facePoset C
│ │ │ │  
│ │ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ │ -                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o8 : Poset
│ │ │ │  The labels on the vertices can be controlled via the optional parameter Labels
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -127,18 +127,18 @@
│ │ │  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 }                                       }
│ │ │ -                      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 }
│ │ │ +                        4   5   5   4    3 2   2 3
│ │ │ +o8 = HashTable{0 => {x*y , x , x , x y, x y , x y }                                       }
│ │ │ +                      5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4
│ │ │ +               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 {} │ │ │ │ @@ -38,20 +38,20 @@ │ │ │ │ i7 : C = cellComplex(S,Delta,Labels=>H) │ │ │ │ │ │ │ │ o7 = C │ │ │ │ │ │ │ │ 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 } │ │ │ │ + 4 5 5 4 3 2 2 3 │ │ │ │ +o8 = HashTable{0 => {x*y , x , x , x y, x y , x y } │ │ │ │ } │ │ │ │ - 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, │ │ │ │ + 5 5 2 5 3 5 4 4 2 4 4 5 3 3 5 2 2 4 │ │ │ │ +5 3 5 4 │ │ │ │ + 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 │ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_face__Poset_lp__Cell__Complex_rp.html │ │ │ @@ -127,18 +127,18 @@ │ │ │ 
    i11 : C = cellComplex(R,{f});
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 |
│ │ │ -                       | 0 1 0 0 0 1 1 0 1 |
│ │ │ -                       | 0 0 1 0 1 0 1 0 1 |
│ │ │ -                       | 0 0 0 1 1 0 0 1 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ +                       | 0 1 0 0 1 0 1 0 1 |
│ │ │ +                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ +                       | 0 0 0 1 0 1 0 1 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
    │ │ │ ├── html2text {} │ │ │ │ @@ -25,18 +25,18 @@ │ │ │ │ i7 : e23 = newCell({v2,v3}); │ │ │ │ i8 : e34 = newCell({v3,v4}); │ │ │ │ i9 : e41 = newCell({v4,v1}); │ │ │ │ i10 : f = newCell({e12,e23,e34,e41}); │ │ │ │ i11 : C = cellComplex(R,{f}); │ │ │ │ i12 : facePoset C │ │ │ │ │ │ │ │ -o12 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 | │ │ │ │ - | 0 1 0 0 0 1 1 0 1 | │ │ │ │ - | 0 0 1 0 1 0 1 0 1 | │ │ │ │ - | 0 0 0 1 1 0 0 1 1 | │ │ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 | │ │ │ │ + | 0 1 0 0 1 0 1 0 1 | │ │ │ │ + | 0 0 1 0 1 0 0 1 1 | │ │ │ │ + | 0 0 0 1 0 1 0 1 1 | │ │ │ │ | 0 0 0 0 1 0 0 0 1 | │ │ │ │ | 0 0 0 0 0 1 0 0 1 | │ │ │ │ | 0 0 0 0 0 0 1 0 1 | │ │ │ │ | 0 0 0 0 0 0 0 1 1 | │ │ │ │ | 0 0 0 0 0 0 0 0 1 | │ │ │ │ │ │ │ │ o12 : Poset │ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html │ │ │ @@ -109,16 +109,16 @@ │ │ │ o4 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       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}
│ │ │ +       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}
│ │ │  
│ │ │  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
│ │ │ │  
│ │ │ │ -       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}
│ │ │ │ +       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}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : cells(2,H)/cellLabel
│ │ │ │  
│ │ │ │         5 4    4 5
│ │ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html
│ │ │ @@ -146,16 +146,16 @@
│ │ │  o13 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2 2   2   2     2    2 2       2
│ │ │ -o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │ +        2 2   2 2       2     2    2   2
│ │ │ +o14 = {a b , b c , a*b*c , a*b c, a b*c }
│ │ │  
│ │ │  o14 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -37,16 +37,16 @@ │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ o13 = {a b, b*c , b , a*c} │ │ │ │ │ │ │ │ o13 : List │ │ │ │ i14 : for c in cells(1,relabeledC) list cellLabel(c) │ │ │ │ │ │ │ │ - 2 2 2 2 2 2 2 2 │ │ │ │ -o14 = {b c , a b*c , a*b c, a b , a*b*c } │ │ │ │ + 2 2 2 2 2 2 2 2 │ │ │ │ +o14 = {a b , b c , a*b*c , a*b c, a b*c } │ │ │ │ │ │ │ │ o14 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_e_l_l_L_a_b_e_l -- return the label of a cell │ │ │ │ * _R_i_n_g_M_a_p_ _*_*_ _C_e_l_l_C_o_m_p_l_e_x -- tensors labels via a ring map │ │ │ │ ********** WWaayyss ttoo uussee rreellaabbeellCCeellllCCoommpplleexx:: ********** │ │ │ │ * relabelCellComplex(CellComplex,HashTable) │ │ ├── ./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 0.31971s (cpu); 0.255498s (thread); 0s (gc) │ │ │ + -- used 0.352481s (cpu); 0.276347s (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 0.0944114s (cpu); 0.0965491s (thread); 0s (gc) │ │ │ + -- used 0.107767s (cpu); 0.106301s (thread); 0s (gc) │ │ │ │ │ │ i9 : time n = cartanEilenbergResolution C; │ │ │ - -- used 0.210681s (cpu); 0.141947s (thread); 0s (gc) │ │ │ + -- used 0.245209s (cpu); 0.165727s (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 1.22085s (cpu); 0.469594s (thread); 0s (gc) │ │ │ │ + -- used 1.32136s (cpu); 0.421016s (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 0.290318s (cpu); 0.0831165s (thread); 0s (gc) │ │ │ │ + -- used 0.109229s (cpu); 0.0700805s (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 0.31971s (cpu); 0.255498s (thread); 0s (gc)
    │ │ │ + -- used 0.352481s (cpu); 0.276347s (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 0.31971s (cpu); 0.255498s (thread); 0s (gc) │ │ │ │ + -- used 0.352481s (cpu); 0.276347s (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 0.0944114s (cpu); 0.0965491s (thread); 0s (gc)
    │ │ │ + -- used 0.107767s (cpu); 0.106301s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.210681s (cpu); 0.141947s (thread); 0s (gc)
    │ │ │ + -- used 0.245209s (cpu); 0.165727s (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 0.0944114s (cpu); 0.0965491s (thread); 0s (gc)
│ │ │ │ + -- used 0.107767s (cpu); 0.106301s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.210681s (cpu); 0.141947s (thread); 0s (gc)
│ │ │ │ + -- used 0.245209s (cpu); 0.165727s (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 1.22085s (cpu); 0.469594s (thread); 0s (gc)
│ │ │ + -- used 1.32136s (cpu); 0.421016s (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 0.290318s (cpu); 0.0831165s (thread); 0s (gc)
│ │ │ + -- used 0.109229s (cpu); 0.0700805s (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 0.232816s (cpu); 0.15335s (thread); 0s (gc)
│ │ │ + -- used 0.232309s (cpu); 0.160305s (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 0.3124s (cpu); 0.267202s (thread); 0s (gc)
│ │ │ + -- used 0.376348s (cpu); 0.306266s (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 0.987671s (cpu); 0.410041s (thread); 0s (gc)
│ │ │ + -- used 1.57236s (cpu); 0.464711s (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 2.11836s (cpu); 1.81442s (thread); 0s (gc)
│ │ │ + -- used 2.17581s (cpu); 2.00385s (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 0.269302s (cpu); 0.183236s (thread); 0s (gc)
│ │ │ + -- used 0.275283s (cpu); 0.203339s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0828103s (cpu); 0.0853985s (thread); 0s (gc)
│ │ │ + -- used 0.0999208s (cpu); 0.0997039s (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 0.0399153s (cpu); 0.0384691s (thread); 0s (gc)
│ │ │ + -- used 0.125453s (cpu); 0.0581131s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.374482s (cpu); 0.175344s (thread); 0s (gc)
│ │ │ + -- used 0.353613s (cpu); 0.192681s (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 0.256077s (cpu); 0.0789773s (thread); 0s (gc)
│ │ │ + -- used 0.265085s (cpu); 0.112407s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.285021s (cpu); 0.0817288s (thread); 0s (gc)
│ │ │ + -- used 0.325386s (cpu); 0.117706s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 3.47382s (cpu); 1.0716s (thread); 0s (gc)
│ │ │ + -- used 6.29037s (cpu); 1.4562s (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 3.79404s (cpu); 1.25533s (thread); 0s (gc)
│ │ │ + -- used 5.5711s (cpu); 1.35947s (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 0.950841s (cpu); 0.48594s (thread); 0s (gc)
│ │ │ + -- used 1.33243s (cpu); 0.532122s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0633156s (cpu); 0.0374833s (thread); 0s (gc)
│ │ │ + -- used 0.0947806s (cpu); 0.0435965s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.12208s (cpu); 0.0384165s (thread); 0s (gc)
│ │ │ + -- used 0.0945323s (cpu); 0.0389938s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 2.19317s (cpu); 0.927318s (thread); 0s (gc)
│ │ │ + -- used 3.84446s (cpu); 1.18569s (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 0.767272s (cpu); 0.237748s (thread); 0s (gc)
│ │ │ + -- used 0.867843s (cpu); 0.262903s (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 1.22085s (cpu); 0.469594s (thread); 0s (gc)
│ │ │ + -- used 1.32136s (cpu); 0.421016s (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 0.290318s (cpu); 0.0831165s (thread); 0s (gc)
│ │ │ + -- used 0.109229s (cpu); 0.0700805s (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 0.232816s (cpu); 0.15335s (thread); 0s (gc)
│ │ │ + -- used 0.232309s (cpu); 0.160305s (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 0.3124s (cpu); 0.267202s (thread); 0s (gc)
│ │ │ + -- used 0.376348s (cpu); 0.306266s (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 0.232816s (cpu); 0.15335s (thread); 0s (gc)
│ │ │ │ + -- used 0.232309s (cpu); 0.160305s (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 0.3124s (cpu); 0.267202s (thread); 0s (gc)
│ │ │ │ + -- used 0.376348s (cpu); 0.306266s (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 0.987671s (cpu); 0.410041s (thread); 0s (gc)
│ │ │ + -- used 1.57236s (cpu); 0.464711s (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 2.11836s (cpu); 1.81442s (thread); 0s (gc)
│ │ │ + -- used 2.17581s (cpu); 2.00385s (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 0.269302s (cpu); 0.183236s (thread); 0s (gc)
│ │ │ + -- used 0.275283s (cpu); 0.203339s (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 0.0828103s (cpu); 0.0853985s (thread); 0s (gc)
│ │ │ + -- used 0.0999208s (cpu); 0.0997039s (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 0.987671s (cpu); 0.410041s (thread); 0s (gc)
│ │ │ │ + -- used 1.57236s (cpu); 0.464711s (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 2.11836s (cpu); 1.81442s (thread); 0s (gc)
│ │ │ │ + -- used 2.17581s (cpu); 2.00385s (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 0.269302s (cpu); 0.183236s (thread); 0s (gc)
│ │ │ │ + -- used 0.275283s (cpu); 0.203339s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0828103s (cpu); 0.0853985s (thread); 0s (gc)
│ │ │ │ + -- used 0.0999208s (cpu); 0.0997039s (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 0.0399153s (cpu); 0.0384691s (thread); 0s (gc)
│ │ │ + -- used 0.125453s (cpu); 0.0581131s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : time Euler I
│ │ │ - -- used 0.374482s (cpu); 0.175344s (thread); 0s (gc)
│ │ │ + -- used 0.353613s (cpu); 0.192681s (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 0.256077s (cpu); 0.0789773s (thread); 0s (gc)
│ │ │ + -- used 0.265085s (cpu); 0.112407s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    │ │ │
    │ │ │ 
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.285021s (cpu); 0.0817288s (thread); 0s (gc)
│ │ │ + -- used 0.325386s (cpu); 0.117706s (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 0.0399153s (cpu); 0.0384691s (thread); 0s (gc) │ │ │ │ + -- used 0.125453s (cpu); 0.0581131s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : time Euler I │ │ │ │ - -- used 0.374482s (cpu); 0.175344s (thread); 0s (gc) │ │ │ │ + -- used 0.353613s (cpu); 0.192681s (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 0.256077s (cpu); 0.0789773s (thread); 0s (gc) │ │ │ │ + -- used 0.265085s (cpu); 0.112407s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = 2 │ │ │ │ i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1}) │ │ │ │ - -- used 0.285021s (cpu); 0.0817288s (thread); 0s (gc) │ │ │ │ + -- used 0.325386s (cpu); 0.117706s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ Now consider an example in \PP^2 \times \PP^2. │ │ │ │ i12 : R=MultiProjCoordRing({2,2}) │ │ │ │ │ │ │ │ o12 = R │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ @@ -70,15 +70,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │
    │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 3.47382s (cpu); 1.0716s (thread); 0s (gc)
│ │ │ + -- used 6.29037s (cpu); 1.4562s (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 3.79404s (cpu); 1.25533s (thread); 0s (gc)
│ │ │ + -- used 5.5711s (cpu); 1.35947s (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 3.47382s (cpu); 1.0716s (thread); 0s (gc)
│ │ │ │ + -- used 6.29037s (cpu); 1.4562s (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 3.79404s (cpu); 1.25533s (thread); 0s (gc)
│ │ │ │ + -- used 5.5711s (cpu); 1.35947s (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 0.950841s (cpu); 0.48594s (thread); 0s (gc)
│ │ │ + -- used 1.33243s (cpu); 0.532122s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -83,15 +83,15 @@
│ │ │         h
│ │ │          1
    │ │ │
    │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0633156s (cpu); 0.0374833s (thread); 0s (gc)
│ │ │ + -- used 0.0947806s (cpu); 0.0435965s (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 0.12208s (cpu); 0.0384165s (thread); 0s (gc)
│ │ │ + -- used 0.0945323s (cpu); 0.0389938s (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 0.950841s (cpu); 0.48594s (thread); 0s (gc)
│ │ │ │ + -- used 1.33243s (cpu); 0.532122s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0633156s (cpu); 0.0374833s (thread); 0s (gc)
│ │ │ │ + -- used 0.0947806s (cpu); 0.0435965s (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 0.12208s (cpu); 0.0384165s (thread); 0s (gc)
│ │ │ │ + -- used 0.0945323s (cpu); 0.0389938s (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 2.19317s (cpu); 0.927318s (thread); 0s (gc)
│ │ │ + -- used 3.84446s (cpu); 1.18569s (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 0.767272s (cpu); 0.237748s (thread); 0s (gc)
│ │ │ + -- used 0.867843s (cpu); 0.262903s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 2.19317s (cpu); 0.927318s (thread); 0s (gc)
│ │ │ │ + -- used 3.84446s (cpu); 1.18569s (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 0.767272s (cpu); 0.237748s (thread); 0s (gc)
│ │ │ │ + -- used 0.867843s (cpu); 0.262903s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cartesian__Code.out
│ │ │ @@ -45,108 +45,108 @@
│ │ │  
│ │ │  i2 : F=GF(4);
│ │ │  
│ │ │  i3 : R=F[x,y];
│ │ │  
│ │ │  i4 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                           }
│ │ │ +o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | a   a   |
│ │ │ +                                             Code => image | a+1 0   |
│ │ │ +                                                           | a+1 0   |
│ │ │ +                                                           | a   a   |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 1   a+1 |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                             GeneratorMatrix => | a a 1   1 0 0 1 a+1 a+1 |
│ │ │ -                                                                | a a a+1 0 0 0 1 0   0   |
│ │ │ -                                             Generators => {{a, a, 1, 1, 0, 0, 1, a + 1, a + 1}, {a, a, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0 0 a   0   0 |
│ │ │ -                                                                  | 0 1 0 0 0 0 a   0   0 |
│ │ │ -                                                                  | 0 0 1 0 0 0 a+1 a+1 0 |
│ │ │ -                                                                  | 0 0 0 1 0 0 0   a   0 |
│ │ │ -                                                                  | 0 0 0 0 1 0 0   0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 1 0   0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 0   1   1 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, a, 0, 0}, {0, 1, 0, 0, 0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ -                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}, {0, a}, {a, 0}}
│ │ │ +                                                           | 1   a+1 |
│ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1 0 0 1 1   |
│ │ │ +                                                                | 0   0   a a 0 0 0 1 a+1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, 1, a + 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 1 a   |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 1 a   |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 a 0   |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 a 0   |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 a a+1 |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 1, a}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, a}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o4 : EvaluationCode
│ │ │  
│ │ │  i5 : C.LinearCode
│ │ │  
│ │ │                                    9
│ │ │ -o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                  BaseField => F
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | a   a   |
│ │ │ +                Code => image | a+1 0   |
│ │ │ +                              | a+1 0   |
│ │ │ +                              | a   a   |
│ │ │                                | a   a   |
│ │ │ -                              | 1   a+1 |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │                                | 1   1   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a+1 0   |
│ │ │ -                GeneratorMatrix => | a a 1   1 0 0 1 a+1 a+1 |
│ │ │ -                                   | a a a+1 0 0 0 1 0   0   |
│ │ │ -                Generators => {{a, a, 1, 1, 0, 0, 1, a + 1, a + 1}, {a, a, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 0 a   0   0 |
│ │ │ -                                     | 0 1 0 0 0 0 a   0   0 |
│ │ │ -                                     | 0 0 1 0 0 0 a+1 a+1 0 |
│ │ │ -                                     | 0 0 0 1 0 0 0   a   0 |
│ │ │ -                                     | 0 0 0 0 1 0 0   0   0 |
│ │ │ -                                     | 0 0 0 0 0 1 0   0   0 |
│ │ │ -                                     | 0 0 0 0 0 0 0   1   1 |
│ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, 0, a, 0, 0}, {0, 1, 0, 0, 0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ +                              | 1   a+1 |
│ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1 1   |
│ │ │ +                                   | 0   0   a a 0 0 0 1 a+1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, 1, a + 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 1 a   |
│ │ │ +                                     | 0 1 0 0 0 0 0 1 a   |
│ │ │ +                                     | 0 0 1 0 0 0 0 a 0   |
│ │ │ +                                     | 0 0 0 1 0 0 0 a 0   |
│ │ │ +                                     | 0 0 0 0 1 0 0 a a+1 |
│ │ │ +                                     | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                     | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 1, a}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │  
│ │ │  o5 : LinearCode
│ │ │  
│ │ │  i6 : F=GF(4);
│ │ │  
│ │ │  i7 : R=F[x,y];
│ │ │  
│ │ │  i8 : C=cartesianCode(F,{{0,1,a},{0,1,a}},matrix{{1,2},{2,3}})
│ │ │  
│ │ │ -o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                   }
│ │ │ +o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                            }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │                                                             | a   a+1 |
│ │ │ +                                                           | a+1 1   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 1   0 0 a+1 a   0 0 0 1 |
│ │ │ -                                                                | a+1 0 0 1   a+1 0 0 0 1 |
│ │ │ -                                             Generators => {{1, 0, 0, a + 1, a, 0, 0, 0, 1}, {a + 1, 0, 0, 1, a + 1, 0, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a 0 0 0 a |
│ │ │ -                                                                  | 0 1 0 0 0 0 0 0 0 |
│ │ │ -                                                                  | 0 0 1 0 0 0 0 0 0 |
│ │ │ -                                                                  | 0 0 0 1 a 0 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 1 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 1 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 0 1 0 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, a, 0, 0, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, a, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {1, a}, {a, 1}, {0, 0}, {0, 1}, {1, 0}, {1, 1}}
│ │ │ +                                                           | 1   a+1 |
│ │ │ +                                             GeneratorMatrix => | 0 0 a   a+1 0 0 0 1 1   |
│ │ │ +                                                                | 0 0 a+1 1   0 0 0 1 a+1 |
│ │ │ +                                             Generators => {{0, 0, a, a + 1, 0, 0, 0, 1, 1}, {0, 0, a + 1, 1, 0, 0, 0, 1, a + 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 0 0   |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 0 0   |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 1 a+1 |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 a 1   |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 1, a + 1}, {0, 0, 0, 1, 0, 0, 0, a, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, a}}
│ │ │                                           2   2 3
│ │ │                      PolynomialSet => {t t , t t }
│ │ │                                         0 1   0 1
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2          3           2
│ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a + 1)t  + a*t )
│ │ │                                                0           0      0   1           1      1
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_messages.out
│ │ │ @@ -2,15 +2,15 @@
│ │ │  
│ │ │  i1 : F=GF(4,Variable=>a);
│ │ │  
│ │ │  i2 : R=linearCode(F,{{1,1,1}});
│ │ │  
│ │ │  i3 : messages R
│ │ │  
│ │ │ -o3 = {{0}, {1}, {a}, {a + 1}}
│ │ │ +o3 = {{1}, {a}, {a + 1}, {0}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : messages hammingCode(2,3)
│ │ │  
│ │ │  o4 = {{1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 0, 1, 1}, {1, 1, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_order__Code.out
│ │ │ @@ -10,39 +10,39 @@
│ │ │  
│ │ │  o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                                                                           }
│ │ │                      Points => {{0, 0}, {a, a}, {a + 1, a}, {1, a}, {a, a + 1}, {a + 1, a + 1}, {1, a + 1}, {0, 1}}
│ │ │                                                                 8
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                                                                    }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1 0   0   0   0 0   0   0   |
│ │ │ -                                                           | 1 a+1 1   a   1 a   a+1 a+1 |
│ │ │ -                                                           | 1 a   a+1 a   1 a+1 a+1 1   |
│ │ │ -                                                           | 1 1   a   a   1 1   a+1 a   |
│ │ │ -                                                           | 1 a+1 a   a+1 1 a   a   1   |
│ │ │ -                                                           | 1 a   1   a+1 1 a+1 a   a   |
│ │ │ -                                                           | 1 1   a+1 a+1 1 1   a   a+1 |
│ │ │ -                                                           | 1 0   0   1   0 0   1   0   |
│ │ │ -                                             GeneratorMatrix => | 1 1   1   1   1   1   1   1 |
│ │ │ +                                             Code => image | 0   1 0   0   0   0 0   0   |
│ │ │ +                                                           | a+1 1 a+1 1   a   1 a   a+1 |
│ │ │ +                                                           | 1   1 a   a+1 a   1 a+1 a+1 |
│ │ │ +                                                           | a   1 1   a   a   1 1   a+1 |
│ │ │ +                                                           | 1   1 a+1 a   a+1 1 a   a   |
│ │ │ +                                                           | a   1 a   1   a+1 1 a+1 a   |
│ │ │ +                                                           | a+1 1 1   a+1 a+1 1 1   a   |
│ │ │ +                                                           | 0   1 0   0   1   0 0   1   |
│ │ │ +                                             GeneratorMatrix => | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ +                                                                | 1 1   1   1   1   1   1   1 |
│ │ │                                                                  | 0 a+1 a   1   a+1 a   1   0 |
│ │ │                                                                  | 0 1   a+1 a   a   1   a+1 0 |
│ │ │                                                                  | 0 a   a   a   a+1 a+1 a+1 1 |
│ │ │                                                                  | 0 1   1   1   1   1   1   0 |
│ │ │                                                                  | 0 a   a+1 1   a   a+1 1   0 |
│ │ │                                                                  | 0 a+1 a+1 a+1 a   a   a   1 |
│ │ │ -                                                                | 0 a+1 1   a   1   a   a+1 0 |
│ │ │ -                                             Generators => {{1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}, {0, a + 1, 1, a, 1, a, a + 1, 0}}
│ │ │ +                                             Generators => {{0, a + 1, 1, a, 1, a, a + 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}}
│ │ │                                               ParityCheckMatrix => | 1 1 1 1 1 1 1 1 |
│ │ │                                               ParityCheckRows => {{1, 1, 1, 1, 1, 1, 1, 1}}
│ │ │                                                  2               3    2        4
│ │ │                      VanishingIdeal => ideal (t t  + t t  + t , t  + t  + t , t  + t )
│ │ │                                                0 1    0 1    0   0    1    1   1    1
│ │ │ -                                          2   2         3       2
│ │ │ -                    PolynomialSet => {1, t , t t , t , t , t , t , t t }
│ │ │ -                                          0   0 1   1   0   0   1   0 1
│ │ │ +                                                2   2         3       2
│ │ │ +                    PolynomialSet => {t t , 1, t , t t , t , t , t , t }
│ │ │ +                                       0 1      0   0 1   1   0   0   1
│ │ │  
│ │ │  i5 : F = GF(4);
│ │ │  
│ │ │  i6 : R = F[x,y];
│ │ │  
│ │ │  i7 : I = ideal(x^3+y^2+y)
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_ring_lp__Linear__Code_rp.out
│ │ │ @@ -2,30 +2,30 @@
│ │ │  
│ │ │  i1 : C = hammingCode(2, 3)
│ │ │  
│ │ │                                         7
│ │ │  o1 = LinearCode{AmbientModule => (GF 2)                                                                                   }
│ │ │                  BaseField => GF 2
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | 1 1 0 1 |
│ │ │ -                              | 1 0 1 1 |
│ │ │ +                Code => image | 1 0 1 1 |
│ │ │ +                              | 1 1 0 1 |
│ │ │                                | 1 1 1 0 |
│ │ │                                | 1 0 0 0 |
│ │ │                                | 0 1 0 0 |
│ │ │                                | 0 0 1 0 |
│ │ │                                | 0 0 0 1 |
│ │ │                  GeneratorMatrix => | 1 1 1 1 0 0 0 |
│ │ │ -                                   | 1 0 1 0 1 0 0 |
│ │ │ -                                   | 0 1 1 0 0 1 0 |
│ │ │ +                                   | 0 1 1 0 1 0 0 |
│ │ │ +                                   | 1 0 1 0 0 1 0 |
│ │ │                                     | 1 1 0 0 0 0 1 |
│ │ │ -                Generators => {{1, 1, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0}, {0, 1, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │ +                Generators => {{1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0}, {1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 0, 1}}
│ │ │                  ParityCheckMatrix => | 1 1 1 1 0 0 0 |
│ │ │                                       | 0 0 1 1 1 1 0 |
│ │ │ -                                     | 0 1 0 1 0 1 1 |
│ │ │ -                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 0, 1, 1}}
│ │ │ +                                     | 0 1 0 1 1 0 1 |
│ │ │ +                ParityCheckRows => {{1, 1, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0, 1}}
│ │ │  
│ │ │  o1 : LinearCode
│ │ │  
│ │ │  i2 : ring(C)
│ │ │  
│ │ │  o2 = GF 2
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/example-output/_toric__Code.out
│ │ │ @@ -30,35 +30,35 @@
│ │ │  
│ │ │  i5 : T.LinearCode
│ │ │  
│ │ │                                         9
│ │ │  o5 = LinearCode{AmbientModule => (GF 4)                                                                                                                                                                                                                                             }
│ │ │                  BaseField => GF 4
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | a   a+1 1   a   a+1 a   |
│ │ │ -                              | a+1 a   a+1 1   a   a   |
│ │ │ -                              | a   a+1 a+1 a+1 a+1 1   |
│ │ │ -                              | 1   1   1   1   1   1   |
│ │ │ +                Code => image | a+1 a   a   a   a   1   |
│ │ │                                | a+1 a   1   a+1 a   a+1 |
│ │ │                                | 1   1   a+1 a   1   a+1 |
│ │ │                                | 1   1   a   a+1 1   a   |
│ │ │ -                              | a+1 a   a   a   a   1   |
│ │ │                                | a   a+1 a   1   a+1 a+1 |
│ │ │ -                GeneratorMatrix => | a   a+1 a   1 a+1 1   1   a+1 a   |
│ │ │ -                                   | a+1 a   a+1 1 a   1   1   a   a+1 |
│ │ │ -                                   | 1   a+1 a+1 1 1   a+1 a   a   a   |
│ │ │ -                                   | a   1   a+1 1 a+1 a   a+1 a   1   |
│ │ │ -                                   | a+1 a   a+1 1 a   1   1   a   a+1 |
│ │ │ -                                   | a   a   1   1 a+1 a+1 a   1   a+1 |
│ │ │ -                Generators => {{a, a + 1, a, 1, a + 1, 1, 1, a + 1, a}, {a + 1, a, a + 1, 1, a, 1, 1, a, a + 1}, {1, a + 1, a + 1, 1, 1, a + 1, a, a, a}, {a, 1, a + 1, 1, a + 1, a, a + 1, a, 1}, {a + 1, a, a + 1, 1, a, 1, 1, a, a + 1}, {a, a, 1, 1, a + 1, a + 1, a, 1, a + 1}}
│ │ │ -                ParityCheckMatrix => | 1 0 0 0 a+1 0   a+1 0   a   |
│ │ │ -                                     | 0 1 0 0 0   a+1 a   0   1   |
│ │ │ -                                     | 0 0 1 0 0   a   0   a   a+1 |
│ │ │ -                                     | 0 0 0 1 a   0   0   a+1 1   |
│ │ │ -                ParityCheckRows => {{1, 0, 0, 0, a + 1, 0, a + 1, 0, a}, {0, 1, 0, 0, 0, a + 1, a, 0, 1}, {0, 0, 1, 0, 0, a, 0, a, a + 1}, {0, 0, 0, 1, a, 0, 0, a + 1, 1}}
│ │ │ +                              | 1   1   1   1   1   1   |
│ │ │ +                              | a   a+1 a+1 a+1 a+1 1   |
│ │ │ +                              | a+1 a   a+1 1   a   a   |
│ │ │ +                              | a   a+1 1   a   a+1 a   |
│ │ │ +                GeneratorMatrix => | a+1 a+1 1   1   a   1 a   a+1 a   |
│ │ │ +                                   | a   a   1   1   a+1 1 a+1 a   a+1 |
│ │ │ +                                   | a   1   a+1 a   a   1 a+1 a+1 1   |
│ │ │ +                                   | a   a+1 a   a+1 1   1 a+1 1   a   |
│ │ │ +                                   | a   a   1   1   a+1 1 a+1 a   a+1 |
│ │ │ +                                   | 1   a+1 a+1 a   a+1 1 1   a   a   |
│ │ │ +                Generators => {{a + 1, a + 1, 1, 1, a, 1, a, a + 1, a}, {a, a, 1, 1, a + 1, 1, a + 1, a, a + 1}, {a, 1, a + 1, a, a, 1, a + 1, a + 1, 1}, {a, a + 1, a, a + 1, 1, 1, a + 1, 1, a}, {a, a, 1, 1, a + 1, 1, a + 1, a, a + 1}, {1, a + 1, a + 1, a, a + 1, 1, 1, a, a}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 a+1 0   0 a+1 a 0   |
│ │ │ +                                     | 0 1 0 1   a+1 0 0   0 a   |
│ │ │ +                                     | 0 0 1 a+1 a   0 0   a 0   |
│ │ │ +                                     | 0 0 0 0   0   1 a   1 a+1 |
│ │ │ +                ParityCheckRows => {{1, 0, 0, a + 1, 0, 0, a + 1, a, 0}, {0, 1, 0, 1, a + 1, 0, 0, 0, a}, {0, 0, 1, a + 1, a, 0, 0, a, 0}, {0, 0, 0, 0, 0, 1, a, 1, a + 1}}
│ │ │  
│ │ │  o5 : LinearCode
│ │ │  
│ │ │  i6 : length T.LinearCode
│ │ │  
│ │ │  o6 = 9
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_cartesian__Code.html
│ │ │ @@ -182,76 +182,76 @@
│ │ │                  
    i3 : R=F[x,y];
    
│ │ │                
    
│ │ │                  
    i4 : C=cartesianCode(F,{{0,1,a},{0,1,a}},{1+x+y,x*y})
│ │ │  
│ │ │ -o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                           }
│ │ │ +o4 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | a   a   |
│ │ │ +                                             Code => image | a+1 0   |
│ │ │ +                                                           | a+1 0   |
│ │ │ +                                                           | a   a   |
│ │ │                                                             | a   a   |
│ │ │ -                                                           | 1   a+1 |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                                           | a+1 0   |
│ │ │ -                                             GeneratorMatrix => | a a 1   1 0 0 1 a+1 a+1 |
│ │ │ -                                                                | a a a+1 0 0 0 1 0   0   |
│ │ │ -                                             Generators => {{a, a, 1, 1, 0, 0, 1, a + 1, a + 1}, {a, a, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 0 0 a   0   0 |
│ │ │ -                                                                  | 0 1 0 0 0 0 a   0   0 |
│ │ │ -                                                                  | 0 0 1 0 0 0 a+1 a+1 0 |
│ │ │ -                                                                  | 0 0 0 1 0 0 0   a   0 |
│ │ │ -                                                                  | 0 0 0 0 1 0 0   0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 1 0   0   0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 0   1   1 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, a, 0, 0}, {0, 1, 0, 0, 0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ -                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0}, {1, 1}, {0, a}, {a, 0}}
│ │ │ +                                                           | 1   a+1 |
│ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1 0 0 1 1   |
│ │ │ +                                                                | 0   0   a a 0 0 0 1 a+1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, 1, a + 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 1 a   |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 1 a   |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 a 0   |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 a 0   |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 a a+1 |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 1, a}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, a}}
│ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2         3           2
│ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a + 1)y  + a*y)
│ │ │  
│ │ │  o4 : EvaluationCode
    
│ │ │                
    
│ │ │                  
    i5 : C.LinearCode
│ │ │  
│ │ │                                    9
│ │ │ -o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                    }
│ │ │ +o5 = LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                  BaseField => F
│ │ │                  cache => CacheTable{}
│ │ │ -                Code => image | a   a   |
│ │ │ +                Code => image | a+1 0   |
│ │ │ +                              | a+1 0   |
│ │ │ +                              | a   a   |
│ │ │                                | a   a   |
│ │ │ -                              | 1   a+1 |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │                                | 1   1   |
│ │ │ -                              | a+1 0   |
│ │ │ -                              | a+1 0   |
│ │ │ -                GeneratorMatrix => | a a 1   1 0 0 1 a+1 a+1 |
│ │ │ -                                   | a a a+1 0 0 0 1 0   0   |
│ │ │ -                Generators => {{a, a, 1, 1, 0, 0, 1, a + 1, a + 1}, {a, a, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 0 a   0   0 |
│ │ │ -                                     | 0 1 0 0 0 0 a   0   0 |
│ │ │ -                                     | 0 0 1 0 0 0 a+1 a+1 0 |
│ │ │ -                                     | 0 0 0 1 0 0 0   a   0 |
│ │ │ -                                     | 0 0 0 0 1 0 0   0   0 |
│ │ │ -                                     | 0 0 0 0 0 1 0   0   0 |
│ │ │ -                                     | 0 0 0 0 0 0 0   1   1 |
│ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, 0, a, 0, 0}, {0, 1, 0, 0, 0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ +                              | 1   a+1 |
│ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1 1   |
│ │ │ +                                   | 0   0   a a 0 0 0 1 a+1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, 1, a + 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 1 a   |
│ │ │ +                                     | 0 1 0 0 0 0 0 1 a   |
│ │ │ +                                     | 0 0 1 0 0 0 0 a 0   |
│ │ │ +                                     | 0 0 0 1 0 0 0 a 0   |
│ │ │ +                                     | 0 0 0 0 1 0 0 a a+1 |
│ │ │ +                                     | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                     | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 1, a}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │  
│ │ │  o5 : LinearCode
    
│ │ │                
    
│ │ │          
│ │ │          
    
│ │ │ @@ -288,40 +288,40 @@
│ │ │                  
    i7 : R=F[x,y];
    
│ │ │
    │ │ │ 
    i8 : C=cartesianCode(F,{{0,1,a},{0,1,a}},matrix{{1,2},{2,3}})
│ │ │  
│ │ │ -o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                   }
│ │ │ +o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │ -                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                            }
│ │ │ +                    LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │ -                                             Code => image | 1   a+1 |
│ │ │ +                                             Code => image | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │                                                             | a   a+1 |
│ │ │ +                                                           | a+1 1   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 1   1   |
│ │ │ -                                             GeneratorMatrix => | 1   0 0 a+1 a   0 0 0 1 |
│ │ │ -                                                                | a+1 0 0 1   a+1 0 0 0 1 |
│ │ │ -                                             Generators => {{1, 0, 0, a + 1, a, 0, 0, 0, 1}, {a + 1, 0, 0, 1, a + 1, 0, 0, 0, 1}}
│ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a 0 0 0 a |
│ │ │ -                                                                  | 0 1 0 0 0 0 0 0 0 |
│ │ │ -                                                                  | 0 0 1 0 0 0 0 0 0 |
│ │ │ -                                                                  | 0 0 0 1 a 0 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 1 0 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 1 0 0 |
│ │ │ -                                                                  | 0 0 0 0 0 0 0 1 0 |
│ │ │ -                                             ParityCheckRows => {{1, 0, 0, 0, a, 0, 0, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, a, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {1, a}, {a, 1}, {0, 0}, {0, 1}, {1, 0}, {1, 1}}
│ │ │ +                                                           | 1   a+1 |
│ │ │ +                                             GeneratorMatrix => | 0 0 a   a+1 0 0 0 1 1   |
│ │ │ +                                                                | 0 0 a+1 1   0 0 0 1 a+1 |
│ │ │ +                                             Generators => {{0, 0, a, a + 1, 0, 0, 0, 1, 1}, {0, 0, a + 1, 1, 0, 0, 0, 1, a + 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 0 0   |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 0 0   |
│ │ │ +                                                                  | 0 0 1 0 0 0 0 1 a+1 |
│ │ │ +                                                                  | 0 0 0 1 0 0 0 a 1   |
│ │ │ +                                                                  | 0 0 0 0 1 0 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 1 0 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 1, a + 1}, {0, 0, 0, 1, 0, 0, 0, a, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0}, {0, 1}, {1, 1}, {a, a}}
│ │ │                                           2   2 3
│ │ │                      PolynomialSet => {t t , t t }
│ │ │                                         0 1   0 1
│ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │                                                3           2          3           2
│ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a + 1)t  + a*t )
│ │ │                                                0           0      0   1           1      1
│ │ │ ├── html2text {}
│ │ │ │ @@ -123,49 +123,49 @@
│ │ │ │  o4 = EvaluationCode{cache => CacheTable{}
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | a   a   |
│ │ │ │ +                                             Code => image | a+1 0   |
│ │ │ │ +                                                           | a+1 0   |
│ │ │ │ +                                                           | a   a   |
│ │ │ │                                                             | a   a   |
│ │ │ │ -                                                           | 1   a+1 |
│ │ │ │                                                             | 1   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 1   1   |
│ │ │ │ -                                                           | a+1 0   |
│ │ │ │ -                                                           | a+1 0   |
│ │ │ │ -                                             GeneratorMatrix => | a a 1   1 0 0
│ │ │ │ -1 a+1 a+1 |
│ │ │ │ -                                                                | a a a+1 0 0 0
│ │ │ │ -1 0   0   |
│ │ │ │ -                                             Generators => {{a, a, 1, 1, 0, 0,
│ │ │ │ -1, a + 1, a + 1}, {a, a, a + 1, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                                                           | 1   a+1 |
│ │ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1
│ │ │ │ +0 0 1 1   |
│ │ │ │ +                                                                | 0   0   a a 0
│ │ │ │ +0 0 1 a+1 |
│ │ │ │ +                                             Generators => {{a + 1, a + 1, a,
│ │ │ │ +a, 1, 0, 0, 1, 1}, {0, 0, a, a, 0, 0, 0, 1, a + 1}}
│ │ │ │                                               ParityCheckMatrix => | 1 0 0 0 0 0
│ │ │ │ -a   0   0 |
│ │ │ │ +0 1 a   |
│ │ │ │                                                                    | 0 1 0 0 0 0
│ │ │ │ -a   0   0 |
│ │ │ │ +0 1 a   |
│ │ │ │                                                                    | 0 0 1 0 0 0
│ │ │ │ -a+1 a+1 0 |
│ │ │ │ +0 a 0   |
│ │ │ │                                                                    | 0 0 0 1 0 0
│ │ │ │ -0   a   0 |
│ │ │ │ +0 a 0   |
│ │ │ │                                                                    | 0 0 0 0 1 0
│ │ │ │ -0   0   0 |
│ │ │ │ +0 a a+1 |
│ │ │ │                                                                    | 0 0 0 0 0 1
│ │ │ │ -0   0   0 |
│ │ │ │ +0 0 0   |
│ │ │ │                                                                    | 0 0 0 0 0 0
│ │ │ │ -0   1   1 |
│ │ │ │ +1 0 0   |
│ │ │ │                                               ParityCheckRows => {{1, 0, 0, 0,
│ │ │ │ -0, 0, a, 0, 0}, {0, 1, 0, 0, 0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1,
│ │ │ │ -0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
│ │ │ │ -1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 1}}
│ │ │ │ -                    Points => {{1, a}, {a, 1}, {a, a}, {0, 0}, {0, 1}, {1, 0},
│ │ │ │ -{1, 1}, {0, a}, {a, 0}}
│ │ │ │ +0, 0, 0, 1, a}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0,
│ │ │ │ +0, 0, 1, 0, 0, 0, a, 0}, {0, 0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0,
│ │ │ │ +0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0},
│ │ │ │ +{0, 1}, {1, 1}, {a, a}}
│ │ │ │                      PolynomialSet => {x + y + 1, x*y}
│ │ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │ │                                                3           2         3
│ │ │ │  2
│ │ │ │                      VanishingIdeal => ideal (x  + (a + 1)x  + a*x, y  + (a +
│ │ │ │  1)y  + a*y)
│ │ │ │  
│ │ │ │ @@ -173,38 +173,38 @@
│ │ │ │  i5 : C.LinearCode
│ │ │ │  
│ │ │ │                                    9
│ │ │ │  o5 = LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                  BaseField => F
│ │ │ │                  cache => CacheTable{}
│ │ │ │ -                Code => image | a   a   |
│ │ │ │ +                Code => image | a+1 0   |
│ │ │ │ +                              | a+1 0   |
│ │ │ │ +                              | a   a   |
│ │ │ │                                | a   a   |
│ │ │ │ -                              | 1   a+1 |
│ │ │ │                                | 1   0   |
│ │ │ │                                | 0   0   |
│ │ │ │                                | 0   0   |
│ │ │ │                                | 1   1   |
│ │ │ │ -                              | a+1 0   |
│ │ │ │ -                              | a+1 0   |
│ │ │ │ -                GeneratorMatrix => | a a 1   1 0 0 1 a+1 a+1 |
│ │ │ │ -                                   | a a a+1 0 0 0 1 0   0   |
│ │ │ │ -                Generators => {{a, a, 1, 1, 0, 0, 1, a + 1, a + 1}, {a, a, a +
│ │ │ │ -1, 0, 0, 0, 1, 0, 0}}
│ │ │ │ -                ParityCheckMatrix => | 1 0 0 0 0 0 a   0   0 |
│ │ │ │ -                                     | 0 1 0 0 0 0 a   0   0 |
│ │ │ │ -                                     | 0 0 1 0 0 0 a+1 a+1 0 |
│ │ │ │ -                                     | 0 0 0 1 0 0 0   a   0 |
│ │ │ │ -                                     | 0 0 0 0 1 0 0   0   0 |
│ │ │ │ -                                     | 0 0 0 0 0 1 0   0   0 |
│ │ │ │ -                                     | 0 0 0 0 0 0 0   1   1 |
│ │ │ │ -                ParityCheckRows => {{1, 0, 0, 0, 0, 0, a, 0, 0}, {0, 1, 0, 0,
│ │ │ │ -0, 0, a, 0, 0}, {0, 0, 1, 0, 0, 0, a + 1, a + 1, 0}, {0, 0, 0, 1, 0, 0, 0, a,
│ │ │ │ -0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0,
│ │ │ │ -0, 0, 1, 1}}
│ │ │ │ +                              | 1   a+1 |
│ │ │ │ +                GeneratorMatrix => | a+1 a+1 a a 1 0 0 1 1   |
│ │ │ │ +                                   | 0   0   a a 0 0 0 1 a+1 |
│ │ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 0, 0, 1, 1}, {0, 0, a,
│ │ │ │ +a, 0, 0, 0, 1, a + 1}}
│ │ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 0 0 1 a   |
│ │ │ │ +                                     | 0 1 0 0 0 0 0 1 a   |
│ │ │ │ +                                     | 0 0 1 0 0 0 0 a 0   |
│ │ │ │ +                                     | 0 0 0 1 0 0 0 a 0   |
│ │ │ │ +                                     | 0 0 0 0 1 0 0 a a+1 |
│ │ │ │ +                                     | 0 0 0 0 0 1 0 0 0   |
│ │ │ │ +                                     | 0 0 0 0 0 0 1 0 0   |
│ │ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 1, a}, {0, 1, 0, 0,
│ │ │ │ +0, 0, 0, 1, a}, {0, 0, 1, 0, 0, 0, 0, a, 0}, {0, 0, 0, 1, 0, 0, 0, a, 0}, {0,
│ │ │ │ +0, 0, 0, 1, 0, 0, a, a + 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1,
│ │ │ │ +0, 0}}
│ │ │ │  
│ │ │ │  o5 : LinearCode
│ │ │ │  ********** aa rriinngg,, aa lliisstt aanndd aa MMaattrriixx aarree ggiivveenn **********
│ │ │ │      *   Usage:
│ │ │ │              cartesianCode(F, L, M)
│ │ │ │      * Inputs:
│ │ │ │            o F, a _r_i_n_g,
│ │ │ │ @@ -223,49 +223,49 @@
│ │ │ │  o8 = EvaluationCode{cache => CacheTable{}
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │ -                                             Code => image | 1   a+1 |
│ │ │ │ -                                                           | 0   0   |
│ │ │ │ +                                             Code => image | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │ -                                                           | a+1 1   |
│ │ │ │                                                             | a   a+1 |
│ │ │ │ +                                                           | a+1 1   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 1   1   |
│ │ │ │ -                                             GeneratorMatrix => | 1   0 0 a+1 a
│ │ │ │ -0 0 0 1 |
│ │ │ │ -                                                                | a+1 0 0 1
│ │ │ │ -a+1 0 0 0 1 |
│ │ │ │ -                                             Generators => {{1, 0, 0, a + 1, a,
│ │ │ │ -0, 0, 0, 1}, {a + 1, 0, 0, 1, a + 1, 0, 0, 0, 1}}
│ │ │ │ -                                             ParityCheckMatrix => | 1 0 0 0 a 0
│ │ │ │ -0 0 a |
│ │ │ │ +                                                           | 1   a+1 |
│ │ │ │ +                                             GeneratorMatrix => | 0 0 a   a+1 0
│ │ │ │ +0 0 1 1   |
│ │ │ │ +                                                                | 0 0 a+1 1   0
│ │ │ │ +0 0 1 a+1 |
│ │ │ │ +                                             Generators => {{0, 0, a, a + 1, 0,
│ │ │ │ +0, 0, 1, 1}, {0, 0, a + 1, 1, 0, 0, 0, 1, a + 1}}
│ │ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0
│ │ │ │ +0 0 0   |
│ │ │ │                                                                    | 0 1 0 0 0 0
│ │ │ │ -0 0 0 |
│ │ │ │ +0 0 0   |
│ │ │ │                                                                    | 0 0 1 0 0 0
│ │ │ │ -0 0 0 |
│ │ │ │ -                                                                  | 0 0 0 1 a 0
│ │ │ │ -0 0 0 |
│ │ │ │ +0 1 a+1 |
│ │ │ │ +                                                                  | 0 0 0 1 0 0
│ │ │ │ +0 a 1   |
│ │ │ │ +                                                                  | 0 0 0 0 1 0
│ │ │ │ +0 0 0   |
│ │ │ │                                                                    | 0 0 0 0 0 1
│ │ │ │ -0 0 0 |
│ │ │ │ -                                                                  | 0 0 0 0 0 0
│ │ │ │ -1 0 0 |
│ │ │ │ +0 0 0   |
│ │ │ │                                                                    | 0 0 0 0 0 0
│ │ │ │ -0 1 0 |
│ │ │ │ +1 0 0   |
│ │ │ │                                               ParityCheckRows => {{1, 0, 0, 0,
│ │ │ │ -a, 0, 0, 0, a}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0,
│ │ │ │ -0, 0, 1, a, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0,
│ │ │ │ -0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ │ -                    Points => {{a, a}, {0, a}, {a, 0}, {1, a}, {a, 1}, {0, 0},
│ │ │ │ -{0, 1}, {1, 0}, {1, 1}}
│ │ │ │ +0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 1, a + 1},
│ │ │ │ +{0, 0, 0, 1, 0, 0, 0, a, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0,
│ │ │ │ +0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {0, 0}, {1, 0},
│ │ │ │ +{0, 1}, {1, 1}, {a, a}}
│ │ │ │                                           2   2 3
│ │ │ │                      PolynomialSet => {t t , t t }
│ │ │ │                                         0 1   0 1
│ │ │ │                      Sets => {{0, 1, a}, {0, 1, a}}
│ │ │ │                                                3           2          3
│ │ │ │  2
│ │ │ │                      VanishingIdeal => ideal (t  + (a + 1)t  + a*t , t  + (a +
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_messages.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │                
    i2 : R=linearCode(F,{{1,1,1}});
    
│ │ │
    │ │ │ 
    i3 : messages R
│ │ │  
│ │ │ -o3 = {{0}, {1}, {a}, {a + 1}}
│ │ │ +o3 = {{1}, {a}, {a + 1}, {0}}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -14,15 +14,15 @@ │ │ │ │ Given a code C of dimension $k$ over a finite field $F$, this function returns │ │ │ │ the list that contains all the elements of $F^k$. Every element of the list can │ │ │ │ be used to encode a message using the linear code C. │ │ │ │ i1 : F=GF(4,Variable=>a); │ │ │ │ i2 : R=linearCode(F,{{1,1,1}}); │ │ │ │ i3 : messages R │ │ │ │ │ │ │ │ -o3 = {{0}, {1}, {a}, {a + 1}} │ │ │ │ +o3 = {{1}, {a}, {a + 1}, {0}} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : messages hammingCode(2,3) │ │ │ │ │ │ │ │ o4 = {{1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 0, 1, 1}, {1, 1, 1, 0}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {1, 1, 1, 1}, {0, 1, 1, 0}, {0, 1, 0, 1}, {0, 1, 0, 0}, {0, 1, 1, 1}, │ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/html/_order__Code.html │ │ │ @@ -122,39 +122,39 @@ │ │ │ │ │ │ o4 = EvaluationCode{cache => CacheTable{} } │ │ │ Points => {{0, 0}, {a, a}, {a + 1, a}, {1, a}, {a, a + 1}, {a + 1, a + 1}, {1, a + 1}, {0, 1}} │ │ │ 8 │ │ │ LinearCode => LinearCode{AmbientModule => F } │ │ │ BaseField => F │ │ │ cache => CacheTable{} │ │ │ - Code => image | 1 0 0 0 0 0 0 0 | │ │ │ - | 1 a+1 1 a 1 a a+1 a+1 | │ │ │ - | 1 a a+1 a 1 a+1 a+1 1 | │ │ │ - | 1 1 a a 1 1 a+1 a | │ │ │ - | 1 a+1 a a+1 1 a a 1 | │ │ │ - | 1 a 1 a+1 1 a+1 a a | │ │ │ - | 1 1 a+1 a+1 1 1 a a+1 | │ │ │ - | 1 0 0 1 0 0 1 0 | │ │ │ - GeneratorMatrix => | 1 1 1 1 1 1 1 1 | │ │ │ + Code => image | 0 1 0 0 0 0 0 0 | │ │ │ + | a+1 1 a+1 1 a 1 a a+1 | │ │ │ + | 1 1 a a+1 a 1 a+1 a+1 | │ │ │ + | a 1 1 a a 1 1 a+1 | │ │ │ + | 1 1 a+1 a a+1 1 a a | │ │ │ + | a 1 a 1 a+1 1 a+1 a | │ │ │ + | a+1 1 1 a+1 a+1 1 1 a | │ │ │ + | 0 1 0 0 1 0 0 1 | │ │ │ + GeneratorMatrix => | 0 a+1 1 a 1 a a+1 0 | │ │ │ + | 1 1 1 1 1 1 1 1 | │ │ │ | 0 a+1 a 1 a+1 a 1 0 | │ │ │ | 0 1 a+1 a a 1 a+1 0 | │ │ │ | 0 a a a a+1 a+1 a+1 1 | │ │ │ | 0 1 1 1 1 1 1 0 | │ │ │ | 0 a a+1 1 a a+1 1 0 | │ │ │ | 0 a+1 a+1 a+1 a a a 1 | │ │ │ - | 0 a+1 1 a 1 a a+1 0 | │ │ │ - Generators => {{1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}, {0, a + 1, 1, a, 1, a, a + 1, 0}} │ │ │ + Generators => {{0, a + 1, 1, a, 1, a, a + 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1}, {0, a + 1, a, 1, a + 1, a, 1, 0}, {0, 1, a + 1, a, a, 1, a + 1, 0}, {0, a, a, a, a + 1, a + 1, a + 1, 1}, {0, 1, 1, 1, 1, 1, 1, 0}, {0, a, a + 1, 1, a, a + 1, 1, 0}, {0, a + 1, a + 1, a + 1, a, a, a, 1}} │ │ │ ParityCheckMatrix => | 1 1 1 1 1 1 1 1 | │ │ │ ParityCheckRows => {{1, 1, 1, 1, 1, 1, 1, 1}} │ │ │ 2 3 2 4 │ │ │ VanishingIdeal => ideal (t t + t t + t , t + t + t , t + t ) │ │ │ 0 1 0 1 0 0 1 1 1 1 │ │ │ - 2 2 3 2 │ │ │ - PolynomialSet => {1, t , t t , t , t , t , t , t t } │ │ │ - 0 0 1 1 0 0 1 0 1 │ │ │ + 2 2 3 2 │ │ │ + PolynomialSet => {t t , 1, t , t t , t , t , t , t } │ │ │ + 0 1 0 0 1 1 0 0 1 │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    given the ideal of the finite algebra associated to the order function and a list of points

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -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 0.980621s (cpu); 0.68151s (thread); 0s (gc) │ │ │ │ + -- used 1.39841s (cpu); 0.807282s (thread); 0s (gc) │ │ │ │ 2 │ │ │ │ Tally{{{1, 1}} => 2 } │ │ │ │ {{2, 2}, {1, 2}} => 4 │ │ │ │ │ │ │ │ - -- used 0.510606s (cpu); 0.3771s (thread); 0s (gc) │ │ │ │ + -- used 0.723122s (cpu); 0.482851s (thread); 0s (gc) │ │ │ │ 3 │ │ │ │ Tally{{{2, 2}, {1, 2}} => 2} │ │ │ │ {{3, 3}, {2, 3}} => 1 │ │ │ │ │ │ │ │ - -- used 0.102776s (cpu); 0.102358s (thread); 0s (gc) │ │ │ │ + -- used 0.188781s (cpu); 0.13224s (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; │ │ │ - -- 2.89707s elapsed │ │ │ + -- 2.43406s elapsed │ │ │ │ │ │ i10 : elapsedTime assert isIntegrable A │ │ │ - -- 5.39791s elapsed │ │ │ + -- 4.20724s 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); │ │ │ - -- .9059s elapsed │ │ │ + -- .482271s elapsed │ │ │ │ │ │ 4 4 │ │ │ o14 : Matrix F <-- F │ │ │ │ │ │ i15 : elapsedTime A1 = gaugeTransform(g, A); │ │ │ - -- 1.39932s elapsed │ │ │ + -- 1.10007s elapsed │ │ │ │ │ │ i16 : elapsedTime assert isIntegrable A1 │ │ │ - -- .921725s elapsed │ │ │ + -- .823957s 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); │ │ │ - -- .4357s elapsed │ │ │ + -- .384379s elapsed │ │ │ │ │ │ i20 : elapsedTime assert isIntegrable A2 │ │ │ - -- .714266s elapsed │ │ │ + -- .659956s 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; │ │ │ - -- .247268s elapsed │ │ │ + -- .199563s elapsed │ │ │ │ │ │ i8 : elapsedTime assert isIntegrable A │ │ │ - -- .286292s elapsed │ │ │ + -- .218137s 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 0.258684s (cpu); 0.1808s (thread); 0s (gc) │ │ │ │ + -- used 0.261441s (cpu); 0.194449s (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 │ │ │ @@ -134,58 +134,58 @@ │ │ │ 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;
│ │ │ - -- 2.89707s elapsed
    │ │ │ + -- 2.43406s elapsed │ │ │
    │ │ │ 
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.39791s elapsed
    │ │ │ + -- 4.20724s elapsed │ │ │
    │ │ │ 
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -167,30 +167,30 @@
│ │ │
    		│ │ │ 
    i13 : B = {1_D,dx,dy,dx*dy};
    │ │ │
    │ │ │ 
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .9059s elapsed
│ │ │ + -- .482271s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    │ │ │
    │ │ │ 
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.39932s elapsed
    │ │ │ + -- 1.10007s elapsed │ │ │
    │ │ │ 
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .921725s elapsed
    │ │ │ + -- .823957s elapsed │ │ │
    │ │ │ 
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -227,21 +227,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    │ │ │
    │ │ │ 
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .4357s elapsed
    │ │ │ + -- .384379s elapsed │ │ │
    │ │ │ 
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .714266s elapsed
    │ │ │ + -- .659956s 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;
│ │ │ │ - -- 2.89707s elapsed
│ │ │ │ + -- 2.43406s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.39791s elapsed
│ │ │ │ + -- 4.20724s 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);
│ │ │ │ - -- .9059s elapsed
│ │ │ │ + -- .482271s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.39932s elapsed
│ │ │ │ + -- 1.10007s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .921725s elapsed
│ │ │ │ + -- .823957s 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);
│ │ │ │ - -- .4357s elapsed
│ │ │ │ + -- .384379s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .714266s elapsed
│ │ │ │ + -- .659956s 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;
│ │ │ - -- .247268s elapsed
    
│ │ │ + -- .199563s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .286292s elapsed
    
│ │ │ + -- .218137s 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;
│ │ │ │ - -- .247268s elapsed
│ │ │ │ + -- .199563s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .286292s elapsed
│ │ │ │ + -- .218137s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.58318s (cpu); 0.935998s (thread); 0s (gc)
│ │ │ + -- used 1.88088s (cpu); 1.18249s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ - -- used 1.30066s (cpu); 0.983759s (thread); 0s (gc)
│ │ │ + -- used 1.4482s (cpu); 0.959392s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │ @@ -62,26 +62,26 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.196051s (cpu); 0.141072s (thread); 0s (gc)
│ │ │ + -- used 0.242895s (cpu); 0.16424s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ - -- used 0.117767s (cpu); 0.056998s (thread); 0s (gc)
│ │ │ + -- used 0.182014s (cpu); 0.054872s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00609207s (cpu); 0.00406524s (thread); 0s (gc)
│ │ │ + -- used 0.00400154s (cpu); 0.00450706s (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 0.131563s (cpu); 0.0830383s (thread); 0s (gc)
│ │ │ + -- used 0.0439231s (cpu); 0.0470118s (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 0.0270197s (cpu); 0.0276383s (thread); 0s (gc)
│ │ │ + -- used 0.0320153s (cpu); 0.0317012s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.538583s (cpu); 0.430648s (thread); 0s (gc)
│ │ │ + -- used 0.604541s (cpu); 0.473326s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0549243s (cpu); 0.0581869s (thread); 0s (gc)
│ │ │ + -- used 0.157936s (cpu); 0.0918695s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000500408s (cpu); 0.00215721s (thread); 0s (gc)
│ │ │ + -- used 0.000157337s (cpu); 0.0023842s (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 0.573193s (cpu); 0.453822s (thread); 0s (gc)
│ │ │ + -- used 0.453919s (cpu); 0.453238s (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 0.0120027s (cpu); 0.00969265s (thread); 0s (gc)
│ │ │ + -- used 0.0120457s (cpu); 0.0118729s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.225754s (cpu); 0.163208s (thread); 0s (gc)
│ │ │ + -- used 0.349374s (cpu); 0.229296s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.20829s (cpu); 4.50938s (thread); 0s (gc)
│ │ │ + -- used 5.20784s (cpu); 4.89261s (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 0.000659066s (cpu); 0.00207662s (thread); 0s (gc)
│ │ │ + -- used 0.00249297s (cpu); 0.00246029s (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 0.0508216s (cpu); 0.0503314s (thread); 0s (gc)
│ │ │ + -- used 0.0639073s (cpu); 0.0608789s (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 0.667941s (cpu); 0.515211s (thread); 0s (gc)
│ │ │ + -- used 0.541903s (cpu); 0.472751s (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 0.392139s (cpu); 0.317841s (thread); 0s (gc)
│ │ │ + -- used 0.443089s (cpu); 0.366583s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0157833s (cpu); 0.0172382s (thread); 0s (gc)
│ │ │ + -- used 0.053748s (cpu); 0.0206246s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00190065s (cpu); 0.00298267s (thread); 0s (gc)
│ │ │ + -- used 0.00321856s (cpu); 0.00354258s (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 0.00674931s (cpu); 0.00981537s (thread); 0s (gc)
│ │ │ + -- used 0.0107303s (cpu); 0.0113342s (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 0.00882568s (cpu); 0.00945813s (thread); 0s (gc)
│ │ │ + -- used 0.00731456s (cpu); 0.010468s (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 1.33692s (cpu); 0.940575s (thread); 0s (gc)
│ │ │ + -- used 1.19405s (cpu); 0.933775s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 0.000189867s (cpu); 1.4877e-05s (thread); 0s (gc)
│ │ │ + -- used 9.9517e-05s (cpu); 1.6071e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 9.2374e-05s (cpu); 0.00151987s (thread); 0s (gc)
│ │ │ + -- used 9.1742e-05s (cpu); 0.00168813s (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 0.318484s (cpu); 0.16471s (thread); 0s (gc)
│ │ │ + -- used 0.302691s (cpu); 0.175561s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.011616s (cpu); 0.0120564s (thread); 0s (gc)
│ │ │ + -- used 0.0259473s (cpu); 0.0142352s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.147873s (cpu); 0.0667188s (thread); 0s (gc)
│ │ │ + -- used 0.169943s (cpu); 0.0835485s (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 0.0376208s (cpu); 0.0384761s (thread); 0s (gc)
│ │ │ + -- used 0.0663537s (cpu); 0.0432652s (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 0.258684s (cpu); 0.1808s (thread); 0s (gc)
│ │ │ + -- used 0.261441s (cpu); 0.194449s (thread); 0s (gc)
│ │ │  
│ │ │                  -e        -d        -c        -b        -a
│ │ │  o6 = ideal (u + --*v, t + --*v, z + --*v, y + --*v, x + --*v)
│ │ │                   f         f         f         f         f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,49 +47,49 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.535432s (cpu); 0.419255s (thread); 0s (gc)
│ │ │ + -- used 0.564081s (cpu); 0.489367s (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 0.406043s (cpu); 0.271651s (thread); 0s (gc)
│ │ │ + -- used 0.473689s (cpu); 0.329225s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ - -- used 0.0250447s (cpu); 0.024676s (thread); 0s (gc)
│ │ │ + -- used 0.0403782s (cpu); 0.0269503s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ - -- used 0.0964579s (cpu); 0.0991384s (thread); 0s (gc)
│ │ │ + -- used 0.264194s (cpu); 0.150719s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │ @@ -105,15 +105,15 @@
│ │ │         ZZ
│ │ │  o9 = ------[x ..x ]
│ │ │       100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.20942s (cpu); 0.134778s (thread); 0s (gc)
│ │ │ + -- used 0.0705414s (cpu); 0.0710159s (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 0.494061s (cpu); 0.339918s (thread); 0s (gc)
│ │ │ + -- used 0.404225s (cpu); 0.256195s (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 0.513534s (cpu); 0.385771s (thread); 0s (gc)
│ │ │ + -- used 0.446911s (cpu); 0.285039s (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 1.30229s (cpu); 0.969049s (thread); 0s (gc)
│ │ │ + -- used 1.06527s (cpu); 0.823376s (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 0.00329632s (cpu); 0.000397406s (thread); 0s (gc)
│ │ │ + -- used 0.000152419s (cpu); 0.00043236s (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 0.204411s (cpu); 0.137253s (thread); 0s (gc)
│ │ │ + -- used 0.216536s (cpu); 0.156827s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.503555s (cpu); 0.363993s (thread); 0s (gc)
│ │ │ + -- used 0.425593s (cpu); 0.361768s (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 0.142169s (cpu); 0.0615885s (thread); 0s (gc)
│ │ │ + -- used 0.0480851s (cpu); 0.0481438s (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 3.24317s (cpu); 1.76112s (thread); 0s (gc)
│ │ │ + -- used 3.7022s (cpu); 2.11836s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 0.000207039s (cpu); 2.7321e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000181497s (cpu); 2.8442e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 5.38785s (cpu); 2.93636s (thread); 0s (gc)
│ │ │ + -- used 6.21311s (cpu); 3.37963s (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 4.38057s (cpu); 2.47597s (thread); 0s (gc)
│ │ │ + -- used 4.98679s (cpu); 2.93795s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │                        x x  - x x
│ │ │                         1 2    0 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
│ │ │  
│ │ │  i3 : time psi = approximateInverseMap phi
│ │ │ - -- used 0.375337s (cpu); 0.235592s (thread); 0s (gc)
│ │ │ + -- used 0.377808s (cpu); 0.262595s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 10
│ │ │  -- approximateInverseMap: step 2 of 10
│ │ │  -- approximateInverseMap: step 3 of 10
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ @@ -106,15 +106,15 @@
│ │ │                       }
│ │ │  
│ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ - -- used 0.234106s (cpu); 0.154149s (thread); 0s (gc)
│ │ │ + -- used 0.228652s (cpu); 0.167555s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │ @@ -189,15 +189,15 @@
│ │ │                         0      1 4      0 5      1 5     2 5      3 5      4 5      5      0 6      1 6      2 6      4 6      5 6      6      0 7      1 7      2 7     3 7      4 7      5 7      6 7     0 8      1 8      3 8     4 8      5 8      6 8     7 8
│ │ │                       }
│ │ │  
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ - -- used 2.48249s (cpu); 1.81057s (thread); 0s (gc)
│ │ │ + -- used 1.97346s (cpu); 1.72259s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ @@ -254,15 +254,15 @@
│ │ │  i9 : -- but...
│ │ │       phi * psi == 1
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ - -- used 3.44756s (cpu); 2.65658s (thread); 0s (gc)
│ │ │ + -- used 3.15177s (cpu); 2.80774s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   2                  2                             2                                       2                                                2                                                           2                                                                       2                                                                              2                                                                                            2         2                 2                             2                                       2                                              2                                                           2                                                                   2                                                                               2                                                                                          2        2                   2                            2                                      2                                                  2                                                          2                                                                      2                                                                               2                                                                                            2        2                   2                             2                                      2                                                 2                                                          2                                                                    2                                                                               2                                                                                          2         2                2                          2                                      2                                                 2                                                           2                                                                       2                                                                            2                                                                                          2        2                  2                         2                                       2                                                2                                                           2                                                                      2                                                                               2                                                                                        2       2                  2                             2                                    2                                                2                                                          2                                                                    2                                                                               2                                                                                       2       2                 2                           2                                      2                                                 2                                                            2                                                                       2                                                                                2                                                                                           2        2                   2                             2                                       2                                                  2                                                            2                                                                   2                                                                            2                                                                                          2      2                 2                            2                                       2                                                2                                                         2                                                                        2                                                                               2                                                                                          2     2                   2                             2                                      2                                                   2                                                          2                                                                     2                                                                               2                                                                                          2         2                2                            2                                       2                                                 2                                                           2                                                                      2                                                                                  2                                                                                              2      2                  2                            2                                    2                                                2                                                            2                                                                    2                                                                                2                                                                                          2       2                  2                            2                                    2                                                   2                                                        2                                                                         2                                                                               2                                                                                           2       2                  2                             2                                       2                                                 2                                                          2                                                                       2                                                                               2                                                                                       2
│ │ │  o4 = map (ringP8, ringP14, {- 95x  + 181x x  + 1028x  - 1384x x  - 1455x x  + 559x  - 502x x  + 1264x x  - 162x x  + 1209x  - 180x x  - 504x x  - 1168x x  - 676x x  + 501x  + 73x x  + 1263x x  + 1035x x  + 844x x  + 1593x x  + 785x  + 982x x  - 412x x  + 1335x x  + 1136x x  + 826x x  + 1078x x  + 1158x  + 335x x  - 982x x  - 1479x x  - 15x x  + 1363x x  + 1397x x  - 575x x  - 71x  + 1255x x  - 1138x x  - 1590x x  + 604x x  + 1182x x  - 63x x  - 1382x x  - 1255x x  - 613x , - 1444x  + 575x x  + 767x  - 1495x x  + 1631x x  - 217x  - 294x x  - 1511x x  - 504x x  - 1284x  - 1459x x  + 152x x  + 141x x  - 10x x  - 95x  + 1056x x  + 654x x  + 1397x x  - 930x x  + 578x x  - 696x  + 759x x  + 733x x  + 505x x  - 609x x  + 526x x  - 659x x  + 846x  + 1253x x  - 1519x x  + 635x x  + 576x x  + 54x x  - 1261x x  - 822x x  - 257x  - 986x x  + 356x x  - 1488x x  - 1561x x  - 850x x  - 85x x  - 1350x x  - 783x x  - 1335x , - 871x  + 1006x x  - 1399x  - 1636x x  - 699x x  - 769x  - 307x x  - 1645x x  - 502x x  - 719x  + 1405x x  + 870x x  - 1133x x  + 425x x  - 1203x  - 1601x x  + 117x x  - 382x x  + 318x x  - 117x x  - 560x  + 1135x x  + 1468x x  + 869x x  - 943x x  - 335x x  - 1218x x  + 201x  - 11x x  + 540x x  - 710x x  - 489x x  + 1605x x  + 1663x x  - 423x x  + 1246x  + 97x x  - 644x x  + 1655x x  + 1219x x  + 1476x x  + 1355x x  + 1594x x  + 893x x  + 1150x , - 143x  + 1240x x  - 1042x  + 1649x x  + 1024x x  + 794x  + 1442x x  - 1263x x  + 537x x  - 82x  - 734x x  - 1569x x  - 798x x  - 366x x  + 1289x  - 569x x  - 254x x  + 237x x  - 1234x x  - 807x x  + 264x  - 202x x  - 616x x  + 44x x  + 1465x x  + 685x x  + 1630x x  - 406x  - 123x x  - 4x x  + 1583x x  + 1235x x  + 162x x  + 1034x x  - 1035x x  + 737x  + 660x x  + 1459x x  - 359x x  - 1291x x  + 1638x x  - 325x x  - 631x x  + 73x x  - 1471x , - 1340x  + 31x x  - 994x  - 880x x  - 89x x  + 574x  + 760x x  - 1054x x  + 772x x  - 239x  - 443x x  + 1240x x  + 637x x  - 1423x x  + 320x  - 1363x x  - 1139x x  - 158x x  - 325x x  - 1578x x  + 32x  + 695x x  + 305x x  + 1012x x  + 1492x x  + 1290x x  + 1579x x  - 342x  - 83x x  - 104x x  + 998x x  - 92x x  + 1554x x  + 201x x  - 237x x  + 160x  - 228x x  - 543x x  - 1147x x  - 376x x  + 1313x x  + 603x x  + 106x x  - 1361x x  + 699x , - 228x  - 1510x x  + 277x  - 4x x  - 22x x  - 1526x  + 234x x  + 969x x  + 1253x x  - 1426x  - 1474x x  + 947x x  + 194x x  - 316x x  - 988x  - 1211x x  + 1087x x  + 536x x  - 491x x  + 870x x  - 659x  + 1490x x  - 469x x  + 1190x x  + 807x x  + 650x x  + 448x x  - 1353x  - 218x x  + 759x x  - 253x x  + 830x x  - 1080x x  - 143x x  - 1313x x  - 374x  - 180x x  + 741x x  + 742x x  - 1254x x  + 458x x  - 345x x  + 597x x  + 1567x x  - 31x , 1120x  + 709x x  - 1538x  - 1048x x  - 162x x  - 1518x  - 73x x  + 380x x  + 533x x  - 286x  + 1374x x  - 74x x  - 22x x  + 1535x x  - 1071x  - 839x x  - 560x x  + 928x x  + 335x x  - 1008x x  + 810x  - 448x x  - 357x x  - 107x x  + 40x x  + 784x x  - 1423x x  + 1276x  + 147x x  + 443x x  - 598x x  - 1077x x  - 1214x x  + 322x x  - 1408x x  + 72x  - 63x x  - 1513x x  - 791x x  + 11x x  + 77x x  + 836x x  - 1100x x  + 1637x x  - 788x , 1331x  + 318x x  - 704x  + 51x x  + 275x x  + 1149x  + 1526x x  + 768x x  + 414x x  - 782x  - 262x x  + 686x x  - 380x x  + 1377x x  + 1077x  + 1650x x  - 1129x x  - 508x x  + 846x x  + 1513x x  + 460x  - 1626x x  - 1024x x  + 862x x  + 1352x x  - 188x x  - 1382x x  - 650x  + 55x x  - 326x x  + 1037x x  + 705x x  - 667x x  + 1483x x  + 1661x x  - 1652x  - 1052x x  - 692x x  - 542x x  + 162x x  + 582x x  - 1369x x  + 934x x  + 1392x x  + 1227x , - 346x  + 1408x x  - 1225x  - 1536x x  - 1028x x  - 985x  - 210x x  - 1312x x  + 915x x  + 1633x  - 202x x  - 1636x x  - 1653x x  - 480x x  - 1260x  - 813x x  - 1623x x  - 1429x x  + 1094x x  - 747x x  + 955x  + 898x x  - 795x x  - 35x x  - 566x x  + 1631x x  - 324x x  + 926x  - 132x x  - 9x x  - 1290x x  - 543x x  + 902x x  + 735x x  - 342x x  - 400x  + 900x x  - 463x x  + 694x x  - 1262x x  - 1449x x  - 448x x  - 1402x x  - 731x x  - 996x , 301x  + 166x x  - 955x  - 739x x  - 1199x x  - 319x  + 1047x x  - 532x x  + 902x x  + 1195x  - 663x x  + 1215x x  - 534x x  - 332x x  - 973x  + 772x x  - 308x x  + 315x x  - 454x x  - 483x x  - 239x  - 1313x x  - 419x x  - 1340x x  - 1388x x  - 1340x x  - 1665x x  - 333x  - 465x x  - 1084x x  + 676x x  - 1612x x  - 288x x  + 11x x  - 1170x x  - 189x  + 498x x  - 889x x  + 693x x  + 1460x x  - 473x x  - 414x x  - 122x x  - 1659x x  - 1421x , 14x  - 1049x x  + 1506x  + 1235x x  + 642x x  - 1034x  + 460x x  + 150x x  + 760x x  - 1246x  - 1407x x  + 1570x x  + 1403x x  - 1610x x  - 431x  + 574x x  + 893x x  - 657x x  + 417x x  + 1362x x  + 224x  + 268x x  + 1097x x  + 1132x x  + 148x x  + 1331x x  - 77x x  - 756x  + 228x x  + 136x x  - 1484x x  - 1478x x  - 13x x  + 1620x x  - 701x x  - 769x  - 760x x  - 492x x  - 1077x x  - 1249x x  - 834x x  - 395x x  - 1358x x  - 988x x  + 113x , - 1634x  - 13x x  + 805x  - 21x x  - 1655x x  + 1479x  - 1510x x  - 646x x  + 225x x  - 1411x  + 1227x x  - 1108x x  + 1291x x  - 59x x  - 142x  + 586x x  - 676x x  + 655x x  - 1476x x  + 453x x  - 1076x  - 1152x x  + 1373x x  - 1191x x  - 416x x  + 699x x  + 317x x  + 825x  - 1560x x  - 488x x  - 1035x x  - 1561x x  - 644x x  - 1178x x  - 1320x x  + 158x  + 889x x  + 1444x x  - 1486x x  - 1211x x  + 1269x x  - 1228x x  + 568x x  + 1591x x  + 1207x , 105x  - 538x x  - 1222x  - 277x x  + 716x x  - 1067x  - 428x x  + 154x x  - 469x x  + 77x  + 538x x  - 179x x  + 921x x  - 223x x  + 1093x  - 262x x  + 1299x x  + 631x x  + 1486x x  - 1280x x  - 121x  - 50x x  - 978x x  - 694x x  - 531x x  + 505x x  + 1412x x  - 1061x  + 1202x x  + 448x x  - 187x x  + 1276x x  - 121x x  + 1361x x  + 697x x  + 682x  + 1592x x  + 705x x  - 227x x  - 7x x  - 1423x x  - 1446x x  - 1578x x  + 1511x x  + 917x , 1270x  - 391x x  - 1116x  - 287x x  + 653x x  + 1643x  + 1623x x  + 514x x  - 14x x  - 90x  + 1232x x  - 1434x x  + 1296x x  + 1522x x  + 136x  - 623x x  - 607x x  + 18x x  + 896x x  - 29x x  + 1059x  - 1053x x  + 1643x x  + 1652x x  - 1190x x  - 1073x x  + 1470x x  - 944x  - 93x x  - 187x x  - 994x x  - 1415x x  - 229x x  - 796x x  + 1642x x  + 1600x  - 344x x  + 905x x  + 1032x x  - 538x x  - 891x x  + 1243x x  + 1290x x  + 490x x  - 1148x , 1613x  + 175x x  - 1346x  - 1000x x  - 1217x x  - 729x  - 1296x x  + 1456x x  + 745x x  + 539x  + 525x x  - 811x x  + 753x x  + 1362x x  + 1629x  - 840x x  + 513x x  + 429x x  + 842x x  + 1414x x  - 308x  + 1415x x  - 1461x x  - 1135x x  + 701x x  + 766x x  + 785x x  + 1503x  + 147x x  + 929x x  - 1220x x  - 853x x  + 493x x  + 226x x  + 1416x x  + 280x  - 7x x  + 1632x x  + 520x x  + 1259x x  + 157x x  + 1596x x  + 655x x  - 42x x  - 586x })
│ │ │                                   0       0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4       1 4        2 4       3 4       4      0 5        1 5        2 5       3 5        4 5       5       0 6       1 6        2 6        3 6       4 6        5 6        6       0 7       1 7        2 7      3 7        4 7        5 7       6 7      7        0 8        1 8        2 8       3 8        4 8      5 8        6 8        7 8       8         0       0 1       1        0 2        1 2       2       0 3        1 3       2 3        3        0 4       1 4       2 4      3 4      4        0 5       1 5        2 5       3 5       4 5       5       0 6       1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8      5 8        6 8       7 8        8        0        0 1        1        0 2       1 2       2       0 3        1 3       2 3       3        0 4       1 4        2 4       3 4        4        0 5       1 5       2 5       3 5       4 5       5        0 6        1 6       2 6       3 6       4 6        5 6       6      0 7       1 7       2 7       3 7        4 7        5 7       6 7        7      0 8       1 8        2 8        3 8        4 8        5 8        6 8       7 8        8        0        0 1        1        0 2        1 2       2        0 3        1 3       2 3      3       0 4        1 4       2 4       3 4        4       0 5       1 5       2 5        3 5       4 5       5       0 6       1 6      2 6        3 6       4 6        5 6       6       0 7     1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8       2 8        3 8        4 8       5 8       6 8      7 8        8         0      0 1       1       0 2      1 2       2       0 3        1 3       2 3       3       0 4        1 4       2 4        3 4       4        0 5        1 5       2 5       3 5        4 5      5       0 6       1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7      3 7        4 7       5 7       6 7       7       0 8       1 8        2 8       3 8        4 8       5 8       6 8        7 8       8        0        0 1       1     0 2      1 2        2       0 3       1 3        2 3        3        0 4       1 4       2 4       3 4       4        0 5        1 5       2 5       3 5       4 5       5        0 6       1 6        2 6       3 6       4 6       5 6        6       0 7       1 7       2 7       3 7        4 7       5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8      8       0       0 1        1        0 2       1 2        2      0 3       1 3       2 3       3        0 4      1 4      2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5       0 6       1 6       2 6      3 6       4 6        5 6        6       0 7       1 7       2 7        3 7        4 7       5 7        6 7      7      0 8        1 8       2 8      3 8      4 8       5 8        6 8        7 8       8       0       0 1       1      0 2       1 2        2        0 3       1 3       2 3       3       0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5        4 5       5        0 6        1 6       2 6        3 6       4 6        5 6       6      0 7       1 7        2 7       3 7       4 7        5 7        6 7        7        0 8       1 8       2 8       3 8       4 8        5 8       6 8        7 8        8        0        0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4        1 4        2 4       3 4        4       0 5        1 5        2 5        3 5       4 5       5       0 6       1 6      2 6       3 6        4 6       5 6       6       0 7     1 7        2 7       3 7       4 7       5 7       6 7       7       0 8       1 8       2 8        3 8        4 8       5 8        6 8       7 8       8      0       0 1       1       0 2        1 2       2        0 3       1 3       2 3        3       0 4        1 4       2 4       3 4       4       0 5       1 5       2 5       3 5       4 5       5        0 6       1 6        2 6        3 6        4 6        5 6       6       0 7        1 7       2 7        3 7       4 7      5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8        8     0        0 1        1        0 2       1 2        2       0 3       1 3       2 3        3        0 4        1 4        2 4        3 4       4       0 5       1 5       2 5       3 5        4 5       5       0 6        1 6        2 6       3 6        4 6      5 6       6       0 7       1 7        2 7        3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8       5 8        6 8       7 8       8         0      0 1       1      0 2        1 2        2        0 3       1 3       2 3        3        0 4        1 4        2 4      3 4       4       0 5       1 5       2 5        3 5       4 5        5        0 6        1 6        2 6       3 6       4 6       5 6       6        0 7       1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8        2 8        3 8        4 8        5 8       6 8        7 8        8      0       0 1        1       0 2       1 2        2       0 3       1 3       2 3      3       0 4       1 4       2 4       3 4        4       0 5        1 5       2 5        3 5        4 5       5      0 6       1 6       2 6       3 6       4 6        5 6        6        0 7       1 7       2 7        3 7       4 7        5 7       6 7       7        0 8       1 8       2 8     3 8        4 8        5 8        6 8        7 8       8       0       0 1        1       0 2       1 2        2        0 3       1 3      2 3      3        0 4        1 4        2 4        3 4       4       0 5       1 5      2 5       3 5      4 5        5        0 6        1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7        3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8       4 8        5 8        6 8       7 8        8       0       0 1        1        0 2        1 2       2        0 3        1 3       2 3       3       0 4       1 4       2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5        0 6        1 6        2 6       3 6       4 6       5 6        6       0 7       1 7        2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2 8        3 8       4 8        5 8       6 8      7 8       8
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.064201s (cpu); 0.0650172s (thread); 0s (gc)
│ │ │ + -- used 0.0518674s (cpu); 0.0512658s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   2                  2                           2                                      2                                                 2                                                           2                                                                   2                                                                              2                                                                                          2        2                  2                              2                                       2                                                2                                                             2                                                                  2                                                                              2                                                                                            2        2                  2                             2                                       2                                                2                                                           2                                                                      2                                                                              2                                                                                         2         2                 2                            2                                       2                                                  2                                                             2                                                                    2                                                                                2                                                                                             2       2                   2                            2                                     2                                                2                                                          2                                                                  2                                                                                   2                                                                                            2        2                2                           2                                      2                                                  2                                                            2                                                                      2                                                                                 2                                                                                          2   2                   2                           2                                     2                                                  2                                                           2                                                                    2                                                                              2                                                                                         2      2                  2                           2                                      2                                                  2                                                             2                                                                       2                                                                              2                                                                                          2         2                  2                            2                                     2                                                 2                                                              2                                                                    2                                                                               2                                                                                        2
│ │ │  o6 = map (ringP8, ringP8, {- 780x  - 506x x  + 1537x  - 132x x  - 928x x  + 386x  - 102x x  + 422x x  + 725x x  - 1073x  - 905x x  - 830x x  + 1500x x  + 276x x  + 1533x  - 653x x  + 1558x x  + 939x x  - 1432x x  + 462x x  - 329x  - 92x x  + 661x x  - 1298x x  - 684x x  + 70x x  - 715x x  + 1093x  + 581x x  + 329x x  + 454x x  - 911x x  - 84x x  - 1452x x  - 809x x  + 1202x  + 1353x x  + 1503x x  + 482x x  + 893x x  - 643x x  + 598x x  + 110x x  + 1064x x  - 472x , - 522x  - 583x x  + 1339x  + 1535x x  - 1317x x  + 1113x  - 169x x  + 1440x x  - 1657x x  + 721x  + 40x x  - 1576x x  - 367x x  + 257x x  - 1454x  + 1612x x  + 1529x x  - 1068x x  + 560x x  - 1441x x  + 608x  - 92x x  - 1006x x  + 285x x  + 102x x  - 397x x  + 66x x  - 643x  - 38x x  + 1380x x  + 1069x x  - 426x x  + 1147x x  + 982x x  + 10x x  - 662x  + 16x x  + 1561x x  + 1597x x  + 512x x  + 1288x x  - 1253x x  + 1317x x  + 1481x x  - 354x , - 640x  - 1551x x  + 469x  + 1482x x  - 1593x x  - 986x  + 471x x  + 612x x  + 1228x x  + 1156x  - 731x x  + 1503x x  - 628x x  + 674x x  - 799x  + 1137x x  + 844x x  + 589x x  - 666x x  + 829x x  - 1024x  - 170x x  + 450x x  + 1497x x  + 1204x x  - 907x x  + 1621x x  - 417x  + 1297x x  + 1444x x  + 4x x  + 398x x  + 996x x  - 1031x x  + 239x x  + 303x  + 1215x x  - 83x x  + 1571x x  - 1543x x  - 925x x  - 694x x  + 151x x  - 520x x  + 880x , - 1210x  - 222x x  + 185x  + 245x x  + 1059x x  - 322x  + 238x x  + 962x x  + 1260x x  - 1581x  + 50x x  + 1352x x  - 1465x x  + 1555x x  + 1333x  + 1362x x  + 1365x x  + 1168x x  - 1401x x  + 149x x  - 652x  + 1378x x  - 557x x  - 112x x  + 26x x  + 315x x  + 111x x  + 1592x  - 283x x  - 1454x x  + 907x x  + 212x x  + 400x x  + 1049x x  - 882x x  - 1429x  - 183x x  + 1571x x  - 1286x x  - 1179x x  + 1319x x  + 240x x  - 1100x x  + 1500x x  - 348x , 1051x  - 1325x x  + 1354x  - 346x x  - 1532x x  - 466x  + 163x x  - 659x x  - 291x x  + 966x  + 789x x  + 393x x  + 403x x  - 1199x x  - 570x  - 93x x  - 492x x  - 418x x  + 713x x  - 1323x x  - 1384x  - 830x x  - 54x x  - 306x x  + 709x x  + 421x x  - 954x x  - 299x  + 1053x x  - 1080x x  + 686x x  + 170x x  - 1272x x  - 1661x x  + 1235x x  + 1553x  - 1454x x  - 1411x x  - 1195x x  - 962x x  + 737x x  - 390x x  + 957x x  + 1538x x  + 1234x , - 509x  + 9x x  - 1563x  - 710x x  - 642x x  + 541x  + 220x x  - 1214x x  - 16x x  + 1008x  - 1088x x  + 755x x  - 886x x  - 1433x x  + 1154x  + 1627x x  - 1547x x  - 951x x  + 866x x  + 163x x  - 1142x  - 668x x  + 1361x x  + 1324x x  - 490x x  + 282x x  - 1133x x  - 612x  + 805x x  - 126x x  + 1296x x  - 973x x  + 1271x x  - 1646x x  + 844x x  + 1073x  - 1452x x  - 1112x x  - 141x x  + 176x x  - 1579x x  - 78x x  + 848x x  - 1365x x  + 711x , x  + 1543x x  - 1076x  + 493x x  - 526x x  + 868x  - 582x x  - 996x x  + 206x x  - 419x  + 1258x x  - 391x x  + 1002x x  - 1539x x  + 931x  - 1504x x  + 810x x  + 324x x  + 1356x x  + 313x x  + 772x  + 299x x  + 1186x x  + 718x x  + 407x x  - 64x x  - 828x x  - 1393x  + 94x x  - 290x x  - 766x x  + 950x x  - 640x x  + 265x x  - 1640x x  - 1403x  - 126x x  + 891x x  - 1519x x  - 927x x  - 1335x x  - 1448x x  - x x  - 1103x x  - 1152x , 821x  + 558x x  - 1174x  - 168x x  + 986x x  + 790x  + 549x x  + 817x x  + 1396x x  + 695x  + 1211x x  + 878x x  - 1061x x  - 1244x x  - 880x  + 1409x x  - 567x x  + 1240x x  + 1126x x  - 1262x x  + 490x  + 1553x x  + 1276x x  + 805x x  + 576x x  - 1076x x  + 1617x x  - 495x  - 750x x  - 277x x  + 544x x  + 1479x x  - 784x x  - 64x x  - 1203x x  + 405x  + 1013x x  + 604x x  + 1301x x  + 1003x x  + 235x x  + 696x x  + 939x x  - 714x x  - 879x , - 1452x  + 727x x  - 1159x  + 449x x  - 1169x x  + 732x  + 575x x  - 600x x  + 924x x  - 837x  + 1298x x  - 860x x  + 1010x x  + 774x x  + 319x  + 1087x x  - 1120x x  + 1439x x  + 1175x x  - 1648x x  + 985x  - 1317x x  - 878x x  + 399x x  - 1339x x  + 70x x  - 463x x  + 470x  - 628x x  - 907x x  + 748x x  + 98x x  + 1150x x  + 1140x x  + 1308x x  + 621x  + 369x x  - 991x x  - 1186x x  + 61x x  - 907x x  - 681x x  - 1528x x  + 717x x  + 854x })
│ │ │                                   0       0 1        1       0 2       1 2       2       0 3       1 3       2 3        3       0 4       1 4        2 4       3 4        4       0 5        1 5       2 5        3 5       4 5       5      0 6       1 6        2 6       3 6      4 6       5 6        6       0 7       1 7       2 7       3 7      4 7        5 7       6 7        7        0 8        1 8       2 8       3 8       4 8       5 8       6 8        7 8       8        0       0 1        1        0 2        1 2        2       0 3        1 3        2 3       3      0 4        1 4       2 4       3 4        4        0 5        1 5        2 5       3 5        4 5       5      0 6        1 6       2 6       3 6       4 6      5 6       6      0 7        1 7        2 7       3 7        4 7       5 7      6 7       7      0 8        1 8        2 8       3 8        4 8        5 8        6 8        7 8       8        0        0 1       1        0 2        1 2       2       0 3       1 3        2 3        3       0 4        1 4       2 4       3 4       4        0 5       1 5       2 5       3 5       4 5        5       0 6       1 6        2 6        3 6       4 6        5 6       6        0 7        1 7     2 7       3 7       4 7        5 7       6 7       7        0 8      1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1       1       0 2        1 2       2       0 3       1 3        2 3        3      0 4        1 4        2 4        3 4        4        0 5        1 5        2 5        3 5       4 5       5        0 6       1 6       2 6      3 6       4 6       5 6        6       0 7        1 7       2 7       3 7       4 7        5 7       6 7        7       0 8        1 8        2 8        3 8        4 8       5 8        6 8        7 8       8       0        0 1        1       0 2        1 2       2       0 3       1 3       2 3       3       0 4       1 4       2 4        3 4       4      0 5       1 5       2 5       3 5        4 5        5       0 6      1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7        4 7        5 7        6 7        7        0 8        1 8        2 8       3 8       4 8       5 8       6 8        7 8        8        0     0 1        1       0 2       1 2       2       0 3        1 3      2 3        3        0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5       4 5        5       0 6        1 6        2 6       3 6       4 6        5 6       6       0 7       1 7        2 7       3 7        4 7        5 7       6 7        7        0 8        1 8       2 8       3 8        4 8      5 8       6 8        7 8       8   0        0 1        1       0 2       1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5       2 5        3 5       4 5       5       0 6        1 6       2 6       3 6      4 6       5 6        6      0 7       1 7       2 7       3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8        4 8        5 8    6 8        7 8        8      0       0 1        1       0 2       1 2       2       0 3       1 3        2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5        2 5        3 5        4 5       5        0 6        1 6       2 6       3 6        4 6        5 6       6       0 7       1 7       2 7        3 7       4 7      5 7        6 7       7        0 8       1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1        1       0 2        1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4       3 4       4        0 5        1 5        2 5        3 5        4 5       5        0 6       1 6       2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3 7        4 7        5 7        6 7       7       0 8       1 8        2 8      3 8       4 8       5 8        6 8       7 8       8
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 0.755879s (cpu); 0.572649s (thread); 0s (gc)
│ │ │ + -- used 0.635552s (cpu); 0.565035s (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 0.00195523s (cpu); 0.000661842s (thread); 0s (gc)
│ │ │ + -- used 0.00131271s (cpu); 0.000770986s (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 0.0159981s (cpu); 0.0168064s (thread); 0s (gc)
│ │ │ + -- used 0.0339625s (cpu); 0.019362s (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 0.0335347s (cpu); 0.0323669s (thread); 0s (gc)
│ │ │ + -- used 0.102125s (cpu); 0.0439899s (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 0.0040011s (cpu); 0.00340796s (thread); 0s (gc)
│ │ │ + -- used 0.00393645s (cpu); 0.00650571s (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 0.19608s (cpu); 0.125864s (thread); 0s (gc)
│ │ │ + -- used 0.190688s (cpu); 0.12684s (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 0.184078s (cpu); 0.0994077s (thread); 0s (gc)
│ │ │ + -- used 0.19616s (cpu); 0.111435s (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 0.359803s (cpu); 0.207848s (thread); 0s (gc)
│ │ │ + -- used 0.372231s (cpu); 0.20062s (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 0.0598042s (cpu); 0.0592642s (thread); 0s (gc)
│ │ │ + -- used 0.0710116s (cpu); 0.0720018s (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 0.0159898s (cpu); 0.0183806s (thread); 0s (gc)
│ │ │ + -- used 0.024018s (cpu); 0.0237251s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0152421s (cpu); 0.0166246s (thread); 0s (gc)
│ │ │ + -- used 0.0315645s (cpu); 0.0194747s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : P8 = ZZ/101[x_0..x_8];
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 2.53653s (cpu); 2.17119s (thread); 0s (gc)
│ │ │ + -- used 2.24976s (cpu); 2.09942s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │ @@ -20,13 +20,13 @@
│ │ │  o5 : Ideal of P7
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 3.92624s (cpu); 2.66358s (thread); 0s (gc)
│ │ │ + -- used 2.84207s (cpu); 2.31617s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
│ │ │                0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.0159152s (cpu); 0.0175494s (thread); 0s (gc)
│ │ │ + -- used 0.0200731s (cpu); 0.0211263s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.545522s (cpu); 0.365991s (thread); 0s (gc)
│ │ │ + -- used 0.439411s (cpu); 0.362208s (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 0.00400069s (cpu); 0.00486114s (thread); 0s (gc)
│ │ │ + -- used 0.00678549s (cpu); 0.00591266s (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 0.583205s (cpu); 0.394744s (thread); 0s (gc)
│ │ │ + -- used 0.637322s (cpu); 0.446514s (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 0.319702s (cpu); 0.186155s (thread); 0s (gc)
│ │ │ + -- used 0.322917s (cpu); 0.172455s (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 0.188222s (cpu); 0.112952s (thread); 0s (gc)
│ │ │ + -- used 0.190654s (cpu); 0.126239s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  o2 = map (GF 109561[t ..t ], GF 109561[x ..x ], {- t  + t t , - t t  + t t , - t  + t t , - t t  + t t , - t t  + t t , - t  + t t , a})
│ │ │                       0   4              0   5       1    0 2     1 2    0 3     2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ - -- used 0.0212853s (cpu); 0.0181898s (thread); 0s (gc)
│ │ │ + -- used 0.0302431s (cpu); 0.0167583s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │ @@ -29,15 +29,15 @@
│ │ │                GF 109561[x ..x ]
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ - -- used 0.106161s (cpu); 0.0288538s (thread); 0s (gc)
│ │ │ + -- used 0.133512s (cpu); 0.0366695s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 0.00301299s (cpu); 4.3121e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00256426s (cpu); 3.8399e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 0.000108975s (cpu); 2.141e-05s (thread); 0s (gc)
│ │ │ + -- used 9.7165e-05s (cpu); 2.0987e-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 0.193965s (cpu); 0.124076s (thread); 0s (gc)
│ │ │ + -- used 0.104089s (cpu); 0.105889s (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 0.031972s (cpu); 0.0293826s (thread); 0s (gc)
│ │ │ + -- used 0.0359736s (cpu); 0.0390032s (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 1.20741s (cpu); 0.6835s (thread); 0s (gc)
│ │ │ + -- used 1.49727s (cpu); 0.674009s (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 1.22284s (cpu); 1.00459s (thread); 0s (gc)
│ │ │ + -- used 1.11158s (cpu); 0.981082s (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 0.0927728s (cpu); 0.0927368s (thread); 0s (gc)
│ │ │ + -- used 0.0920643s (cpu); 0.0928535s (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 0.0206327s (cpu); 0.0200933s (thread); 0s (gc)
│ │ │ + -- used 0.0200111s (cpu); 0.0197175s (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 0.0694952s (cpu); 0.0681984s (thread); 0s (gc)
│ │ │ + -- used 0.0800774s (cpu); 0.0833029s (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 0.00399937s (cpu); 0.00613956s (thread); 0s (gc)
│ │ │ + -- used 0.00814783s (cpu); 0.00682803s (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 0.0398632s (cpu); 0.0388171s (thread); 0s (gc)
│ │ │ + -- used 0.0239395s (cpu); 0.0255035s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.098716s (cpu); 0.0321057s (thread); 0s (gc)
│ │ │ + -- used 0.00443214s (cpu); 0.00657179s (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 0.00607096s (cpu); 0.00709152s (thread); 0s (gc)
│ │ │ + -- used 0.00143266s (cpu); 0.00514517s (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
│ │ │ @@ -97,29 +97,29 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.58318s (cpu); 0.935998s (thread); 0s (gc)
│ │ │ + -- used 1.88088s (cpu); 1.18249s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ - -- used 1.30066s (cpu); 0.983759s (thread); 0s (gc)
│ │ │ + -- used 1.4482s (cpu); 0.959392s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │ @@ -167,29 +167,29 @@
│ │ │  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 0.196051s (cpu); 0.141072s (thread); 0s (gc)
│ │ │ + -- used 0.242895s (cpu); 0.16424s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │              
    
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │ - -- used 0.117767s (cpu); 0.056998s (thread); 0s (gc)
│ │ │ + -- used 0.182014s (cpu); 0.054872s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,25 +39,25 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.58318s (cpu); 0.935998s (thread); 0s (gc)
│ │ │ │ + -- used 1.88088s (cpu); 1.18249s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │ - -- used 1.30066s (cpu); 0.983759s (thread); 0s (gc)
│ │ │ │ + -- used 1.4482s (cpu); 0.959392s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │ @@ -88,25 +88,25 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.196051s (cpu); 0.141072s (thread); 0s (gc)
│ │ │ │ + -- used 0.242895s (cpu); 0.16424s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │ - -- used 0.117767s (cpu); 0.056998s (thread); 0s (gc)
│ │ │ │ + -- used 0.182014s (cpu); 0.054872s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -85,23 +85,23 @@
│ │ │  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 0.318484s (cpu); 0.16471s (thread); 0s (gc)
│ │ │ + -- used 0.302691s (cpu); 0.175561s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.011616s (cpu); 0.0120564s (thread); 0s (gc)
│ │ │ + -- used 0.0259473s (cpu); 0.0142352s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,19 +31,19 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.318484s (cpu); 0.16471s (thread); 0s (gc)
│ │ │ │ + -- used 0.302691s (cpu); 0.175561s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │ - -- used 0.011616s (cpu); 0.0120564s (thread); 0s (gc)
│ │ │ │ + -- used 0.0259473s (cpu); 0.0142352s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    │ │ │
    │ │ │ 
    i4 : time phi! ;
│ │ │ - -- used 0.147873s (cpu); 0.0667188s (thread); 0s (gc)
│ │ │ + -- used 0.169943s (cpu); 0.0835485s (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 0.0376208s (cpu); 0.0384761s (thread); 0s (gc)
│ │ │ + -- used 0.0663537s (cpu); 0.0432652s (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 0.147873s (cpu); 0.0667188s (thread); 0s (gc)
│ │ │ │ + -- used 0.169943s (cpu); 0.0835485s (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 0.0376208s (cpu); 0.0384761s (thread); 0s (gc)
│ │ │ │ + -- used 0.0663537s (cpu); 0.0432652s (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 0.258684s (cpu); 0.1808s (thread); 0s (gc)
│ │ │ + -- used 0.261441s (cpu); 0.194449s (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 0.535432s (cpu); 0.419255s (thread); 0s (gc)
│ │ │ + -- used 0.564081s (cpu); 0.489367s (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 0.406043s (cpu); 0.271651s (thread); 0s (gc)
│ │ │ + -- used 0.473689s (cpu); 0.329225s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    │ │ │
    │ │ │ 
    i6 : time SegreClass(X,Certify=>true)
│ │ │ - -- used 0.0250447s (cpu); 0.024676s (thread); 0s (gc)
│ │ │ + -- used 0.0403782s (cpu); 0.0269503s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    │ │ │
    │ │ │ 
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ - -- used 0.0964579s (cpu); 0.0991384s (thread); 0s (gc)
│ │ │ + -- used 0.264194s (cpu); 0.150719s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │ @@ -213,15 +213,15 @@
│ │ │  
│ │ │  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 0.20942s (cpu); 0.134778s (thread); 0s (gc)
│ │ │ + -- used 0.0705414s (cpu); 0.0710159s (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 0.494061s (cpu); 0.339918s (thread); 0s (gc)
│ │ │ + -- used 0.404225s (cpu); 0.256195s (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 0.513534s (cpu); 0.385771s (thread); 0s (gc)
│ │ │ + -- used 0.446911s (cpu); 0.285039s (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 1.30229s (cpu); 0.969049s (thread); 0s (gc)
│ │ │ + -- used 1.06527s (cpu); 0.823376s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,46 +81,46 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x
│ │ │ │  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1
│ │ │ │  6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.535432s (cpu); 0.419255s (thread); 0s (gc)
│ │ │ │ + -- used 0.564081s (cpu); 0.489367s (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 0.406043s (cpu); 0.271651s (thread); 0s (gc)
│ │ │ │ + -- used 0.473689s (cpu); 0.329225s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │ - -- used 0.0250447s (cpu); 0.024676s (thread); 0s (gc)
│ │ │ │ + -- used 0.0403782s (cpu); 0.0269503s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │ - -- used 0.0964579s (cpu); 0.0991384s (thread); 0s (gc)
│ │ │ │ + -- used 0.264194s (cpu); 0.150719s (thread); 0s (gc)
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │ @@ -141,15 +141,15 @@
│ │ │ │         ZZ
│ │ │ │  o9 = ------[x ..x ]
│ │ │ │       100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.20942s (cpu); 0.134778s (thread); 0s (gc)
│ │ │ │ + -- used 0.0705414s (cpu); 0.0710159s (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 0.494061s (cpu); 0.339918s (thread); 0s (gc)
│ │ │ │ + -- used 0.404225s (cpu); 0.256195s (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 0.513534s (cpu); 0.385771s (thread); 0s (gc)
│ │ │ │ + -- used 0.446911s (cpu); 0.285039s (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 1.30229s (cpu); 0.969049s (thread); 0s (gc)
│ │ │ │ + -- used 1.06527s (cpu); 0.823376s (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 0.00329632s (cpu); 0.000397406s (thread); 0s (gc)
│ │ │ + -- used 0.000152419s (cpu); 0.00043236s (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 0.204411s (cpu); 0.137253s (thread); 0s (gc)
│ │ │ + -- used 0.216536s (cpu); 0.156827s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │
    │ │ │ 
    i6 : time rationalMap psi
│ │ │ - -- used 0.503555s (cpu); 0.363993s (thread); 0s (gc)
│ │ │ + -- used 0.425593s (cpu); 0.361768s (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 0.142169s (cpu); 0.0615885s (thread); 0s (gc)
│ │ │ + -- used 0.0480851s (cpu); 0.0481438s (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 3.24317s (cpu); 1.76112s (thread); 0s (gc)
│ │ │ + -- used 3.7022s (cpu); 2.11836s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i16 : time T2 = T * T
│ │ │ - -- used 0.000207039s (cpu); 2.7321e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000181497s (cpu); 2.8442e-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 5.38785s (cpu); 2.93636s (thread); 0s (gc)
│ │ │ + -- used 6.21311s (cpu); 3.37963s (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:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -292,15 +292,15 @@ │ │ │ o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i21 : time f = rationalMap T
│ │ │ - -- used 4.38057s (cpu); 2.47597s (thread); 0s (gc)
│ │ │ + -- used 4.98679s (cpu); 2.93795s (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 0.00329632s (cpu); 0.000397406s (thread); 0s (gc)
│ │ │ │ + -- used 0.000152419s (cpu); 0.00043236s (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 0.204411s (cpu); 0.137253s (thread); 0s (gc)
│ │ │ │ + -- used 0.216536s (cpu); 0.156827s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.503555s (cpu); 0.363993s (thread); 0s (gc)
│ │ │ │ + -- used 0.425593s (cpu); 0.361768s (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 0.142169s (cpu); 0.0615885s (thread); 0s (gc)
│ │ │ │ + -- used 0.0480851s (cpu); 0.0481438s (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 3.24317s (cpu); 1.76112s (thread); 0s (gc)
│ │ │ │ + -- used 3.7022s (cpu); 2.11836s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 0.000207039s (cpu); 2.7321e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000181497s (cpu); 2.8442e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 5.38785s (cpu); 2.93636s (thread); 0s (gc)
│ │ │ │ + -- used 6.21311s (cpu); 3.37963s (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 4.38057s (cpu); 2.47597s (thread); 0s (gc)
│ │ │ │ + -- used 4.98679s (cpu); 2.93795s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
    │ │ │
    │ │ │ 
    i3 : time psi = approximateInverseMap phi
│ │ │ - -- used 0.375337s (cpu); 0.235592s (thread); 0s (gc)
│ │ │ + -- used 0.377808s (cpu); 0.262595s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 10
│ │ │  -- approximateInverseMap: step 2 of 10
│ │ │  -- approximateInverseMap: step 3 of 10
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ @@ -197,15 +197,15 @@
│ │ │
    		│ │ │ 
    i4 : assert(phi * psi == 1 and psi * phi == 1)
    │ │ │
    │ │ │ 
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ - -- used 0.234106s (cpu); 0.154149s (thread); 0s (gc)
│ │ │ + -- used 0.228652s (cpu); 0.167555s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    │ │ │
    │ │ │ 
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ - -- used 2.48249s (cpu); 1.81057s (thread); 0s (gc)
│ │ │ + -- used 1.97346s (cpu); 1.72259s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ @@ -363,15 +363,15 @@
│ │ │  o9 = false
    │ │ │
    │ │ │ 
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ - -- used 3.44756s (cpu); 2.65658s (thread); 0s (gc)
│ │ │ + -- used 3.15177s (cpu); 2.80774s (thread); 0s (gc)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │ ├── html2text {}
│ │ │ │ @@ -125,15 +125,15 @@
│ │ │ │  
│ │ │ │                        x x  - x x
│ │ │ │                         1 2    0 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^9 to PP^8)
│ │ │ │  i3 : time psi = approximateInverseMap phi
│ │ │ │ - -- used 0.375337s (cpu); 0.235592s (thread); 0s (gc)
│ │ │ │ + -- used 0.377808s (cpu); 0.262595s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 10
│ │ │ │  -- approximateInverseMap: step 2 of 10
│ │ │ │  -- approximateInverseMap: step 3 of 10
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │ @@ -249,15 +249,15 @@
│ │ │ │  0 6     3 6      6      0 7      1 7      3 7      4 7      6 7      7      0 8
│ │ │ │  1 8      2 8      3 8      4 8      5 8      6 8      7 8     8
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │ - -- used 0.234106s (cpu); 0.154149s (thread); 0s (gc)
│ │ │ │ + -- used 0.228652s (cpu); 0.167555s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  4 6      5 6      6      0 7      1 7      2 7     3 7      4 7      5 7      6 7     0 8      1 8      3 8     4 8
│ │ │ │  5 8      6 8     7 8
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │ - -- used 2.48249s (cpu); 1.81057s (thread); 0s (gc)
│ │ │ │ + -- used 1.97346s (cpu); 1.72259s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │ @@ -522,15 +522,15 @@
│ │ │ │  o8 : RationalMap (quadratic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │ │  i9 : -- but...
│ │ │ │       phi * psi == 1
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │ - -- used 3.44756s (cpu); 2.65658s (thread); 0s (gc)
│ │ │ │ + -- used 3.15177s (cpu); 2.80774s (thread); 0s (gc)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    │ │ │
    │ │ │ 
    i5 : time degreeMap phi
│ │ │ - -- used 0.064201s (cpu); 0.0650172s (thread); 0s (gc)
│ │ │ + -- used 0.0518674s (cpu); 0.0512658s (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 0.755879s (cpu); 0.572649s (thread); 0s (gc)
│ │ │ + -- used 0.635552s (cpu); 0.565035s (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 0.064201s (cpu); 0.0650172s (thread); 0s (gc) │ │ │ │ + -- used 0.0518674s (cpu); 0.0512658s (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 0.755879s (cpu); 0.572649s (thread); 0s (gc) │ │ │ │ + -- used 0.635552s (cpu); 0.565035s (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 0.00195523s (cpu); 0.000661842s (thread); 0s (gc)
    │ │ │ + -- used 0.00131271s (cpu); 0.000770986s (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 0.00195523s (cpu); 0.000661842s (thread); 0s (gc) │ │ │ │ + -- used 0.00131271s (cpu); 0.000770986s (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 0.0159981s (cpu); 0.0168064s (thread); 0s (gc)
    │ │ │ + -- used 0.0339625s (cpu); 0.019362s (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 0.184078s (cpu); 0.0994077s (thread); 0s (gc)
│ │ │ │ + -- used 0.19616s (cpu); 0.111435s (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 0.359803s (cpu); 0.207848s (thread); 0s (gc)
│ │ │ │ + -- used 0.372231s (cpu); 0.20062s (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 0.0335347s (cpu); 0.0323669s (thread); 0s (gc)
    
│ │ │ + -- used 0.102125s (cpu); 0.0439899s (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 0.0159981s (cpu); 0.0168064s (thread); 0s (gc)
│ │ │ │ + -- used 0.0339625s (cpu); 0.019362s (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 0.0335347s (cpu); 0.0323669s (thread); 0s (gc)
│ │ │ │ + -- used 0.102125s (cpu); 0.0439899s (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 0.0040011s (cpu); 0.00340796s (thread); 0s (gc)
│ │ │ + -- used 0.00393645s (cpu); 0.00650571s (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 0.19608s (cpu); 0.125864s (thread); 0s (gc)
│ │ │ + -- used 0.190688s (cpu); 0.12684s (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 0.0040011s (cpu); 0.00340796s (thread); 0s (gc)
│ │ │ │ + -- used 0.00393645s (cpu); 0.00650571s (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 0.19608s (cpu); 0.125864s (thread); 0s (gc)
│ │ │ │ + -- used 0.190688s (cpu); 0.12684s (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 0.184078s (cpu); 0.0994077s (thread); 0s (gc)
│ │ │ + -- used 0.19616s (cpu); 0.111435s (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 0.359803s (cpu); 0.207848s (thread); 0s (gc)
│ │ │ + -- used 0.372231s (cpu); 0.20062s (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 0.0598042s (cpu); 0.0592642s (thread); 0s (gc)
│ │ │ + -- used 0.0710116s (cpu); 0.0720018s (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 0.0598042s (cpu); 0.0592642s (thread); 0s (gc)
│ │ │ │ + -- used 0.0710116s (cpu); 0.0720018s (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
│ │ │ @@ -123,23 +123,23 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0159898s (cpu); 0.0183806s (thread); 0s (gc)
│ │ │ + -- used 0.024018s (cpu); 0.0237251s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0152421s (cpu); 0.0166246s (thread); 0s (gc)
│ │ │ + -- used 0.0315645s (cpu); 0.0194747s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -58,19 +58,19 @@ │ │ │ │ - t + t t │ │ │ │ 3 2 4 │ │ │ │ } │ │ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in │ │ │ │ PP^5) │ │ │ │ i3 : time isBirational phi │ │ │ │ - -- used 0.0159898s (cpu); 0.0183806s (thread); 0s (gc) │ │ │ │ + -- used 0.024018s (cpu); 0.0237251s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : time isBirational(phi,Certify=>true) │ │ │ │ - -- used 0.0152421s (cpu); 0.0166246s (thread); 0s (gc) │ │ │ │ + -- used 0.0315645s (cpu); 0.0194747s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant │ │ │ │ ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: ********** │ │ │ │ * isBirational(RationalMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 2.53653s (cpu); 2.17119s (thread); 0s (gc)
│ │ │ + -- used 2.24976s (cpu); 2.09942s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -114,15 +114,15 @@ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │ - -- used 3.92624s (cpu); 2.66358s (thread); 0s (gc)
│ │ │ + -- used 2.84207s (cpu); 2.31617s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o7 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ be to perform the command kernel map phi == 0. │ │ │ │ i1 : P8 = ZZ/101[x_0..x_8]; │ │ │ │ i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7}, │ │ │ │ {x_4..x_8}}; │ │ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ i3 : time isDominant(phi,Certify=>true) │ │ │ │ - -- used 2.53653s (cpu); 2.17119s (thread); 0s (gc) │ │ │ │ + -- used 2.24976s (cpu); 2.09942s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : P7 = ZZ/101[x_0..x_7]; │ │ │ │ i5 : -- hyperelliptic curve of genus 3 │ │ │ │ C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6- │ │ │ │ 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5- │ │ │ │ @@ -64,15 +64,15 @@ │ │ │ │ 47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7); │ │ │ │ │ │ │ │ o5 : Ideal of P7 │ │ │ │ i6 : phi = rationalMap(C,3,2); │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ i7 : time isDominant(phi,Certify=>true) │ │ │ │ - -- used 3.92624s (cpu); 2.66358s (thread); 0s (gc) │ │ │ │ + -- used 2.84207s (cpu); 2.31617s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o7 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational │ │ │ │ ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: ********** │ │ │ │ * isDominant(RationalMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ @@ -90,26 +90,26 @@ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0159152s (cpu); 0.0175494s (thread); 0s (gc)
│ │ │ + -- used 0.0200731s (cpu); 0.0211263s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.545522s (cpu); 0.365991s (thread); 0s (gc)
│ │ │ + -- used 0.439411s (cpu); 0.362208s (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 0.0159152s (cpu); 0.0175494s (thread); 0s (gc)
│ │ │ │ + -- used 0.0200731s (cpu); 0.0211263s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.545522s (cpu); 0.365991s (thread); 0s (gc)
│ │ │ │ + -- used 0.439411s (cpu); 0.362208s (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 0.00400069s (cpu); 0.00486114s (thread); 0s (gc)
│ │ │ + -- used 0.00678549s (cpu); 0.00591266s (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 0.583205s (cpu); 0.394744s (thread); 0s (gc)
│ │ │ + -- used 0.637322s (cpu); 0.446514s (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 0.00400069s (cpu); 0.00486114s (thread); 0s (gc)
│ │ │ │ + -- used 0.00678549s (cpu); 0.00591266s (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 0.583205s (cpu); 0.394744s (thread); 0s (gc)
│ │ │ │ + -- used 0.637322s (cpu); 0.446514s (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 0.319702s (cpu); 0.186155s (thread); 0s (gc)
│ │ │ + -- used 0.322917s (cpu); 0.172455s (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 0.188222s (cpu); 0.112952s (thread); 0s (gc)
│ │ │ + -- used 0.190654s (cpu); 0.126239s (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 0.319702s (cpu); 0.186155s (thread); 0s (gc) │ │ │ │ + -- used 0.322917s (cpu); 0.172455s (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 0.188222s (cpu); 0.112952s (thread); 0s (gc) │ │ │ │ + -- used 0.190654s (cpu); 0.126239s (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 │ │ │ @@ -88,15 +88,15 @@ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ 0 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ - -- used 0.0212853s (cpu); 0.0181898s (thread); 0s (gc)
│ │ │ + -- used 0.0302431s (cpu); 0.0167583s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ @@ -116,15 +116,15 @@ │ │ │ x x - x x + x x 0 4 │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ - -- used 0.106161s (cpu); 0.0288538s (thread); 0s (gc)
│ │ │ + -- used 0.133512s (cpu); 0.0366695s (thread); 0s (gc)
│ │ │  Certify: output certified!
│ │ │  
│ │ │  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 0.00301299s (cpu); 4.3121e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00256426s (cpu); 3.8399e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 0.000108975s (cpu); 2.141e-05s (thread); 0s (gc)
│ │ │ + -- used 9.7165e-05s (cpu); 2.0987e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -52,15 +52,15 @@ │ │ │ │ t + t t , - t t + t t , - t t + t t , - t + t t , a}) │ │ │ │ 0 4 0 5 1 0 2 1 2 0 3 │ │ │ │ 2 1 3 1 3 0 4 2 3 1 4 3 2 4 │ │ │ │ │ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ │ 0 4 0 5 │ │ │ │ i3 : time projectiveDegrees(phi,Certify=>true) │ │ │ │ - -- used 0.0212853s (cpu); 0.0181898s (thread); 0s (gc) │ │ │ │ + -- used 0.0302431s (cpu); 0.0167583s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o3 = {1, 2, 4, 4, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : psi=inverseMap(toMap(phi,Dominant=>infinity)) │ │ │ │ │ │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ │ │ │ │ GF 109561[x ..x ] │ │ │ │ 0 5 │ │ │ │ o4 : RingMap ------------------ <-- GF 109561[t ..t ] │ │ │ │ x x - x x + x x 0 4 │ │ │ │ 2 3 1 4 0 5 │ │ │ │ i5 : time projectiveDegrees(psi,Certify=>true) │ │ │ │ - -- used 0.106161s (cpu); 0.0288538s (thread); 0s (gc) │ │ │ │ + -- used 0.133512s (cpu); 0.0366695s (thread); 0s (gc) │ │ │ │ Certify: output certified! │ │ │ │ │ │ │ │ o5 = {2, 4, 4, 2, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : -- Cremona transformation of P^6 defined by the quadrics through a │ │ │ │ rational octic surface │ │ │ │ @@ -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 0.00301299s (cpu); 4.3121e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00256426s (cpu); 3.8399e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 0.000108975s (cpu); 2.141e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.7165e-05s (cpu); 2.0987e-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 0.193965s (cpu); 0.124076s (thread); 0s (gc)
│ │ │ + -- used 0.104089s (cpu); 0.105889s (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 0.193965s (cpu); 0.124076s (thread); 0s (gc)
│ │ │ │ + -- used 0.104089s (cpu); 0.105889s (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 0.031972s (cpu); 0.0293826s (thread); 0s (gc)
│ │ │ + -- used 0.0359736s (cpu); 0.0390032s (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 1.20741s (cpu); 0.6835s (thread); 0s (gc)
│ │ │ + -- used 1.49727s (cpu); 0.674009s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    See also the package Divisor, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -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 0.031972s (cpu); 0.0293826s (thread); 0s (gc)
│ │ │ │ + -- used 0.0359736s (cpu); 0.0390032s (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 1.20741s (cpu); 0.6835s (thread); 0s (gc)
│ │ │ │ + -- used 1.49727s (cpu); 0.674009s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _D_i_v_i_s_o_r, which provides general tools for working with
│ │ │ │  divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.22284s (cpu); 1.00459s (thread); 0s (gc)
│ │ │ + -- used 1.11158s (cpu); 0.981082s (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 1.22284s (cpu); 1.00459s (thread); 0s (gc)
│ │ │ │ + -- used 1.11158s (cpu); 0.981082s (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 0.0927728s (cpu); 0.0927368s (thread); 0s (gc)
│ │ │ + -- used 0.0920643s (cpu); 0.0928535s (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 0.0206327s (cpu); 0.0200933s (thread); 0s (gc)
│ │ │ + -- used 0.0200111s (cpu); 0.0197175s (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 0.0927728s (cpu); 0.0927368s (thread); 0s (gc)
│ │ │ │ + -- used 0.0920643s (cpu); 0.0928535s (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 0.0206327s (cpu); 0.0200933s (thread); 0s (gc)
│ │ │ │ + -- used 0.0200111s (cpu); 0.0197175s (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 0.0694952s (cpu); 0.0681984s (thread); 0s (gc)
│ │ │ + -- used 0.0800774s (cpu); 0.0833029s (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 0.00399937s (cpu); 0.00613956s (thread); 0s (gc)
│ │ │ + -- used 0.00814783s (cpu); 0.00682803s (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 0.0694952s (cpu); 0.0681984s (thread); 0s (gc)
│ │ │ │ + -- used 0.0800774s (cpu); 0.0833029s (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 0.00399937s (cpu); 0.00613956s (thread); 0s (gc)
│ │ │ │ + -- used 0.00814783s (cpu); 0.00682803s (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 0.0398632s (cpu); 0.0388171s (thread); 0s (gc)
│ │ │ + -- used 0.0239395s (cpu); 0.0255035s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.098716s (cpu); 0.0321057s (thread); 0s (gc)
│ │ │ + -- used 0.00443214s (cpu); 0.00657179s (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 0.00607096s (cpu); 0.00709152s (thread); 0s (gc)
│ │ │ + -- used 0.00143266s (cpu); 0.00514517s (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 0.0398632s (cpu); 0.0388171s (thread); 0s (gc)
│ │ │ │ + -- used 0.0239395s (cpu); 0.0255035s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.098716s (cpu); 0.0321057s (thread); 0s (gc)
│ │ │ │ + -- used 0.00443214s (cpu); 0.00657179s (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 0.00607096s (cpu); 0.00709152s (thread); 0s (gc)
│ │ │ │ + -- used 0.00143266s (cpu); 0.00514517s (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 0.00609207s (cpu); 0.00406524s (thread); 0s (gc)
│ │ │ + -- used 0.00400154s (cpu); 0.00450706s (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 0.131563s (cpu); 0.0830383s (thread); 0s (gc)
│ │ │ + -- used 0.0439231s (cpu); 0.0470118s (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 0.0270197s (cpu); 0.0276383s (thread); 0s (gc)
│ │ │ + -- used 0.0320153s (cpu); 0.0317012s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.538583s (cpu); 0.430648s (thread); 0s (gc)
│ │ │ + -- used 0.604541s (cpu); 0.473326s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0549243s (cpu); 0.0581869s (thread); 0s (gc)
│ │ │ + -- used 0.157936s (cpu); 0.0918695s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000500408s (cpu); 0.00215721s (thread); 0s (gc)
│ │ │ + -- used 0.000157337s (cpu); 0.0023842s (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 0.573193s (cpu); 0.453822s (thread); 0s (gc)
│ │ │ + -- used 0.453919s (cpu); 0.453238s (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 0.0120027s (cpu); 0.00969265s (thread); 0s (gc)
│ │ │ + -- used 0.0120457s (cpu); 0.0118729s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.225754s (cpu); 0.163208s (thread); 0s (gc)
│ │ │ + -- used 0.349374s (cpu); 0.229296s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.20829s (cpu); 4.50938s (thread); 0s (gc)
│ │ │ + -- used 5.20784s (cpu); 4.89261s (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)
│ │ │ │ - -- used 0.0249854s (cpu); 0.0212598s (thread); 0s (gc)
│ │ │ │ + -- used 0.0635645s (cpu); 0.0250012s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ │ │  
│ │ │ │  o8 = X X
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -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 0.000659066s (cpu); 0.00207662s (thread); 0s (gc)
│ │ │ + -- used 0.00249297s (cpu); 0.00246029s (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 0.0508216s (cpu); 0.0503314s (thread); 0s (gc)
│ │ │ + -- used 0.0639073s (cpu); 0.0608789s (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 0.667941s (cpu); 0.515211s (thread); 0s (gc)
│ │ │ + -- used 0.541903s (cpu); 0.472751s (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 0.392139s (cpu); 0.317841s (thread); 0s (gc)
│ │ │ + -- used 0.443089s (cpu); 0.366583s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0157833s (cpu); 0.0172382s (thread); 0s (gc)
│ │ │ + -- used 0.053748s (cpu); 0.0206246s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00190065s (cpu); 0.00298267s (thread); 0s (gc)
│ │ │ + -- used 0.00321856s (cpu); 0.00354258s (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 0.00674931s (cpu); 0.00981537s (thread); 0s (gc)
│ │ │ + -- used 0.0107303s (cpu); 0.0113342s (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 0.00882568s (cpu); 0.00945813s (thread); 0s (gc)
│ │ │ + -- used 0.00731456s (cpu); 0.010468s (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 1.33692s (cpu); 0.940575s (thread); 0s (gc)
│ │ │ + -- used 1.19405s (cpu); 0.933775s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 0.000189867s (cpu); 1.4877e-05s (thread); 0s (gc)
│ │ │ + -- used 9.9517e-05s (cpu); 1.6071e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 9.2374e-05s (cpu); 0.00151987s (thread); 0s (gc)
│ │ │ + -- used 9.1742e-05s (cpu); 0.00168813s (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 0.00609207s (cpu); 0.00406524s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400154s (cpu); 0.00450706s (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 0.131563s (cpu); 0.0830383s (thread); 0s (gc)
│ │ │ │ + -- used 0.0439231s (cpu); 0.0470118s (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 0.0270197s (cpu); 0.0276383s (thread); 0s (gc)
│ │ │ │ + -- used 0.0320153s (cpu); 0.0317012s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.538583s (cpu); 0.430648s (thread); 0s (gc)
│ │ │ │ + -- used 0.604541s (cpu); 0.473326s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.0549243s (cpu); 0.0581869s (thread); 0s (gc)
│ │ │ │ + -- used 0.157936s (cpu); 0.0918695s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.000500408s (cpu); 0.00215721s (thread); 0s (gc)
│ │ │ │ + -- used 0.000157337s (cpu); 0.0023842s (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 0.573193s (cpu); 0.453822s (thread); 0s (gc)
│ │ │ │ + -- used 0.453919s (cpu); 0.453238s (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 0.0120027s (cpu); 0.00969265s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120457s (cpu); 0.0118729s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.225754s (cpu); 0.163208s (thread); 0s (gc)
│ │ │ │ + -- used 0.349374s (cpu); 0.229296s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.20829s (cpu); 4.50938s (thread); 0s (gc)
│ │ │ │ + -- used 5.20784s (cpu); 4.89261s (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 0.000659066s (cpu); 0.00207662s (thread); 0s (gc)
│ │ │ │ + -- used 0.00249297s (cpu); 0.00246029s (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 0.0508216s (cpu); 0.0503314s (thread); 0s (gc)
│ │ │ │ + -- used 0.0639073s (cpu); 0.0608789s (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 0.667941s (cpu); 0.515211s (thread); 0s (gc)
│ │ │ │ + -- used 0.541903s (cpu); 0.472751s (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 0.392139s (cpu); 0.317841s (thread); 0s (gc)
│ │ │ │ + -- used 0.443089s (cpu); 0.366583s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0157833s (cpu); 0.0172382s (thread); 0s (gc)
│ │ │ │ + -- used 0.053748s (cpu); 0.0206246s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00190065s (cpu); 0.00298267s (thread); 0s (gc)
│ │ │ │ + -- used 0.00321856s (cpu); 0.00354258s (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 0.00674931s (cpu); 0.00981537s (thread); 0s (gc)
│ │ │ │ + -- used 0.0107303s (cpu); 0.0113342s (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 0.00882568s (cpu); 0.00945813s (thread); 0s (gc)
│ │ │ │ + -- used 0.00731456s (cpu); 0.010468s (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 1.33692s (cpu); 0.940575s (thread); 0s (gc)
│ │ │ │ + -- used 1.19405s (cpu); 0.933775s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 0.000189867s (cpu); 1.4877e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.9517e-05s (cpu); 1.6071e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 9.2374e-05s (cpu); 0.00151987s (thread); 0s (gc)
│ │ │ │ + -- used 9.1742e-05s (cpu); 0.00168813s (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
│ │ │ - -- used 0.0239101s (cpu); 0.0222625s (thread); 0s (gc)
│ │ │ + -- used 0.0287287s (cpu); 0.0165499s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S ..S ]
│ │ │                                 1   4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ - -- used 0.0199996s (cpu); 0.0197932s (thread); 0s (gc)
│ │ │ + -- used 0.104733s (cpu); 0.0295565s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ - -- used 0.13227s (cpu); 0.0670378s (thread); 0s (gc)
│ │ │ + -- used 0.241666s (cpu); 0.0621057s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -55,15 +55,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ - -- used 0.0202003s (cpu); 0.0184683s (thread); 0s (gc)
│ │ │ + -- used 0.0429231s (cpu); 0.020714s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -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
│ │ │ - -- used 0.0204534s (cpu); 0.0206094s (thread); 0s (gc)
│ │ │ + -- used 0.204505s (cpu); 0.0472196s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  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'
│ │ │ - -- used 0.610386s (cpu); 0.563929s (thread); 0s (gc)
│ │ │ + -- used 0.807018s (cpu); 0.710379s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0192386s (cpu); 0.020534s (thread); 0s (gc)
│ │ │ + -- used 0.0351196s (cpu); 0.0207979s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0222343s (cpu); 0.0205029s (thread); 0s (gc)
│ │ │ + -- used 0.204411s (cpu); 0.0452512s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.226988s (cpu); 0.149508s (thread); 0s (gc)
│ │ │ + -- used 0.343683s (cpu); 0.147065s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -114,15 +114,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0558776s (cpu); 0.0551242s (thread); 0s (gc)
│ │ │ + -- used 0.166581s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -151,14 +151,14 @@
│ │ │         Underlying algebra => S[T ..T ]
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ - -- used 0.0200023s (cpu); 0.018836s (thread); 0s (gc)
│ │ │ + -- used 0.0965859s (cpu); 0.0306974s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ - -- used 0.0249854s (cpu); 0.0212598s (thread); 0s (gc)
│ │ │ + -- used 0.0635645s (cpu); 0.0250012s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.114386s (cpu); 0.113134s (thread); 0s (gc)
│ │ │ + -- used 0.206258s (cpu); 0.139603s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T T T  - x x y T T T )
│ │ │               2 2 1 2 3    1 2 2 2 3 4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ - -- used 0.539478s (cpu); 0.45502s (thread); 0s (gc)
│ │ │ + -- used 0.912139s (cpu); 0.655582s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1   4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ - -- used 0.166074s (cpu); 0.161795s (thread); 0s (gc)
│ │ │ + -- used 0.234207s (cpu); 0.208317s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.486629s (cpu); 0.423626s (thread); 0s (gc)
│ │ │ + -- used 0.668699s (cpu); 0.551386s (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
│ │ │ - -- used 0.0239101s (cpu); 0.0222625s (thread); 0s (gc)
│ │ │ + -- used 0.0287287s (cpu); 0.0165499s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -210,15 +210,15 @@
│ │ │ │  a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │ │                                      1     2     3         1       4        6
│ │ │ │  5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ - -- used 0.0239101s (cpu); 0.0222625s (thread); 0s (gc)
│ │ │ │ + -- used 0.0287287s (cpu); 0.0165499s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -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
│ │ │ │ - -- used 0.0199996s (cpu); 0.0197932s (thread); 0s (gc)
│ │ │ │ + -- used 0.104733s (cpu); 0.0295565s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │        ZZ
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ - -- used 0.13227s (cpu); 0.0670378s (thread); 0s (gc)
│ │ │ │ + -- used 0.241666s (cpu); 0.0621057s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i11 : C = killCycles(B)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ - -- used 0.0199996s (cpu); 0.0197932s (thread); 0s (gc)
│ │ │ + -- used 0.104733s (cpu); 0.0295565s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ - -- used 0.13227s (cpu); 0.0670378s (thread); 0s (gc)
│ │ │ + -- used 0.241666s (cpu); 0.0621057s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i7 : HHg = HH g
│ │ │ - -- used 0.0202003s (cpu); 0.0184683s (thread); 0s (gc)
│ │ │ + -- used 0.0429231s (cpu); 0.020714s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ - -- used 0.0202003s (cpu); 0.0184683s (thread); 0s (gc)
│ │ │ │ + -- used 0.0429231s (cpu); 0.020714s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -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
│ │ │ │ - -- used 0.0204534s (cpu); 0.0206094s (thread); 0s (gc)
│ │ │ │ + -- used 0.204505s (cpu); 0.0472196s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │  o8 = ideal ()
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-
│ │ │ │  c^2*d^2}
│ │ │ │  
│ │ │ │  o9 = R'
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ │ - -- used 0.610386s (cpu); 0.563929s (thread); 0s (gc)
│ │ │ │ + -- used 0.807018s (cpu); 0.710379s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ │ │  
│ │ │ │  o11 = 34
│ │ ├── ./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)
│ │ │ │ - -- used 0.0192386s (cpu); 0.020534s (thread); 0s (gc)
│ │ │ │ + -- used 0.0351196s (cpu); 0.0207979s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ │ │  
│ │ │ │  o5 = 4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ - -- used 0.0204534s (cpu); 0.0206094s (thread); 0s (gc)
│ │ │ + -- used 0.204505s (cpu); 0.0472196s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ - -- used 0.610386s (cpu); 0.563929s (thread); 0s (gc)
│ │ │ + -- used 0.807018s (cpu); 0.710379s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0192386s (cpu); 0.020534s (thread); 0s (gc)
│ │ │ + -- used 0.0351196s (cpu); 0.0207979s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0222343s (cpu); 0.0205029s (thread); 0s (gc)
│ │ │ + -- used 0.204411s (cpu); 0.0452512s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.226988s (cpu); 0.149508s (thread); 0s (gc)
│ │ │ + -- used 0.343683s (cpu); 0.147065s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ - -- used 0.0558776s (cpu); 0.0551242s (thread); 0s (gc)
│ │ │ + -- used 0.166581s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -302,15 +302,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ - -- used 0.0200023s (cpu); 0.018836s (thread); 0s (gc)
│ │ │ + -- used 0.0965859s (cpu); 0.0306974s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.0222343s (cpu); 0.0205029s (thread); 0s (gc)
│ │ │ │ + -- used 0.204411s (cpu); 0.0452512s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.226988s (cpu); 0.149508s (thread); 0s (gc)
│ │ │ │ + -- used 0.343683s (cpu); 0.147065s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │  o9 = 19
│ │ │ │ @@ -120,15 +120,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ - -- used 0.0558776s (cpu); 0.0551242s (thread); 0s (gc)
│ │ │ │ + -- used 0.166581s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │  options GenDegreeLimit and RelDegreeLimit.
│ │ │ │ @@ -155,15 +155,15 @@
│ │ │ │  o20 = {Ring => S                      }
│ │ │ │         Underlying algebra => S[T ..T ]
│ │ │ │                                  1   4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ - -- used 0.0200023s (cpu); 0.018836s (thread); 0s (gc)
│ │ │ │ + -- used 0.0965859s (cpu); 0.0306974s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ - -- used 0.0249854s (cpu); 0.0212598s (thread); 0s (gc)
│ │ │ + -- used 0.0635645s (cpu); 0.0250012s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.114386s (cpu); 0.113134s (thread); 0s (gc)
│ │ │ + -- used 0.206258s (cpu); 0.139603s (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 0.114386s (cpu); 0.113134s (thread); 0s (gc)
│ │ │ │ + -- used 0.206258s (cpu); 0.139603s (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 {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T ..T ]
│ │ │ │                                 1   4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ - -- used 0.166074s (cpu); 0.161795s (thread); 0s (gc)
│ │ │ │ + -- used 0.234207s (cpu); 0.208317s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ │ │  
│ │ │ │                5       343
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -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 0.486629s (cpu); 0.423626s (thread); 0s (gc)
│ │ │ │ + -- used 0.668699s (cpu); 0.551386s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0439526s (cpu); 0.0447774s (thread); 0s (gc)
│ │ │ + -- used 0.0519767s (cpu); 0.0516847s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.385643s (cpu); 0.388115s (thread); 0s (gc)
│ │ │ + -- used 0.447721s (cpu); 0.449846s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.525154s (cpu); 0.445251s (thread); 0s (gc)
│ │ │ + -- used 0.564567s (cpu); 0.496548s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0001174s (cpu); 0.0027735s (thread); 0s (gc)
│ │ │ + -- used 0.000603854s (cpu); 0.00317336s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.000636203s (cpu); 0.00217933s (thread); 0s (gc)
│ │ │ + -- used 0.00132949s (cpu); 0.0028677s (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 0.269663s (cpu); 0.12401s (thread); 0s (gc)
│ │ │ + -- used 0.265804s (cpu); 0.125181s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00254324s (cpu); 0.00608307s (thread); 0s (gc)
│ │ │ + -- used 0.00414621s (cpu); 0.0070433s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_is__Reduced.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 6263371580478090172
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │  
│ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor(x * y * z)
│ │ │  
│ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isReduced( D1 )
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_map__To__Projective__Space.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │                          1    2
│ │ │  
│ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │  
│ │ │  i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(x) + Div(y)
│ │ │ +o5 = Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │  
│ │ │               ZZ            2             2        2
│ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out
│ │ │ @@ -103,104 +103,104 @@
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time reflexify(J);
│ │ │ - -- used 0.251838s (cpu); 0.200054s (thread); 0s (gc)
│ │ │ + -- used 0.306827s (cpu); 0.250559s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.502812s (cpu); 0.370051s (thread); 0s (gc)
│ │ │ + -- used 0.518393s (cpu); 0.457975s (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 0.23827s (cpu); 0.116225s (thread); 0s (gc)
│ │ │ + -- used 0.277571s (cpu); 0.142668s (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 6.93321s (cpu); 4.71801s (thread); 0s (gc)
│ │ │ + -- used 5.40183s (cpu); 4.38205s (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 6.02799s (cpu); 4.65655s (thread); 0s (gc)
│ │ │ + -- used 5.48686s (cpu); 4.33613s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.583225s (cpu); 0.432695s (thread); 0s (gc)
│ │ │ + -- used 0.528445s (cpu); 0.373128s (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 1.13348s (cpu); 0.386359s (thread); 0s (gc)
│ │ │ + -- used 1.00741s (cpu); 0.36931s (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 0.241588s (cpu); 0.0854298s (thread); 0s (gc)
│ │ │ + -- used 0.208831s (cpu); 0.0497825s (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 0.0414755s (cpu); 0.0406028s (thread); 0s (gc)
│ │ │ + -- used 0.0440287s (cpu); 0.0466211s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00385702s (cpu); 0.00696376s (thread); 0s (gc)
│ │ │ + -- used 0.00512406s (cpu); 0.00829492s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexive__Power.out
│ │ │ @@ -23,44 +23,44 @@
│ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.114823s (cpu); 0.0479161s (thread); 0s (gc)
│ │ │ + -- used 0.107493s (cpu); 0.0500456s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.119986s (cpu); 0.121517s (thread); 0s (gc)
│ │ │ + -- used 0.134134s (cpu); 0.133814s (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 0.115835s (cpu); 0.0518117s (thread); 0s (gc)
│ │ │ + -- used 0.112693s (cpu); 0.0528006s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0648921s (cpu); 0.0651929s (thread); 0s (gc)
│ │ │ + -- used 0.0576231s (cpu); 0.0599791s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_dualize.html
│ │ │ @@ -163,43 +163,43 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ - -- used 0.539478s (cpu); 0.45502s (thread); 0s (gc)
│ │ │ + -- used 0.912139s (cpu); 0.655582s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -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)
│ │ │ │ - -- used 0.539478s (cpu); 0.45502s (thread); 0s (gc)
│ │ │ │ + -- used 0.912139s (cpu); 0.655582s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │               2
│ │ │ │  o10 = x x y z T T T T
│ │ │ │         1 2 2   2 3 4 5
│ │ │ │  
│ │ │ │  o10 : R[T ..T ]
│ │ │ │           1   5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ - -- used 0.166074s (cpu); 0.161795s (thread); 0s (gc)
│ │ │ + -- used 0.234207s (cpu); 0.208317s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.486629s (cpu); 0.423626s (thread); 0s (gc)
│ │ │ + -- used 0.668699s (cpu); 0.551386s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i11 : M = J*R^1;
    
│ │ │              
    
│ │ │                
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0439526s (cpu); 0.0447774s (thread); 0s (gc)
│ │ │ + -- used 0.0519767s (cpu); 0.0516847s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
    
│ │ │                
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.385643s (cpu); 0.388115s (thread); 0s (gc)
│ │ │ + -- used 0.447721s (cpu); 0.449846s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
    
│ │ │                
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.525154s (cpu); 0.445251s (thread); 0s (gc)
    
│ │ │ + -- used 0.564567s (cpu); 0.496548s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0001174s (cpu); 0.0027735s (thread); 0s (gc)
    
│ │ │ + -- used 0.000603854s (cpu); 0.00317336s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.000636203s (cpu); 0.00217933s (thread); 0s (gc)
│ │ │ + -- used 0.00132949s (cpu); 0.0028677s (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 0.269663s (cpu); 0.12401s (thread); 0s (gc)
│ │ │ + -- used 0.265804s (cpu); 0.125181s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    
│ │ │              
    
│ │ │                
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00254324s (cpu); 0.00608307s (thread); 0s (gc)
│ │ │ + -- used 0.00414621s (cpu); 0.0070433s (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 0.0439526s (cpu); 0.0447774s (thread); 0s (gc)
│ │ │ │ + -- used 0.0519767s (cpu); 0.0516847s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.385643s (cpu); 0.388115s (thread); 0s (gc)
│ │ │ │ + -- used 0.447721s (cpu); 0.449846s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.525154s (cpu); 0.445251s (thread); 0s (gc)
│ │ │ │ + -- used 0.564567s (cpu); 0.496548s (thread); 0s (gc)
│ │ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.0001174s (cpu); 0.0027735s (thread); 0s (gc)
│ │ │ │ + -- used 0.000603854s (cpu); 0.00317336s (thread); 0s (gc)
│ │ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.000636203s (cpu); 0.00217933s (thread); 0s (gc)
│ │ │ │ + -- used 0.00132949s (cpu); 0.0028677s (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 0.269663s (cpu); 0.12401s (thread); 0s (gc)
│ │ │ │ + -- used 0.265804s (cpu); 0.125181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : Ideal of R
│ │ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00254324s (cpu); 0.00608307s (thread); 0s (gc)
│ │ │ │ + -- used 0.00414621s (cpu); 0.0070433s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  KnownDomain is an option for dualize. If it is false (default is true), then
│ │ │ │  the computer will first check whether the ring is a domain, if it is not then
│ │ │ │  it will revert to ModuleStrategy. If KnownDomain is set to true for a non-
│ │ │ │  domain, then the function can return an incorrect answer.
│ │ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_is__Reduced.html
│ │ │ @@ -76,24 +76,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
    
│ │ │                
    i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │  
│ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
    
│ │ │                
    i3 : D2 = divisor(x * y * z)
│ │ │  
│ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
    
│ │ │                
    i4 : isReduced( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,20 +12,20 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is reduced (all coefficients equal to
│ │ │ │  1), otherwise it returns false.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │ │  
│ │ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor(x * y * z)
│ │ │ │  
│ │ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isReduced( D1 )
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : isReduced( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_map__To__Projective__Space.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │                
    i4 : R = ZZ/7[x,y,z];
    
│ │ │              
    
│ │ │                
    i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(x) + Div(y)
│ │ │ +o5 = Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
    
│ │ │                
    i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │ │                          1    2
│ │ │ │  The user may also specify the variable name of the new projective space.
│ │ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │ │  i5 : D = divisor(x*y)
│ │ │ │  
│ │ │ │ -o5 = Div(x) + Div(y)
│ │ │ │ +o5 = Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ │  
│ │ │ │               ZZ            2             2        2
│ │ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ │ │                7  1   6
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_reflexify.html
│ │ │ @@ -267,23 +267,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
    
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.251838s (cpu); 0.200054s (thread); 0s (gc)
│ │ │ + -- used 0.306827s (cpu); 0.250559s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.502812s (cpu); 0.370051s (thread); 0s (gc)
    
│ │ │ + -- used 0.518393s (cpu); 0.457975s (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 0.23827s (cpu); 0.116225s (thread); 0s (gc)
│ │ │ + -- used 0.277571s (cpu); 0.142668s (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 6.93321s (cpu); 4.71801s (thread); 0s (gc)
│ │ │ + -- used 5.40183s (cpu); 4.38205s (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 6.02799s (cpu); 4.65655s (thread); 0s (gc)
    │ │ │ + -- used 5.48686s (cpu); 4.33613s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.583225s (cpu); 0.432695s (thread); 0s (gc)
    │ │ │ + -- used 0.528445s (cpu); 0.373128s (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 1.13348s (cpu); 0.386359s (thread); 0s (gc)
│ │ │ + -- used 1.00741s (cpu); 0.36931s (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 0.241588s (cpu); 0.0854298s (thread); 0s (gc)
│ │ │ + -- used 0.208831s (cpu); 0.0497825s (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 0.0414755s (cpu); 0.0406028s (thread); 0s (gc)
    │ │ │ + -- used 0.0440287s (cpu); 0.0466211s (thread); 0s (gc) │ │ │
    │ │ │ 
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00385702s (cpu); 0.00696376s (thread); 0s (gc)
    │ │ │ + -- used 0.00512406s (cpu); 0.00829492s (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 0.251838s (cpu); 0.200054s (thread); 0s (gc) │ │ │ │ + -- used 0.306827s (cpu); 0.250559s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ │ - -- used 0.502812s (cpu); 0.370051s (thread); 0s (gc) │ │ │ │ + -- used 0.518393s (cpu); 0.457975s (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 0.23827s (cpu); 0.116225s (thread); 0s (gc) │ │ │ │ + -- used 0.277571s (cpu); 0.142668s (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 6.93321s (cpu); 4.71801s (thread); 0s (gc) │ │ │ │ + -- used 5.40183s (cpu); 4.38205s (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 6.02799s (cpu); 4.65655s (thread); 0s (gc) │ │ │ │ + -- used 5.48686s (cpu); 4.33613s (thread); 0s (gc) │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.583225s (cpu); 0.432695s (thread); 0s (gc) │ │ │ │ + -- used 0.528445s (cpu); 0.373128s (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 1.13348s (cpu); 0.386359s (thread); 0s (gc) │ │ │ │ + -- used 1.00741s (cpu); 0.36931s (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 0.241588s (cpu); 0.0854298s (thread); 0s (gc) │ │ │ │ + -- used 0.208831s (cpu); 0.0497825s (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 0.0414755s (cpu); 0.0406028s (thread); 0s (gc) │ │ │ │ + -- used 0.0440287s (cpu); 0.0466211s (thread); 0s (gc) │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.00385702s (cpu); 0.00696376s (thread); 0s (gc) │ │ │ │ + -- used 0.00512406s (cpu); 0.00829492s (thread); 0s (gc) │ │ │ │ For ideals, if KnownDomain is false (default value is true), then the function │ │ │ │ will check whether it is a domain. If it is a domain (or assumed to be a │ │ │ │ domain), it will reflexify using a strategy which can speed up computation, if │ │ │ │ not it will compute using a sometimes slower method which is essentially │ │ │ │ reflexifying it as a module. │ │ │ │ Consider the following example showing the importance of making the correct │ │ │ │ assumption about the ring being a domain. │ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_reflexive__Power.html │ │ │ @@ -124,30 +124,30 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.114823s (cpu); 0.0479161s (thread); 0s (gc)
│ │ │ + -- used 0.107493s (cpu); 0.0500456s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.119986s (cpu); 0.121517s (thread); 0s (gc)
│ │ │ + -- used 0.134134s (cpu); 0.133814s (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 0.115835s (cpu); 0.0518117s (thread); 0s (gc)
│ │ │ + -- used 0.112693s (cpu); 0.0528006s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0648921s (cpu); 0.0651929s (thread); 0s (gc)
│ │ │ + -- used 0.0576231s (cpu); 0.0599791s (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 0.114823s (cpu); 0.0479161s (thread); 0s (gc)
│ │ │ │ + -- used 0.107493s (cpu); 0.0500456s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.119986s (cpu); 0.121517s (thread); 0s (gc)
│ │ │ │ + -- used 0.134134s (cpu); 0.133814s (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 0.115835s (cpu); 0.0518117s (thread); 0s (gc)
│ │ │ │ + -- used 0.112693s (cpu); 0.0528006s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.0648921s (cpu); 0.0651929s (thread); 0s (gc)
│ │ │ │ + -- used 0.0576231s (cpu); 0.0599791s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : J1 == J2
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M,
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_ring_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ring( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_complement__Graph.out
│ │ │ @@ -2,15 +2,15 @@
│ │ │  
│ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │  
│ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │  
│ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -4,10 +4,18 @@
│ │ │  
│ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │ +o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ +                              1   2   4     3   4     1   2   3   5
│ │ │ +                "ring" => R
│ │ │ +                "vertices" => {x , x , x , x , x }
│ │ │ +                                1   2   3   4   5
│ │ │ +
│ │ │ +o4 : HyperGraph
│ │ │ +
│ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_complement__Graph.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │                
    i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  When applied to a graph, complementGraph returns the graph whose edge set is
│ │ │ │  the set of edges not in G. When applied to a hypergraph, the edge set is found
│ │ │ │  by taking the complement of each edge of H in the vertex set.
│ │ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │ │  
│ │ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -90,15 +90,23 @@
│ │ │            
│ │ │              
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ -              
    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   2   4     3   4     1   2   3   5
│ │ │ +                "ring" => R
│ │ │ +                "vertices" => {x , x , x , x , x }
│ │ │ +                                1   2   3   4   5
│ │ │ +
│ │ │ +o4 : HyperGraph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,14 +24,22 @@
│ │ │ │  number of recursive steps taken (see _B_r_a_n_c_h_L_i_m_i_t) and on the time taken (see
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │ +
│ │ │ │ +o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ +                              1   2   4     3   4     1   2   3   5
│ │ │ │ +                "ring" => R
│ │ │ │ +                "vertices" => {x , x , x , x , x }
│ │ │ │ +                                1   2   3   4   5
│ │ │ │ +
│ │ │ │ +o4 : HyperGraph
│ │ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch
│ │ │ │  limit reached
│ │ │ │  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;
│ │ │ - -- .208555s elapsed
│ │ │ + -- .2339s 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]
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .208555s elapsed
    
│ │ │ + -- .2339s 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; │ │ │ │ - -- .208555s elapsed │ │ │ │ + -- .2339s 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 0.000968686s (cpu); 0.00245205s (thread); 0s (gc) │ │ │ + -- used 0.000652404s (cpu); 0.00281989s (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 0.000579497s (cpu); 0.00143783s (thread); 0s (gc) │ │ │ + -- used 0.000497955s (cpu); 0.00167079s (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 0.0039999s (cpu); 0.0038425s (thread); 0s (gc) │ │ │ + -- used 0.00405684s (cpu); 0.00292176s (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 0.000478307s (cpu); 0.0022107s (thread); 0s (gc) │ │ │ + -- used 0.000410733s (cpu); 0.00171976s (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 1.56597s (cpu); 1.33739s (thread); 0s (gc) │ │ │ + -- used 1.50461s (cpu); 1.37834s (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 0.0199488s (cpu); 0.0161765s (thread); 0s (gc) │ │ │ + -- used 0.015974s (cpu); 0.0164702s (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 1.69366s (cpu); 1.46212s (thread); 0s (gc) │ │ │ + -- used 1.47998s (cpu); 1.35688s (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 0.0159994s (cpu); 0.0160116s (thread); 0s (gc) │ │ │ + -- used 0.0160017s (cpu); 0.016859s (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 0.000968686s (cpu); 0.00245205s (thread); 0s (gc)
│ │ │ + -- used 0.000652404s (cpu); 0.00281989s (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 0.000579497s (cpu); 0.00143783s (thread); 0s (gc)
│ │ │ + -- used 0.000497955s (cpu); 0.00167079s (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
    │ │ │ │ │ │ │ │ │ ├── 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 0.000968686s (cpu); 0.00245205s (thread); 0s (gc) │ │ │ │ + -- used 0.000652404s (cpu); 0.00281989s (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 0.000579497s (cpu); 0.00143783s (thread); 0s (gc) │ │ │ │ + -- used 0.000497955s (cpu); 0.00167079s (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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.0039999s (cpu); 0.0038425s (thread); 0s (gc)
│ │ │ + -- used 0.00405684s (cpu); 0.00292176s (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 0.000478307s (cpu); 0.0022107s (thread); 0s (gc)
│ │ │ + -- used 0.000410733s (cpu); 0.00171976s (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
    │ │ │ │ │ │ │ │ │ ├── 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 0.0039999s (cpu); 0.0038425s (thread); 0s (gc) │ │ │ │ + -- used 0.00405684s (cpu); 0.00292176s (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 0.000478307s (cpu); 0.0022107s (thread); 0s (gc) │ │ │ │ + -- used 0.000410733s (cpu); 0.00171976s (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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.56597s (cpu); 1.33739s (thread); 0s (gc)
│ │ │ + -- used 1.50461s (cpu); 1.37834s (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 0.0199488s (cpu); 0.0161765s (thread); 0s (gc)
│ │ │ + -- used 0.015974s (cpu); 0.0164702s (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 1.56597s (cpu); 1.33739s (thread); 0s (gc)
│ │ │ │ + -- used 1.50461s (cpu); 1.37834s (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 0.0199488s (cpu); 0.0161765s (thread); 0s (gc)
│ │ │ │ + -- used 0.015974s (cpu); 0.0164702s (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 1.69366s (cpu); 1.46212s (thread); 0s (gc)
│ │ │ + -- used 1.47998s (cpu); 1.35688s (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 0.0159994s (cpu); 0.0160116s (thread); 0s (gc)
│ │ │ + -- used 0.0160017s (cpu); 0.016859s (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 1.69366s (cpu); 1.46212s (thread); 0s (gc)
│ │ │ │ + -- used 1.47998s (cpu); 1.35688s (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 0.0159994s (cpu); 0.0160116s (thread); 0s (gc)
│ │ │ │ + -- used 0.0160017s (cpu); 0.016859s (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/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 0.027998s (cpu); 0.0285419s (thread); 0s (gc)
│ │ │ + -- used 0.0280221s (cpu); 0.0292185s (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 0.32462s (cpu); 0.267136s (thread); 0s (gc)
│ │ │ + -- used 0.332248s (cpu); 0.285695s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.214054s (cpu); 0.155731s (thread); 0s (gc)
│ │ │ + -- used 0.132562s (cpu); 0.135612s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.90969s (cpu); 5.10379s (thread); 0s (gc)
│ │ │ + -- used 4.94458s (cpu); 4.42781s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232672s (cpu); 0.170714s (thread); 0s (gc)
│ │ │ + -- used 0.132866s (cpu); 0.136387s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.05671s (cpu); 5.29131s (thread); 0s (gc)
│ │ │ + -- used 5.08951s (cpu); 4.51999s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 2.38172s (cpu); 2.12608s (thread); 0s (gc)
│ │ │ + -- used 1.59411s (cpu); 1.40827s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.74763s (cpu); 6.10052s (thread); 0s (gc)
│ │ │ + -- used 6.5758s (cpu); 5.58167s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.0547s (cpu); 1.77214s (thread); 0s (gc)
│ │ │ + -- used 1.57246s (cpu); 1.40313s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.79917s (cpu); 5.95769s (thread); 0s (gc)
│ │ │ + -- used 6.74141s (cpu); 5.70709s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.68806s (cpu); 5.99871s (thread); 0s (gc)
│ │ │ + -- used 6.74249s (cpu); 5.73446s (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 0.027998s (cpu); 0.0285419s (thread); 0s (gc) │ │ │ │ + -- used 0.0280221s (cpu); 0.0292185s (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 0.027998s (cpu); 0.0285419s (thread); 0s (gc)
│ │ │ + -- used 0.0280221s (cpu); 0.0292185s (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 0.32462s (cpu); 0.267136s (thread); 0s (gc)
│ │ │ + -- used 0.332248s (cpu); 0.285695s (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 0.214054s (cpu); 0.155731s (thread); 0s (gc)
│ │ │ + -- used 0.132562s (cpu); 0.135612s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    │ │ │
    │ │ │ 
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.90969s (cpu); 5.10379s (thread); 0s (gc)
│ │ │ + -- used 4.94458s (cpu); 4.42781s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232672s (cpu); 0.170714s (thread); 0s (gc)
│ │ │ + -- used 0.132866s (cpu); 0.136387s (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 6.05671s (cpu); 5.29131s (thread); 0s (gc)
│ │ │ + -- used 5.08951s (cpu); 4.51999s (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 2.38172s (cpu); 2.12608s (thread); 0s (gc)
│ │ │ + -- used 1.59411s (cpu); 1.40827s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    │ │ │
    │ │ │ 
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.74763s (cpu); 6.10052s (thread); 0s (gc)
│ │ │ + -- used 6.5758s (cpu); 5.58167s (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 2.0547s (cpu); 1.77214s (thread); 0s (gc)
│ │ │ + -- used 1.57246s (cpu); 1.40313s (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 7.79917s (cpu); 5.95769s (thread); 0s (gc)
│ │ │ + -- used 6.74141s (cpu); 5.70709s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    │ │ │
    │ │ │ 
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.68806s (cpu); 5.99871s (thread); 0s (gc)
│ │ │ + -- used 6.74249s (cpu); 5.73446s (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 0.32462s (cpu); 0.267136s (thread); 0s (gc) │ │ │ │ + -- used 0.332248s (cpu); 0.285695s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {609250, 92288, 52812, 22428, 9728} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ For rational curves of degree 3: │ │ │ │ i8 : time rationalCurve(3) │ │ │ │ - -- used 0.214054s (cpu); 0.155731s (thread); 0s (gc) │ │ │ │ + -- used 0.132562s (cpu); 0.135612s (thread); 0s (gc) │ │ │ │ │ │ │ │ 8564575000 │ │ │ │ o8 = ---------- │ │ │ │ 27 │ │ │ │ │ │ │ │ o8 : QQ │ │ │ │ i9 : time for D in T list rationalCurve(3,D) │ │ │ │ - -- used 5.90969s (cpu); 5.10379s (thread); 0s (gc) │ │ │ │ + -- used 4.94458s (cpu); 4.42781s (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 0.232672s (cpu); 0.170714s (thread); 0s (gc) │ │ │ │ + -- used 0.132866s (cpu); 0.136387s (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 6.05671s (cpu); 5.29131s (thread); 0s (gc) │ │ │ │ + -- used 5.08951s (cpu); 4.51999s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = {317206375, 15655168, 6424326, 1611504, 416256} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ For rational curves of degree 4: │ │ │ │ i12 : time rationalCurve(4) │ │ │ │ - -- used 2.38172s (cpu); 2.12608s (thread); 0s (gc) │ │ │ │ + -- used 1.59411s (cpu); 1.40827s (thread); 0s (gc) │ │ │ │ │ │ │ │ 15517926796875 │ │ │ │ o12 = -------------- │ │ │ │ 64 │ │ │ │ │ │ │ │ o12 : QQ │ │ │ │ i13 : time rationalCurve(4,{4,2}) │ │ │ │ - -- used 7.74763s (cpu); 6.10052s (thread); 0s (gc) │ │ │ │ + -- used 6.5758s (cpu); 5.58167s (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 2.0547s (cpu); 1.77214s (thread); 0s (gc) │ │ │ │ + -- used 1.57246s (cpu); 1.40313s (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 7.79917s (cpu); 5.95769s (thread); 0s (gc) │ │ │ │ + -- used 6.74141s (cpu); 5.70709s (thread); 0s (gc) │ │ │ │ │ │ │ │ o15 = 3883902528 │ │ │ │ │ │ │ │ o15 : QQ │ │ │ │ i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8 │ │ │ │ - -- used 7.68806s (cpu); 5.99871s (thread); 0s (gc) │ │ │ │ + -- used 6.74249s (cpu); 5.73446s (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 .0030289 seconds │ │ │ - -- used .00400072 seconds │ │ │ + -- used .00501337 seconds │ │ │ + -- used 0 seconds │ │ │ (9, 9) │ │ │ new stuff found │ │ │ 4 │ │ │ - -- used .0058332 seconds │ │ │ - -- used .00392708 seconds │ │ │ + -- used .00579407 seconds │ │ │ + -- used .00399833 seconds │ │ │ (16, 26) │ │ │ new stuff found │ │ │ 5 │ │ │ - -- used .00535985 seconds │ │ │ - -- used .0237807 seconds │ │ │ + -- used .00969253 seconds │ │ │ + -- used .0240343 seconds │ │ │ (25, 60) │ │ │ 6 │ │ │ - -- used .0167658 seconds │ │ │ - -- used .193291 seconds │ │ │ + -- used .0182164 seconds │ │ │ + -- used .215493 seconds │ │ │ (36, 120) │ │ │ 7 │ │ │ - -- used .0398612 seconds │ │ │ - -- used .764167 seconds │ │ │ + -- used .0434872 seconds │ │ │ + -- used .880778 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 .0030289 seconds
│ │ │ -     -- used .00400072 seconds
│ │ │ +     -- used .00501337 seconds
│ │ │ +     -- used 0 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .0058332 seconds
│ │ │ -     -- used .00392708 seconds
│ │ │ +     -- used .00579407 seconds
│ │ │ +     -- used .00399833 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00535985 seconds
│ │ │ -     -- used .0237807 seconds
│ │ │ +     -- used .00969253 seconds
│ │ │ +     -- used .0240343 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0167658 seconds
│ │ │ -     -- used .193291 seconds
│ │ │ +     -- used .0182164 seconds
│ │ │ +     -- used .215493 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0398612 seconds
│ │ │ -     -- used .764167 seconds
│ │ │ +     -- used .0434872 seconds
│ │ │ +     -- used .880778 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 .0030289 seconds
│ │ │ │ -     -- used .00400072 seconds
│ │ │ │ +     -- used .00501337 seconds
│ │ │ │ +     -- used 0 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .0058332 seconds
│ │ │ │ -     -- used .00392708 seconds
│ │ │ │ +     -- used .00579407 seconds
│ │ │ │ +     -- used .00399833 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00535985 seconds
│ │ │ │ -     -- used .0237807 seconds
│ │ │ │ +     -- used .00969253 seconds
│ │ │ │ +     -- used .0240343 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0167658 seconds
│ │ │ │ -     -- used .193291 seconds
│ │ │ │ +     -- used .0182164 seconds
│ │ │ │ +     -- used .215493 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0398612 seconds
│ │ │ │ -     -- used .764167 seconds
│ │ │ │ +     -- used .0434872 seconds
│ │ │ │ +     -- used .880778 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 0.106935s (cpu); 0.110412s (thread); 0s (gc)
│ │ │ + -- used 0.143996s (cpu); 0.143422s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.422481s (cpu); 0.226786s (thread); 0s (gc)
│ │ │ + -- used 0.416492s (cpu); 0.260919s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.423989s (cpu); 0.327246s (thread); 0s (gc)
│ │ │ + -- used 0.432117s (cpu); 0.3222s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.324135s (cpu); 0.21431s (thread); 0s (gc)
│ │ │ + -- used 0.304576s (cpu); 0.201292s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.305684s (cpu); 0.177955s (thread); 0s (gc)
│ │ │ + -- used 0.303685s (cpu); 0.199135s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.346151s (cpu); 0.221249s (thread); 0s (gc)
│ │ │ + -- used 0.370265s (cpu); 0.252981s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.400882s (cpu); 0.195018s (thread); 0s (gc)
│ │ │ + -- used 0.383191s (cpu); 0.223702s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 14.833s (cpu); 9.09213s (thread); 0s (gc)
│ │ │ + -- used 15.0325s (cpu); 9.61821s (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 0.541238s (cpu); 0.367994s (thread); 0s (gc)
│ │ │ + -- used 0.582927s (cpu); 0.394837s (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 0.903508s (cpu); 0.503353s (thread); 0s (gc)
│ │ │ + -- used 0.901823s (cpu); 0.554419s (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 0.783973s (cpu); 0.422158s (thread); 0s (gc)
│ │ │ + -- used 0.86857s (cpu); 0.505204s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 4.90765s (cpu); 3.43182s (thread); 0s (gc)
│ │ │ + -- used 5.58969s (cpu); 4.1272s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.12756s (cpu); 0.746302s (thread); 0s (gc)
│ │ │ + -- used 1.14848s (cpu); 0.776737s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.27762s (cpu); 2.71297s (thread); 0s (gc)
│ │ │ + -- used 3.95098s (cpu); 3.39455s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.92673s (cpu); 2.37161s (thread); 0s (gc)
│ │ │ + -- used 4.2378s (cpu); 2.54341s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.15777s (cpu); 0.848218s (thread); 0s (gc)
│ │ │ + -- used 1.19171s (cpu); 0.912889s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 4.42178s (cpu); 2.70985s (thread); 0s (gc)
│ │ │ + -- used 4.29674s (cpu); 2.66277s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.92785s (cpu); 3.38408s (thread); 0s (gc)
│ │ │ + -- used 5.23507s (cpu); 3.39197s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 15.3109s (cpu); 9.17344s (thread); 0s (gc)
│ │ │ + -- used 16.5723s (cpu); 9.99055s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 11.1976s (cpu); 6.62743s (thread); 0s (gc)
│ │ │ + -- used 12.9097s (cpu); 7.48919s (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 0.935047s (cpu); 0.593527s (thread); 0s (gc)
│ │ │ + -- used 1.10321s (cpu); 0.718668s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1904s (cpu); 7.90121s (thread); 0s (gc)
│ │ │ + -- used 12.4351s (cpu); 8.94171s (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 1.4308s (cpu); 0.98174s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.60128s (cpu); 1.14468s (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 0.174195s (cpu); 0.115117s (thread); 0s (gc)
│ │ │ + -- used 0.196566s (cpu); 0.136221s (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 0.152483s (cpu); 0.103272s (thread); 0s (gc)
│ │ │ + -- used 0.174448s (cpu); 0.116332s (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 0.551377s (cpu); 0.426674s (thread); 0s (gc)
│ │ │ + -- used 1.32203s (cpu); 0.559637s (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 0.647081s (cpu); 0.470509s (thread); 0s (gc)
│ │ │ + -- used 0.84532s (cpu); 0.613056s (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 0.622782s (cpu); 0.485649s (thread); 0s (gc)
│ │ │ + -- used 0.507658s (cpu); 0.349885s (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)
│ │ │ - -- 1.97671s elapsed
│ │ │ + -- 1.45167s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.14804s elapsed
│ │ │ + -- .905116s 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 0.000345979s (cpu); 0.00261885s (thread); 0s (gc)
│ │ │ + -- used 0.00404699s (cpu); 0.00300157s (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 0.000719891s (cpu); 0.00257196s (thread); 0s (gc)
│ │ │ + -- used 0.00246918s (cpu); 0.00281421s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.000681358s (cpu); 0.00243255s (thread); 0s (gc)
│ │ │ + -- used 0.00145432s (cpu); 0.00283402s (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 0.2863s (cpu); 0.152614s (thread); 0s (gc)
│ │ │ + -- used 0.291481s (cpu); 0.169761s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0141386s (cpu); 0.0139774s (thread); 0s (gc)
│ │ │ + -- used 0.02122s (cpu); 0.0223047s (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 0.507301s (cpu); 0.442118s (thread); 0s (gc)
│ │ │ + -- used 0.551521s (cpu); 0.498549s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.55737s (cpu); 1.32009s (thread); 0s (gc)
│ │ │ + -- used 1.59272s (cpu); 1.46536s (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 0.611243s (cpu); 0.47468s (thread); 0s (gc)
│ │ │ + -- used 0.774094s (cpu); 0.610195s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.27567s (cpu); 4.69485s (thread); 0s (gc)
│ │ │ + -- used 7.61019s (cpu); 5.81798s (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 0.0199259s (cpu); 0.0181406s (thread); 0s (gc)
│ │ │ + -- used 0.0199993s (cpu); 0.0199557s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.186992s (cpu); 0.143604s (thread); 0s (gc)
│ │ │ + -- used 0.200126s (cpu); 0.145398s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.916373s (cpu); 0.57922s (thread); 0s (gc)
│ │ │ + -- used 0.964483s (cpu); 0.666383s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.33274s (cpu); 0.858883s (thread); 0s (gc)
│ │ │ + -- used 1.56706s (cpu); 1.04061s (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 6.68388s (cpu); 4.94067s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 7.46589s (cpu); 5.73719s (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 1.40114s (cpu); 1.04615s (thread); 0s (gc)
│ │ │ + -- used 1.55213s (cpu); 1.20534s (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 0.337618s (cpu); 0.216115s (thread); 0s (gc)
│ │ │ + -- used 0.34079s (cpu); 0.236578s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.137319s (cpu); 0.081577s (thread); 0s (gc)
│ │ │ + -- used 0.130106s (cpu); 0.0806068s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.395652s (cpu); 0.27815s (thread); 0s (gc)
│ │ │ + -- used 0.419922s (cpu); 0.311807s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.67106s (cpu); 1.22082s (thread); 0s (gc)
│ │ │ + -- used 1.89581s (cpu); 1.48409s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.35818s (cpu); 1.60227s (thread); 0s (gc)
│ │ │ + -- used 2.61852s (cpu); 1.87391s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.36132s (cpu); 1.60964s (thread); 0s (gc)
│ │ │ + -- used 2.54432s (cpu); 1.83148s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.385538s (cpu); 0.299079s (thread); 0s (gc)
│ │ │ + -- used 0.485417s (cpu); 0.368857s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.738869s (cpu); 0.583954s (thread); 0s (gc)
│ │ │ + -- used 0.883264s (cpu); 0.708604s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.09565s (cpu); 0.824678s (thread); 0s (gc)
│ │ │ + -- used 1.24786s (cpu); 0.971149s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.6321s (cpu); 1.24368s (thread); 0s (gc)
│ │ │ + -- used 1.94521s (cpu); 1.53822s (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).
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.106935s (cpu); 0.110412s (thread); 0s (gc)
│ │ │ + -- used 0.143996s (cpu); 0.143422s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.422481s (cpu); 0.226786s (thread); 0s (gc)
│ │ │ + -- used 0.416492s (cpu); 0.260919s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.423989s (cpu); 0.327246s (thread); 0s (gc)
│ │ │ + -- used 0.432117s (cpu); 0.3222s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.324135s (cpu); 0.21431s (thread); 0s (gc)
│ │ │ + -- used 0.304576s (cpu); 0.201292s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.305684s (cpu); 0.177955s (thread); 0s (gc)
│ │ │ + -- used 0.303685s (cpu); 0.199135s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.346151s (cpu); 0.221249s (thread); 0s (gc)
│ │ │ + -- used 0.370265s (cpu); 0.252981s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.400882s (cpu); 0.195018s (thread); 0s (gc)
│ │ │ + -- used 0.383191s (cpu); 0.223702s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 14.833s (cpu); 9.09213s (thread); 0s (gc)
│ │ │ + -- used 15.0325s (cpu); 9.61821s (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:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.541238s (cpu); 0.367994s (thread); 0s (gc)
│ │ │ + -- used 0.582927s (cpu); 0.394837s (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 0.903508s (cpu); 0.503353s (thread); 0s (gc)
│ │ │ + -- used 0.901823s (cpu); 0.554419s (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 0.783973s (cpu); 0.422158s (thread); 0s (gc)
│ │ │ + -- used 0.86857s (cpu); 0.505204s (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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 4.90765s (cpu); 3.43182s (thread); 0s (gc)
    
│ │ │ + -- used 5.58969s (cpu); 4.1272s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.12756s (cpu); 0.746302s (thread); 0s (gc)
│ │ │ + -- used 1.14848s (cpu); 0.776737s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.27762s (cpu); 2.71297s (thread); 0s (gc)
    
│ │ │ + -- used 3.95098s (cpu); 3.39455s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.92673s (cpu); 2.37161s (thread); 0s (gc)
    
│ │ │ + -- used 4.2378s (cpu); 2.54341s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.15777s (cpu); 0.848218s (thread); 0s (gc)
│ │ │ + -- used 1.19171s (cpu); 0.912889s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 4.42178s (cpu); 2.70985s (thread); 0s (gc)
    
│ │ │ + -- used 4.29674s (cpu); 2.66277s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.92785s (cpu); 3.38408s (thread); 0s (gc)
    
│ │ │ + -- used 5.23507s (cpu); 3.39197s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 15.3109s (cpu); 9.17344s (thread); 0s (gc)
│ │ │ + -- used 16.5723s (cpu); 9.99055s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 11.1976s (cpu); 6.62743s (thread); 0s (gc)
│ │ │ + -- used 12.9097s (cpu); 7.48919s (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 0.106935s (cpu); 0.110412s (thread); 0s (gc)
│ │ │ │ + -- used 0.143996s (cpu); 0.143422s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.422481s (cpu); 0.226786s (thread); 0s (gc)
│ │ │ │ + -- used 0.416492s (cpu); 0.260919s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.423989s (cpu); 0.327246s (thread); 0s (gc)
│ │ │ │ + -- used 0.432117s (cpu); 0.3222s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.324135s (cpu); 0.21431s (thread); 0s (gc)
│ │ │ │ + -- used 0.304576s (cpu); 0.201292s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.305684s (cpu); 0.177955s (thread); 0s (gc)
│ │ │ │ + -- used 0.303685s (cpu); 0.199135s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.346151s (cpu); 0.221249s (thread); 0s (gc)
│ │ │ │ + -- used 0.370265s (cpu); 0.252981s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.400882s (cpu); 0.195018s (thread); 0s (gc)
│ │ │ │ + -- used 0.383191s (cpu); 0.223702s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 14.833s (cpu); 9.09213s (thread); 0s (gc)
│ │ │ │ + -- used 15.0325s (cpu); 9.61821s (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 0.541238s (cpu); 0.367994s (thread); 0s (gc)
│ │ │ │ + -- used 0.582927s (cpu); 0.394837s (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 0.903508s (cpu); 0.503353s (thread); 0s (gc)
│ │ │ │ + -- used 0.901823s (cpu); 0.554419s (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 0.783973s (cpu); 0.422158s (thread); 0s (gc)
│ │ │ │ + -- used 0.86857s (cpu); 0.505204s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 4.90765s (cpu); 3.43182s (thread); 0s (gc)
│ │ │ │ + -- used 5.58969s (cpu); 4.1272s (thread); 0s (gc)
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 1.12756s (cpu); 0.746302s (thread); 0s (gc)
│ │ │ │ + -- used 1.14848s (cpu); 0.776737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 3.27762s (cpu); 2.71297s (thread); 0s (gc)
│ │ │ │ + -- used 3.95098s (cpu); 3.39455s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 3.92673s (cpu); 2.37161s (thread); 0s (gc)
│ │ │ │ + -- used 4.2378s (cpu); 2.54341s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 1.15777s (cpu); 0.848218s (thread); 0s (gc)
│ │ │ │ + -- used 1.19171s (cpu); 0.912889s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 4.42178s (cpu); 2.70985s (thread); 0s (gc)
│ │ │ │ + -- used 4.29674s (cpu); 2.66277s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 4.92785s (cpu); 3.38408s (thread); 0s (gc)
│ │ │ │ + -- used 5.23507s (cpu); 3.39197s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 15.3109s (cpu); 9.17344s (thread); 0s (gc)
│ │ │ │ + -- used 16.5723s (cpu); 9.99055s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 11.1976s (cpu); 6.62743s (thread); 0s (gc)
│ │ │ │ + -- used 12.9097s (cpu); 7.48919s (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 0.935047s (cpu); 0.593527s (thread); 0s (gc)
│ │ │ + -- used 1.10321s (cpu); 0.718668s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1904s (cpu); 7.90121s (thread); 0s (gc)
    
│ │ │ + -- used 12.4351s (cpu); 8.94171s (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 1.4308s (cpu); 0.98174s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.60128s (cpu); 1.14468s (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 0.174195s (cpu); 0.115117s (thread); 0s (gc)
│ │ │ + -- used 0.196566s (cpu); 0.136221s (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 0.152483s (cpu); 0.103272s (thread); 0s (gc)
│ │ │ + -- used 0.174448s (cpu); 0.116332s (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 0.551377s (cpu); 0.426674s (thread); 0s (gc)
│ │ │ + -- used 1.32203s (cpu); 0.559637s (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 0.647081s (cpu); 0.470509s (thread); 0s (gc)
│ │ │ + -- used 0.84532s (cpu); 0.613056s (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 0.622782s (cpu); 0.485649s (thread); 0s (gc)
│ │ │ + -- used 0.507658s (cpu); 0.349885s (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 0.935047s (cpu); 0.593527s (thread); 0s (gc)
│ │ │ │ + -- used 1.10321s (cpu); 0.718668s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 11.1904s (cpu); 7.90121s (thread); 0s (gc)
│ │ │ │ + -- used 12.4351s (cpu); 8.94171s (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 1.4308s (cpu); 0.98174s (thread); 0s (gc)
│ │ │ │ +-- used 1.60128s (cpu); 1.14468s (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 0.174195s (cpu); 0.115117s (thread); 0s (gc)
│ │ │ │ + -- used 0.196566s (cpu); 0.136221s (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 0.152483s (cpu); 0.103272s (thread); 0s (gc)
│ │ │ │ + -- used 0.174448s (cpu); 0.116332s (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 0.551377s (cpu); 0.426674s (thread); 0s (gc)
│ │ │ │ + -- used 1.32203s (cpu); 0.559637s (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 0.647081s (cpu); 0.470509s (thread); 0s (gc)
│ │ │ │ + -- used 0.84532s (cpu); 0.613056s (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 0.622782s (cpu); 0.485649s (thread); 0s (gc)
│ │ │ │ + -- used 0.507658s (cpu); 0.349885s (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)
│ │ │ - -- 1.97671s elapsed
│ │ │ + -- 1.45167s 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)
│ │ │ - -- 1.14804s elapsed
│ │ │ + -- .905116s 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)
│ │ │ │ - -- 1.97671s elapsed
│ │ │ │ + -- 1.45167s 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)
│ │ │ │ - -- 1.14804s elapsed
│ │ │ │ + -- .905116s 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-
│ │ │ │  1.25.06+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 0.000345979s (cpu); 0.00261885s (thread); 0s (gc)
│ │ │ + -- used 0.00404699s (cpu); 0.00300157s (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 0.000719891s (cpu); 0.00257196s (thread); 0s (gc)
│ │ │ + -- used 0.00246918s (cpu); 0.00281421s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.000681358s (cpu); 0.00243255s (thread); 0s (gc)
│ │ │ + -- used 0.00145432s (cpu); 0.00283402s (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 0.000345979s (cpu); 0.00261885s (thread); 0s (gc)
│ │ │ │ + -- used 0.00404699s (cpu); 0.00300157s (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 0.000719891s (cpu); 0.00257196s (thread); 0s (gc)
│ │ │ │ + -- used 0.00246918s (cpu); 0.00281421s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 0.000681358s (cpu); 0.00243255s (thread); 0s (gc)
│ │ │ │ + -- used 0.00145432s (cpu); 0.00283402s (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 0.2863s (cpu); 0.152614s (thread); 0s (gc)
│ │ │ + -- used 0.291481s (cpu); 0.169761s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0141386s (cpu); 0.0139774s (thread); 0s (gc)
│ │ │ + -- used 0.02122s (cpu); 0.0223047s (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 0.2863s (cpu); 0.152614s (thread); 0s (gc)
│ │ │ │ + -- used 0.291481s (cpu); 0.169761s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0141386s (cpu); 0.0139774s (thread); 0s (gc)
│ │ │ │ + -- used 0.02122s (cpu); 0.0223047s (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 0.507301s (cpu); 0.442118s (thread); 0s (gc)
│ │ │ + -- used 0.551521s (cpu); 0.498549s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.55737s (cpu); 1.32009s (thread); 0s (gc)
│ │ │ + -- used 1.59272s (cpu); 1.46536s (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 0.507301s (cpu); 0.442118s (thread); 0s (gc)
│ │ │ │ + -- used 0.551521s (cpu); 0.498549s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.55737s (cpu); 1.32009s (thread); 0s (gc)
│ │ │ │ + -- used 1.59272s (cpu); 1.46536s (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 0.611243s (cpu); 0.47468s (thread); 0s (gc)
│ │ │ + -- used 0.774094s (cpu); 0.610195s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.27567s (cpu); 4.69485s (thread); 0s (gc)
    
│ │ │ + -- used 7.61019s (cpu); 5.81798s (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 0.0199259s (cpu); 0.0181406s (thread); 0s (gc)
│ │ │ + -- used 0.0199993s (cpu); 0.0199557s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.186992s (cpu); 0.143604s (thread); 0s (gc)
│ │ │ + -- used 0.200126s (cpu); 0.145398s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.916373s (cpu); 0.57922s (thread); 0s (gc)
│ │ │ + -- used 0.964483s (cpu); 0.666383s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.33274s (cpu); 0.858883s (thread); 0s (gc)
│ │ │ + -- used 1.56706s (cpu); 1.04061s (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 6.68388s (cpu); 4.94067s (thread); 0s (gc) │ │ │ +internalChooseMinor: Ch -- used 7.46589s (cpu); 5.73719s (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 1.40114s (cpu); 1.04615s (thread); 0s (gc)
│ │ │ + -- used 1.55213s (cpu); 1.20534s (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 0.337618s (cpu); 0.216115s (thread); 0s (gc)
│ │ │ + -- used 0.34079s (cpu); 0.236578s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    │ │ │
    │ │ │ 
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.137319s (cpu); 0.081577s (thread); 0s (gc)
│ │ │ + -- used 0.130106s (cpu); 0.0806068s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    │ │ │
    │ │ │ 
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.395652s (cpu); 0.27815s (thread); 0s (gc)
│ │ │ + -- used 0.419922s (cpu); 0.311807s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │ 
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.67106s (cpu); 1.22082s (thread); 0s (gc)
│ │ │ + -- used 1.89581s (cpu); 1.48409s (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 2.35818s (cpu); 1.60227s (thread); 0s (gc)
    │ │ │ + -- used 2.61852s (cpu); 1.87391s (thread); 0s (gc) │ │ │
    │ │ │ 
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.36132s (cpu); 1.60964s (thread); 0s (gc)
│ │ │ + -- used 2.54432s (cpu); 1.83148s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    │ │ │
    │ │ │ 
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.385538s (cpu); 0.299079s (thread); 0s (gc)
│ │ │ + -- used 0.485417s (cpu); 0.368857s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    │ │ │
    │ │ │ 
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.738869s (cpu); 0.583954s (thread); 0s (gc)
│ │ │ + -- used 0.883264s (cpu); 0.708604s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │
    │ │ │ 
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.09565s (cpu); 0.824678s (thread); 0s (gc)
│ │ │ + -- used 1.24786s (cpu); 0.971149s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    │ │ │
    │ │ │ 
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.6321s (cpu); 1.24368s (thread); 0s (gc)
│ │ │ + -- used 1.94521s (cpu); 1.53822s (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 0.611243s (cpu); 0.47468s (thread); 0s (gc) │ │ │ │ + -- used 0.774094s (cpu); 0.610195s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : time regularInCodimension(2, S) │ │ │ │ - -- used 6.27567s (cpu); 4.69485s (thread); 0s (gc) │ │ │ │ + -- used 7.61019s (cpu); 5.81798s (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 0.0199259s (cpu); 0.0181406s (thread); 0s (gc) │ │ │ │ + -- used 0.0199993s (cpu); 0.0199557s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = -1 │ │ │ │ i13 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.186992s (cpu); 0.143604s (thread); 0s (gc) │ │ │ │ + -- used 0.200126s (cpu); 0.145398s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.916373s (cpu); 0.57922s (thread); 0s (gc) │ │ │ │ + -- used 0.964483s (cpu); 0.666383s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : time regularInCodimension(2, R) │ │ │ │ - -- used 1.33274s (cpu); 0.858883s (thread); 0s (gc) │ │ │ │ + -- used 1.56706s (cpu); 1.04061s (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 6.68388s (cpu); 4.94067s (thread); 0s (gc) │ │ │ │ +internalChooseMinor: Ch -- used 7.46589s (cpu); 5.73719s (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 1.40114s (cpu); 1.04615s (thread); 0s (gc) │ │ │ │ + -- used 1.55213s (cpu); 1.20534s (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 0.337618s (cpu); 0.216115s (thread); 0s (gc) │ │ │ │ + -- used 0.34079s (cpu); 0.236578s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = true │ │ │ │ i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.137319s (cpu); 0.081577s (thread); 0s (gc) │ │ │ │ + -- used 0.130106s (cpu); 0.0806068s (thread); 0s (gc) │ │ │ │ │ │ │ │ o22 = true │ │ │ │ i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.395652s (cpu); 0.27815s (thread); 0s (gc) │ │ │ │ + -- used 0.419922s (cpu); 0.311807s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = true │ │ │ │ i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 1.67106s (cpu); 1.22082s (thread); 0s (gc) │ │ │ │ + -- used 1.89581s (cpu); 1.48409s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = true │ │ │ │ i25 : StrategyCurrent#LexSmallest = 0; │ │ │ │ i26 : StrategyCurrent#LexSmallestTerm = 100; │ │ │ │ i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 2.35818s (cpu); 1.60227s (thread); 0s (gc) │ │ │ │ + -- used 2.61852s (cpu); 1.87391s (thread); 0s (gc) │ │ │ │ i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 2.36132s (cpu); 1.60964s (thread); 0s (gc) │ │ │ │ + -- used 2.54432s (cpu); 1.83148s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = true │ │ │ │ i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.385538s (cpu); 0.299079s (thread); 0s (gc) │ │ │ │ + -- used 0.485417s (cpu); 0.368857s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = true │ │ │ │ i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.738869s (cpu); 0.583954s (thread); 0s (gc) │ │ │ │ + -- used 0.883264s (cpu); 0.708604s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.09565s (cpu); 0.824678s (thread); 0s (gc) │ │ │ │ + -- used 1.24786s (cpu); 0.971149s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.6321s (cpu); 1.24368s (thread); 0s (gc) │ │ │ │ + -- used 1.94521s (cpu); 1.53822s (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 0.000751029s (cpu); 0.00285057s (thread); 0s (gc) │ │ │ + -- used 0.0007059s (cpu); 0.00303536s (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 0.00513402s (cpu); 0.00777858s (thread); 0s (gc) │ │ │ + -- used 0.00400183s (cpu); 0.00669621s (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 0.000751029s (cpu); 0.00285057s (thread); 0s (gc)
│ │ │ + -- used 0.0007059s (cpu); 0.00303536s (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 0.00513402s (cpu); 0.00777858s (thread); 0s (gc)
│ │ │ + -- used 0.00400183s (cpu); 0.00669621s (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 0.000751029s (cpu); 0.00285057s (thread); 0s (gc)
│ │ │ │ + -- used 0.0007059s (cpu); 0.00303536s (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 0.00513402s (cpu); 0.00777858s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400183s (cpu); 0.00669621s (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 => 0x7fb4aa11c3f0}
│ │ │ +o2 = int32{Address => 0x7fd9da48cee0}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fb4aa11c3f0
│ │ │ +o3 = 0x7fd9da48cee0
│ │ │  
│ │ │  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 = {0x7f7dadf549a0, 0x7f7dadf54960, 0x7f7dadf54950}
│ │ │ +o3 = {0x7f21c30d1200, 0x7f21c30d11f0, 0x7f21c30d11e0}
│ │ │  
│ │ │  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 = {0x7f5ca44c2e20, 0x7f5ca44c2e10, 0x7f5ca44c2e00}
│ │ │ +o2 = {0x7f39068879b0, 0x7f39068879a0, 0x7f3906887990}
│ │ │  
│ │ │  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 = 0x7fdf2dab9480
│ │ │ +o1 = 0x7fdbf3889880
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fdf2dab9480
│ │ │ +o2 = 0x7fdbf3889880
│ │ │  
│ │ │  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 = 0x7f8029adc420
│ │ │ +o2 = 0x7f81a34d1d80
│ │ │  
│ │ │  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 = 0x7f6d872d9860
│ │ │ +o1 = 0x7f0a4c9764e0
│ │ │  
│ │ │  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" => 6.91707e-310}
│ │ │ +o2 = HashTable{"bar" => 6.9402e-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 => 0x7fa582cd8c00}
│ │ │ +o2 = int32{Address => 0x7ff16e467380}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fa582cd8c00
│ │ │ +o3 = 0x7ff16e467380
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fa582cd8c05
│ │ │ +o4 = 0x7ff16e467385
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fa582cd8bfd
│ │ │ +o5 = 0x7ff16e46737d
│ │ │  
│ │ │  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{0x7fd7a2d17550, mpfr}
│ │ │ +o2 = SharedLibrary{0x7fb681c04550, 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 = 0x7fa1ab48bf10
│ │ │ +o2 = 0x7f2dcc8452c0
│ │ │  
│ │ │  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 = 0x556a51cdaac0
│ │ │ +o1 = 0x5609d3669ac0
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f2f0b0182f0
│ │ │ +o2 = 0x7f05253338f0
│ │ │  
│ │ │  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 = 0x7f638406a460
│ │ │ +o17 = 0x7f2adc06a460
│ │ │  
│ │ │  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 = 0x7f44f19ac370
│ │ │ +o1 = 0x7fdf1edc9360
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f44efee5000
│ │ │ +o2 = 0x7fdf1caebc10
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f44eff02f20
│ │ │ +o3 = 0x7fdf1caebb00
│ │ │  
│ │ │  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 0x7fa978071f50
│ │ │ -freeing memory at 0x7fa978065e80
│ │ │ -freeing memory at 0x7fa97806a460
│ │ │ -freeing memory at 0x7fa978070610
│ │ │ -freeing memory at 0x7fa9780701c0
│ │ │ -freeing memory at 0x7fa9780705f0
│ │ │ -freeing memory at 0x7fa9780708d0
│ │ │ -freeing memory at 0x7fa978004b00
│ │ │ -freeing memory at 0x7fa97806a650
│ │ │ +freeing memory at 0x7f14580701c0
│ │ │ +freeing memory at 0x7f14580708d0
│ │ │ +freeing memory at 0x7f14580705f0
│ │ │ +freeing memory at 0x7f145806a650
│ │ │ +freeing memory at 0x7f1458065e80
│ │ │ +freeing memory at 0x7f1458071f50
│ │ │ +freeing memory at 0x7f1458004b00
│ │ │ +freeing memory at 0x7f1458070610
│ │ │ +freeing memory at 0x7f145806a460
│ │ │  
│ │ │  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 = 0x7fc980306dd0
│ │ │ +o5 = 0x7fac4f079ba0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fc980306dd0
│ │ │ +o6 = 0x7fac4f079ba0
│ │ │  
│ │ │  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 => 0x7fb4aa11c3f0}
    │ │ │ +o2 = int32{Address => 0x7fd9da48cee0} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    To get this, use address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fb4aa11c3f0
│ │ │ +o3 = 0x7fd9da48cee0
│ │ │  
│ │ │  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 => 0x7fb4aa11c3f0} │ │ │ │ +o2 = int32{Address => 0x7fd9da48cee0} │ │ │ │ To get this, use _a_d_d_r_e_s_s. │ │ │ │ i3 : address x │ │ │ │ │ │ │ │ -o3 = 0x7fb4aa11c3f0 │ │ │ │ +o3 = 0x7fd9da48cee0 │ │ │ │ │ │ │ │ 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 = {0x7f7dadf549a0, 0x7f7dadf54960, 0x7f7dadf54950}
│ │ │ +o3 = {0x7f21c30d1200, 0x7f21c30d11f0, 0x7f21c30d11e0}
│ │ │  
│ │ │  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 = {0x7f7dadf549a0, 0x7f7dadf54960, 0x7f7dadf54950} │ │ │ │ +o3 = {0x7f21c30d1200, 0x7f21c30d11f0, 0x7f21c30d11e0} │ │ │ │ │ │ │ │ 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 = {0x7f5ca44c2e20, 0x7f5ca44c2e10, 0x7f5ca44c2e00}
│ │ │ +o2 = {0x7f39068879b0, 0x7f39068879a0, 0x7f3906887990}
│ │ │  
│ │ │  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 = {0x7f5ca44c2e20, 0x7f5ca44c2e10, 0x7f5ca44c2e00}
│ │ │ │ +o2 = {0x7f39068879b0, 0x7f39068879a0, 0x7f3906887990}
│ │ │ │  
│ │ │ │  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 = 0x7fdf2dab9480
│ │ │ +o1 = 0x7fdbf3889880
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │ 
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fdf2dab9480
│ │ │ +o2 = 0x7fdbf3889880
│ │ │  
│ │ │  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 = 0x7fdf2dab9480 │ │ │ │ +o1 = 0x7fdbf3889880 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7fdf2dab9480 │ │ │ │ +o2 = 0x7fdbf3889880 │ │ │ │ │ │ │ │ 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 = 0x7f8029adc420
│ │ │ +o2 = 0x7f81a34d1d80
│ │ │  
│ │ │  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 = 0x7f8029adc420
│ │ │ │ +o2 = 0x7f81a34d1d80
│ │ │ │  
│ │ │ │  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 = 0x7f6d872d9860
│ │ │ +o1 = 0x7f0a4c9764e0
│ │ │  
│ │ │  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 = 0x7f6d872d9860
│ │ │ │ +o1 = 0x7f0a4c9764e0
│ │ │ │  
│ │ │ │  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" => 6.91707e-310}
│ │ │ +o2 = HashTable{"bar" => 6.9402e-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" => 6.91707e-310} │ │ │ │ +o2 = HashTable{"bar" => 6.9402e-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 => 0x7fa582cd8c00}
    │ │ │ +o2 = int32{Address => 0x7ff16e467380} │ │ │
    │ │ │
    │ │ │

    These pointers can be accessed using address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fa582cd8c00
│ │ │ +o3 = 0x7ff16e467380
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Simple arithmetic can be performed on pointers.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fa582cd8c05
│ │ │ +o4 = 0x7ff16e467385
│ │ │  
│ │ │  o4 : Pointer
    │ │ │
    │ │ │ 
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fa582cd8bfd
│ │ │ +o5 = 0x7ff16e46737d
│ │ │  
│ │ │  o5 : Pointer
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7fa582cd8c00} │ │ │ │ +o2 = int32{Address => 0x7ff16e467380} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7fa582cd8c00 │ │ │ │ +o3 = 0x7ff16e467380 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7fa582cd8c05 │ │ │ │ +o4 = 0x7ff16e467385 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7fa582cd8bfd │ │ │ │ +o5 = 0x7ff16e46737d │ │ │ │ │ │ │ │ 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{0x7fd7a2d17550, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7fb681c04550, mpfr} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Menu

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ i1 : mpfr = openSharedLibrary "mpfr" │ │ │ │ │ │ │ │ o1 = mpfr │ │ │ │ │ │ │ │ o1 : SharedLibrary │ │ │ │ i2 : peek mpfr │ │ │ │ │ │ │ │ -o2 = SharedLibrary{0x7fd7a2d17550, mpfr} │ │ │ │ +o2 = SharedLibrary{0x7fb681c04550, mpfr} │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) -- see │ │ │ │ _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ @@ -78,15 +78,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fa1ab48bf10
│ │ │ +o2 = 0x7f2dcc8452c0
│ │ │  
│ │ │  o2 : Pointer
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : *ptr = int 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7fa1ab48bf10
│ │ │ │ +o2 = 0x7f2dcc8452c0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : *ptr = int 6
│ │ │ │  
│ │ │ │  o3 = 6
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html
│ │ │ @@ -71,29 +71,29 @@
│ │ │            
      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 = 0x556a51cdaac0
│ │ │ +o1 = 0x5609d3669ac0
│ │ │  
│ │ │  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 = 0x7f2f0b0182f0
│ │ │ +o2 = 0x7f05253338f0
│ │ │  
│ │ │  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 = 0x556a51cdaac0 │ │ │ │ +o1 = 0x5609d3669ac0 │ │ │ │ │ │ │ │ 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 = 0x7f2f0b0182f0 │ │ │ │ +o2 = 0x7f05253338f0 │ │ │ │ │ │ │ │ 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 = 0x7f638406a460
│ │ │ +o17 = 0x7f2adc06a460
│ │ │  
│ │ │  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 = 0x7f638406a460 │ │ │ │ +o17 = 0x7f2adc06a460 │ │ │ │ │ │ │ │ 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 = 0x7f44f19ac370
│ │ │ +o1 = 0x7fdf1edc9360
│ │ │  
│ │ │  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 = 0x7f44efee5000
│ │ │ +o2 = 0x7fdf1caebc10
│ │ │  
│ │ │  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 = 0x7f44eff02f20
│ │ │ +o3 = 0x7fdf1caebb00
│ │ │  
│ │ │  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 = 0x7f44f19ac370 │ │ │ │ +o1 = 0x7fdf1edc9360 │ │ │ │ │ │ │ │ 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 = 0x7f44efee5000 │ │ │ │ +o2 = 0x7fdf1caebc10 │ │ │ │ │ │ │ │ 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 = 0x7f44eff02f20 │ │ │ │ +o3 = 0x7fdf1caebb00 │ │ │ │ │ │ │ │ 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 0x7fa978071f50
│ │ │ -freeing memory at 0x7fa978065e80
│ │ │ -freeing memory at 0x7fa97806a460
│ │ │ -freeing memory at 0x7fa978070610
│ │ │ -freeing memory at 0x7fa9780701c0
│ │ │ -freeing memory at 0x7fa9780705f0
│ │ │ -freeing memory at 0x7fa9780708d0
│ │ │ -freeing memory at 0x7fa978004b00
│ │ │ -freeing memory at 0x7fa97806a650
    │ │ │ +freeing memory at 0x7f14580701c0 │ │ │ +freeing memory at 0x7f14580708d0 │ │ │ +freeing memory at 0x7f14580705f0 │ │ │ +freeing memory at 0x7f145806a650 │ │ │ +freeing memory at 0x7f1458065e80 │ │ │ +freeing memory at 0x7f1458071f50 │ │ │ +freeing memory at 0x7f1458004b00 │ │ │ +freeing memory at 0x7f1458070610 │ │ │ +freeing memory at 0x7f145806a460 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7fc980306dd0
│ │ │ +o5 = 0x7fac4f079ba0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fc980306dd0
│ │ │ +o6 = 0x7fac4f079ba0
│ │ │  
│ │ │  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 = 0x7fc980306dd0 │ │ │ │ +o5 = 0x7fac4f079ba0 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7fc980306dd0 │ │ │ │ +o6 = 0x7fac4f079ba0 │ │ │ │ │ │ │ │ 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/M2-26869-0/0 │ │ │ +o2 = /tmp/M2-43514-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-26869-0/0 │ │ │ +o3 = /tmp/M2-43514-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-26869-0/0 │ │ │ +o5 = /tmp/M2-43514-0/0 │ │ │ │ │ │ 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/M2-26869-0/0
    │ │ │ +o2 = /tmp/M2-43514-0/0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-26869-0/0
│ │ │ +o3 = /tmp/M2-43514-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-26869-0/0
│ │ │ +o5 = /tmp/M2-43514-0/0
│ │ │  
│ │ │  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/M2-26869-0/0
│ │ │ │ +o2 = /tmp/M2-43514-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-26869-0/0
│ │ │ │ +o3 = /tmp/M2-43514-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-26869-0/0
│ │ │ │ +o5 = /tmp/M2-43514-0/0
│ │ │ │  
│ │ │ │  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 2.42287s (cpu); 1.23399s (thread); 0s (gc)
│ │ │ + -- used 2.91212s (cpu); 1.56735s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.41983s (cpu); 0.772417s (thread); 0s (gc)
│ │ │ + -- used 2.07823s (cpu); 0.972581s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.17549s (cpu); 0.594626s (thread); 0s (gc)
│ │ │ + -- used 1.50122s (cpu); 0.811078s (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 0.00400179s (cpu); 0.00398934s (thread); 0s (gc)
│ │ │ + -- used 0.0039835s (cpu); 0.00577737s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.380218s (cpu); 0.261369s (thread); 0s (gc)
│ │ │ + -- used 0.429648s (cpu); 0.294958s (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 0.364449s (cpu); 0.200145s (thread); 0s (gc)
│ │ │ + -- used 0.415852s (cpu); 0.232632s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.48133s (cpu); 1.12098s (thread); 0s (gc)
│ │ │ + -- used 1.37888s (cpu); 1.15886s (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 0.958467s (cpu); 0.59754s (thread); 0s (gc)
│ │ │ + -- used 1.15257s (cpu); 0.693431s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.43909s (cpu); 0.808893s (thread); 0s (gc)
│ │ │ + -- used 1.95171s (cpu); 1.18442s (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 1.75044s (cpu); 1.3853s (thread); 0s (gc)
│ │ │ + -- used 1.81716s (cpu); 1.4907s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.626563s (cpu); 0.496851s (thread); 0s (gc)
│ │ │ + -- used 0.621714s (cpu); 0.552502s (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 2.42287s (cpu); 1.23399s (thread); 0s (gc)
│ │ │ │ + -- used 2.91212s (cpu); 1.56735s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.41983s (cpu); 0.772417s (thread); 0s (gc)
│ │ │ │ + -- used 2.07823s (cpu); 0.972581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.17549s (cpu); 0.594626s (thread); 0s (gc)
│ │ │ │ + -- used 1.50122s (cpu); 0.811078s (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 2.42287s (cpu); 1.23399s (thread); 0s (gc)
│ │ │ + -- used 2.91212s (cpu); 1.56735s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.41983s (cpu); 0.772417s (thread); 0s (gc)
│ │ │ + -- used 2.07823s (cpu); 0.972581s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.17549s (cpu); 0.594626s (thread); 0s (gc)
│ │ │ + -- used 1.50122s (cpu); 0.811078s (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 0.00400179s (cpu); 0.00398934s (thread); 0s (gc)
│ │ │ + -- used 0.0039835s (cpu); 0.00577737s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.380218s (cpu); 0.261369s (thread); 0s (gc)
│ │ │ + -- used 0.429648s (cpu); 0.294958s (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 0.364449s (cpu); 0.200145s (thread); 0s (gc)
│ │ │ + -- used 0.415852s (cpu); 0.232632s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.48133s (cpu); 1.12098s (thread); 0s (gc)
│ │ │ + -- used 1.37888s (cpu); 1.15886s (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 0.958467s (cpu); 0.59754s (thread); 0s (gc)
│ │ │ + -- used 1.15257s (cpu); 0.693431s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.43909s (cpu); 0.808893s (thread); 0s (gc)
│ │ │ + -- used 1.95171s (cpu); 1.18442s (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 1.75044s (cpu); 1.3853s (thread); 0s (gc)
│ │ │ + -- used 1.81716s (cpu); 1.4907s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.626563s (cpu); 0.496851s (thread); 0s (gc)
│ │ │ + -- used 0.621714s (cpu); 0.552502s (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 0.00400179s (cpu); 0.00398934s (thread); 0s (gc)
│ │ │ │ + -- used 0.0039835s (cpu); 0.00577737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.380218s (cpu); 0.261369s (thread); 0s (gc)
│ │ │ │ + -- used 0.429648s (cpu); 0.294958s (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 0.364449s (cpu); 0.200145s (thread); 0s (gc)
│ │ │ │ + -- used 0.415852s (cpu); 0.232632s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.48133s (cpu); 1.12098s (thread); 0s (gc)
│ │ │ │ + -- used 1.37888s (cpu); 1.15886s (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 0.958467s (cpu); 0.59754s (thread); 0s (gc)
│ │ │ │ + -- used 1.15257s (cpu); 0.693431s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.43909s (cpu); 0.808893s (thread); 0s (gc)
│ │ │ │ + -- used 1.95171s (cpu); 1.18442s (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 1.75044s (cpu); 1.3853s (thread); 0s (gc)
│ │ │ │ + -- used 1.81716s (cpu); 1.4907s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.626563s (cpu); 0.496851s (thread); 0s (gc)
│ │ │ │ + -- used 0.621714s (cpu); 0.552502s (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 0.719968s (cpu); 0.404075s (thread); 0s (gc)
│ │ │ + -- used 1.12156s (cpu); 0.37365s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.65623s (cpu); 0.997981s (thread); 0s (gc)
│ │ │ + -- used 2.37237s (cpu); 0.971681s (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 0.719968s (cpu); 0.404075s (thread); 0s (gc)
│ │ │ + -- used 1.12156s (cpu); 0.37365s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.65623s (cpu); 0.997981s (thread); 0s (gc)
│ │ │ + -- used 2.37237s (cpu); 0.971681s (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 0.719968s (cpu); 0.404075s (thread); 0s (gc) │ │ │ │ + -- used 1.12156s (cpu); 0.37365s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 1.65623s (cpu); 0.997981s (thread); 0s (gc) │ │ │ │ + -- used 2.37237s (cpu); 0.971681s (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/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ @@ -243,72 +243,67 @@ │ │ │ | 2 2 2 2 2 2 2 2 2 2 2 2 | │ │ │ |t - t + t t t - t t + t t t + t t t - t t t t + 25t t + 4t t t - 2t t t t - 2t t t t - 2t t t t + t t t t + t t t t - 3t t + 3t t t t - 26t t t | │ │ │ | 6 23 16 20 22 14 22 23 22 13 23 16 19 16 22 13 19 16 19 23 20 21 20 22 13 21 16 20 19 21 23 13 19 21 22 13 19 21 16 13 19 21 20 21 20 13 19 21 13 19 21| │ │ │ +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11 │ │ │ +o12 = | -36 -30 42 21 -22 23 23 16 -7 -30 -11 -40 19 19 -41 -22 24 -36 -8 33 │ │ │ ----------------------------------------------------------------------- │ │ │ - -30 -29 24 -29 | │ │ │ + -29 -10 -29 -29 | │ │ │ │ │ │ 1 24 │ │ │ o12 : Matrix kk <-- kk │ │ │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39 │ │ │ +o13 = | 24 -7 -25 32 -28 13 -49 11 -17 -17 -2 -50 30 19 -41 -21 -38 -24 21 39 │ │ │ ----------------------------------------------------------------------- │ │ │ - 0 -24 34 | │ │ │ + 34 0 -16 -47 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a - 12b*c - 4c - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c + │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a - 7b*c + 21c - 30a*d - 22b*d - 30c*d - 36d , a*b - 41b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 2 │ │ │ - 23c + 19a*d + 38b*d + 20c*d - 19d , b - 30b*c - 22c - 29a*d - 29b*d │ │ │ + 11c - 22a*d - 40b*d + 23c*d + 42d , b - 29b*c + 24c - 10a*d - 36b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - - 36c*d + 50d , a*c + 24b*c + 19c - 29a*d + 11b*d - 10c*d - 34d ) │ │ │ + 2 2 2 │ │ │ + + 19c*d + 23d , a*c - 29b*c - 8c - 29a*d + 33b*d + 19c*d + 16d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = ideal (a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c - │ │ │ + 2 2 2 │ │ │ +o15 = ideal (a - 17b*c + 32c - 17a*d - 28b*d - 7c*d + 24d , a*b - 41b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 17c + 3a*d - 2b*d - 8c*d - 29d , b + 39b*c - 47c + 19b*d - 49c*d - │ │ │ + 2 2 2 2 │ │ │ + 2c - 21a*d - 50b*d + 13c*d - 25d , b + 34b*c - 38c - 24b*d + 30c*d - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ - 47d , a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d ) │ │ │ + 49d , a*c - 16b*c + 21c - 47a*d + 39b*d + 19c*d + 11d ) │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = {ideal (a + 24b + 19c + 36d, b - 30b*c - 22c - 40b*d + 10c*d - 17d ), │ │ │ + 2 2 2 │ │ │ +o16 = {ideal (a - 29b - 8c - 11d, b - 29b*c + 24c - 23b*d + 40c*d + 14d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 29d, b + 30d, a - 38d)} │ │ │ + ideal (c - 29d, b - 35d, a + 43d)} │ │ │ │ │ │ o16 : List │ │ │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ - 2 2 2 │ │ │ -o17 = {ideal (a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d , b + 39b*c - │ │ │ +o17 = {ideal (b - 6c - 42d, a + 26c - 5d), ideal (b + 40c + 18d, a - 46c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 47c + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c + 3a*d - 2b*d - 8c*d - │ │ │ - ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 29d , a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d )} │ │ │ + 19d)} │ │ │ │ │ │ o17 : List │ │ │ │ │ │ i18 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ @@ -103,46 +103,46 @@ │ │ │ │ │ │ i13 : #compsJ │ │ │ │ │ │ o13 = 2 │ │ │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18 │ │ │ ----------------------------------------------------------------------- │ │ │ - 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ + 24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 | │ │ │ │ │ │ 1 36 │ │ │ o14 : Matrix kk <-- kk │ │ │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ ----------------------------------------------------------------------- │ │ │ - 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ + -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ │ │ │ 1 36 │ │ │ o15 : Matrix kk <-- kk │ │ │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ + 2 2 2 │ │ │ +o16 = ideal (a + 31b*c - 17c + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ + 2 2 2 │ │ │ + c + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c + 39a*d - 18b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ + 2 2 2 2 2 │ │ │ + 10c*d - 14d , b - 29b*c + 19c - 29a*d + 24b*d + 22c*d + 35d , b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 │ │ │ - 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ + 2 2 2 2 3 3 2 │ │ │ + 8b*c*d - 30c d - 36a*d + b*d - 14c*d - 39d , c - 38b*c*d + 21c d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 3 │ │ │ - + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ + 2 2 2 3 │ │ │ + 22a*d - 24b*d + 19c*d - 2d ) │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ 0 1 2 3 │ │ │ o17 = total: 1 6 8 3 │ │ │ @@ -150,28 +150,28 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o17 : BettiTally │ │ │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ + 2 2 2 │ │ │ +o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ + 2 2 2 │ │ │ + 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ + 2 2 2 2 2 │ │ │ + - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , b*c - 43b*c*d + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ - - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ + 19c d + 34a*d + 21b*d + 46c*d + 20d , c + 38b*c*d + 22c d - 15a*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 3 │ │ │ - + 38b*d - 39c*d - d ) │ │ │ + 2 2 3 │ │ │ + - 47b*d - 39c*d + 40d ) │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ 0 1 2 3 │ │ │ o19 = total: 1 6 8 3 │ │ │ @@ -179,30 +179,24 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o19 : BettiTally │ │ │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ - +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ -o20 = |ideal (c + 39d, b + 27d, a - 18d) | │ │ │ - +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ - | 2 2 2 3 2 2 2 3 2 2 2 3 | │ │ │ - |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - 24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d )| │ │ │ - +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c + 25b*d + 24c*d - 38d )| │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i21 : netList decompose F2 │ │ │ │ │ │ - +-------------------------------------------------------+ │ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 24d, b - 49d, a) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ - +-------------------------------------------------------+ │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o21 = |ideal (a*c + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - 47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d )| │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ @@ -200,67 +200,67 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24 │ │ │ +o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22 │ │ │ ----------------------------------------------------------------------- │ │ │ - -10 -29 -22 19 -16 | │ │ │ + -24 -29 -38 19 -16 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ 2 2 2 │ │ │ -o14 = ideal (a - 45b*c - 17c + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c + │ │ │ +o14 = ideal (a - 45b*c + 49c + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 11c - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c - 22a*d + 32b*d │ │ │ + 2 2 2 │ │ │ + 32c - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c - 38a*d - 9b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - - 29c*d + 29d , b + 19b*c - 24c - 16a*d - 10b*d - 8c*d - 49d ) │ │ │ + 2 2 2 2 │ │ │ + 10c*d + 38d , b + 19b*c - 22c - 16a*d - 24b*d - 29c*d - 46d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b + 19b*c - 24c + 31b*d + 33c*d - 15d ), │ │ │ + 2 2 2 │ │ │ +o15 = {ideal (a - 29b - 29c - d, b + 19b*c - 22c + 17b*d + 12c*d + 39d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 22d, b + 22d, a + 2d)} │ │ │ + ideal (c - 38d, b - 49d, a + 41d)} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ ----------------------------------------------------------------------- │ │ │ - -47 -39 34 0 | │ │ │ + -13 -18 34 0 | │ │ │ │ │ │ 1 24 │ │ │ o16 : Matrix kk <-- kk │ │ │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ + 2 2 2 │ │ │ +o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ + 2 2 2 │ │ │ + 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 2 │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ + + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 26d)} │ │ │ + 5d)} │ │ │ │ │ │ o18 : List │ │ │ │ │ │ i19 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ @@ -68,54 +68,54 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : netList randomPointsOnRationalVariety(compsJ_0, 10) │ │ │ │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 || │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ - || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 | | │ │ │ - +----------------------------------------------------------------------------------------+ │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ +o13 = || 47 5 40 -9 20 3 -12 -18 -3 -16 -15 48 21 34 -32 19 -38 2 -47 -18 -39 -13 39 -43 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -21 -22 -24 -14 27 -16 12 15 4 -28 20 20 38 2 -21 16 -15 -39 22 -34 45 -48 -47 -47 || │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 28 33 -36 10 19 39 -7 43 7 19 40 37 7 15 28 -23 47 32 39 -17 43 -11 -16 48 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 8 25 35 -38 -10 4 27 1 -34 35 -1 39 -38 33 35 40 36 46 11 -28 46 1 11 -3 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -3 -23 48 -50 -12 35 9 49 23 -47 -2 29 -7 2 -37 29 22 7 -47 -37 15 -13 -23 -10 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -25 4 -28 -16 -27 -32 33 7 42 -18 15 -50 27 -22 -35 32 30 40 -9 -20 -32 24 39 -30 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 41 44 9 24 48 -20 10 33 34 -15 -21 18 0 33 -14 -49 -48 -24 -33 17 -19 -20 39 44 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -44 -29 36 35 32 7 -3 41 12 36 -34 -47 -39 4 -35 13 -39 -40 -26 -49 22 -11 9 -8 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -24 5 -42 -39 -44 13 -15 16 -38 -8 -50 -8 -3 -22 -28 -30 43 5 41 -28 16 -6 36 35 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -13 14 8 19 16 -50 -46 -39 -12 -35 50 32 40 3 3 -31 -9 -3 25 -41 -2 -49 6 -13 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ │ │ │ i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 || │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 || │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ @@ -232,52 +232,52 @@ │ │ │ o13 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18
│ │ │        -----------------------------------------------------------------------
│ │ │ -      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │ +      24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │        -----------------------------------------------------------------------
│ │ │ -      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │ +      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │ +              2              2                              2               
│ │ │ +o16 = ideal (a  + 31b*c - 17c  + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │ +       2                             2                   2                  
│ │ │ +      c  + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c  + 39a*d - 18b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2   2              2                              2     2  
│ │ │ -      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │ +                 2   2              2                              2     2  
│ │ │ +      10c*d - 14d , b  - 29b*c + 19c  - 29a*d + 24b*d + 22c*d + 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │ +                  2         2      2        2      3   3                2   
│ │ │ +      8b*c*d - 30c d - 36a*d  + b*d  - 14c*d  - 39d , c  - 38b*c*d + 21c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │ +           2        2        2     3
│ │ │ +      22a*d  - 24b*d  + 19c*d  - 2d )
│ │ │  
│ │ │  o16 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : betti res F1
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o17 : BettiTally
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │ +              2              2                              2               
│ │ │ +o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                
│ │ │ -      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │ +         2                              2                  2                
│ │ │ +      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                      2     2          
│ │ │ -      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │ +                   2   2              2                   2     2            
│ │ │ +      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2       2      3   3               2         2
│ │ │ -      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │ +         2         2        2        2      3   3                2         2
│ │ │ +      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2    3
│ │ │ -      + 38b*d  - 39c*d  - d )
│ │ │ +             2        2      3
│ │ │ +      - 47b*d  - 39c*d  + 40d )
│ │ │  
│ │ │  o18 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : betti res F2
│ │ │ @@ -331,38 +331,32 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ -      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c + 25b*d + 24c*d - 38d )| │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ -      +-------------------------------------------------------+
    │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o21 = |ideal (a*c + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - 47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d )| │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │
    │ │ │

    We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.

    │ │ │
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -170,71 +170,71 @@ │ │ │ │ 32 13 21 33 19 31 │ │ │ │ i12 : compsJ = decompose J; │ │ │ │ i13 : #compsJ │ │ │ │ │ │ │ │ o13 = 2 │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ │ + 24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o14 : Matrix kk <-- kk │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ + -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o15 : Matrix kk <-- kk │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o16 = ideal (a + 31b*c - 17c + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ │ + 2 2 2 │ │ │ │ + c + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c + 39a*d - 18b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ │ + 2 2 2 2 2 │ │ │ │ + 10c*d - 14d , b - 29b*c + 19c - 29a*d + 24b*d + 22c*d + 35d , b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 8b*c*d - 30c d - 36a*d + b*d - 14c*d - 39d , c - 38b*c*d + 21c d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + 22a*d - 24b*d + 19c*d - 2d ) │ │ │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o17 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o17 : BettiTally │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ │ + 2 2 2 │ │ │ │ + 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ │ + 2 2 2 2 2 │ │ │ │ + - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , b*c - 43b*c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + 19c d + 34a*d + 21b*d + 46c*d + 20d , c + 38b*c*d + 22c d - 15a*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - + 38b*d - 39c*d - d ) │ │ │ │ + 2 2 3 │ │ │ │ + - 47b*d - 39c*d + 40d ) │ │ │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o19 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ @@ -242,43 +242,50 @@ │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o19 : BettiTally │ │ │ │ What are the ideals F1 and F2? │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ ---+ │ │ │ │ -o20 = |ideal (c + 39d, b + 27d, a - 18d) │ │ │ │ +------------------------+ │ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) │ │ │ │ +| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ + | 2 2 2 2 │ │ │ │ +2 | │ │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c │ │ │ │ ++ 25b*d + 24c*d - 38d )| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ +i21 : netList decompose F2 │ │ │ │ + │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ---+ │ │ │ │ - | 2 2 2 3 │ │ │ │ -2 2 2 3 2 2 2 3 | │ │ │ │ - |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - │ │ │ │ -24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +-+ │ │ │ │ + | 2 2 2 │ │ │ │ +2 2 2 2 2 │ │ │ │ +2 2 3 2 2 2 │ │ │ │ +2 3 2 2 2 2 2 3 | │ │ │ │ +o21 = |ideal (a*c + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - │ │ │ │ +28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a │ │ │ │ +- 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - │ │ │ │ +47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d │ │ │ │ )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ---+ │ │ │ │ -i21 : netList decompose F2 │ │ │ │ - │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 24d, b - 49d, a) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +-+ │ │ │ │ We can determine what these represent. One should be a set of 6 points, where 5 │ │ │ │ lie on a plane. The other should be 6 points with 3 points on one line, and the │ │ │ │ other 3 points on a skew line. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can │ │ │ │ be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ @@ -318,90 +318,90 @@ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24
│ │ │ +o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -10 -29 -22 19 -16 |
│ │ │ +      -24 -29 -38 19 -16 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o14 = ideal (a  - 45b*c - 17c  + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c +
│ │ │ +o14 = ideal (a  - 45b*c + 49c  + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      11c  - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c  - 22a*d + 32b*d
│ │ │ +         2                             2                   2                 
│ │ │ +      32c  - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c  - 38a*d - 9b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                             2
│ │ │ -      - 29c*d + 29d , b  + 19b*c - 24c  - 16a*d - 10b*d - 8c*d - 49d )
│ │ │ +                 2   2              2                              2
│ │ │ +      10c*d + 38d , b  + 19b*c - 22c  - 16a*d - 24b*d - 29c*d - 46d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b  + 19b*c - 24c  + 31b*d + 33c*d - 15d ),
│ │ │ +                                  2              2                      2
│ │ │ +o15 = {ideal (a - 29b - 29c - d, b  + 19b*c - 22c  + 17b*d + 12c*d + 39d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 22d, b + 22d, a + 2d)}
│ │ │ +      ideal (c - 38d, b - 49d, a + 41d)}
│ │ │  
│ │ │  o15 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      -13 -18 34 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ +              2             2                             2               
│ │ │ +o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +        2                              2                   2                
│ │ │ +      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │        -----------------------------------------------------------------------
│ │ │                     2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │  
│ │ │  o17 : Ideal of S
    │ │ │
    │ │ │ 
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +      5d)}
│ │ │  
│ │ │  o18 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -212,67 +212,67 @@ │ │ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ There are 2 components. We attempt to find a point on the first component │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24 │ │ │ │ +o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -10 -29 -22 19 -16 | │ │ │ │ + -24 -29 -38 19 -16 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o14 = ideal (a - 45b*c - 17c + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c + │ │ │ │ +o14 = ideal (a - 45b*c + 49c + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 11c - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c - 22a*d + 32b*d │ │ │ │ + 2 2 2 │ │ │ │ + 32c - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c - 38a*d - 9b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - - 29c*d + 29d , b + 19b*c - 24c - 16a*d - 10b*d - 8c*d - 49d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + 10c*d + 38d , b + 19b*c - 22c - 16a*d - 24b*d - 29c*d - 46d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b + 19b*c - 24c + 31b*d + 33c*d - 15d ), │ │ │ │ + 2 2 2 │ │ │ │ +o15 = {ideal (a - 29b - 29c - d, b + 19b*c - 22c + 17b*d + 12c*d + 39d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 22d, b + 22d, a + 2d)} │ │ │ │ + ideal (c - 38d, b - 49d, a + 41d)} │ │ │ │ │ │ │ │ o15 : List │ │ │ │ We attempt to find a point on the second component in parameter space, and its │ │ │ │ corresponding ideal. │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -47 -39 34 0 | │ │ │ │ + -13 -18 34 0 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o16 : Matrix kk <-- kk │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ │ + 2 2 2 │ │ │ │ + 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 │ │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ │ + + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 26d)} │ │ │ │ + 5d)} │ │ │ │ │ │ │ │ o18 : List │ │ │ │ It turns out that this is the ideal of 2 skew lines, just not defined over this │ │ │ │ field. │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ @@ -186,62 +186,62 @@ │ │ │

    There are 2 components. We attempt to find points on each of these two components. We are successful. This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 |  |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 |     |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 |          |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 |       |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 |      |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 ||
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 |   |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 |       |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 |       |
│ │ │ -      +----------------------------------------------------------------------------------------+
│ │ │ -      || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 |      |
│ │ │ -      +----------------------------------------------------------------------------------------+
    │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ +o13 = || 47 5 40 -9 20 3 -12 -18 -3 -16 -15 48 21 34 -32 19 -38 2 -47 -18 -39 -13 39 -43 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -21 -22 -24 -14 27 -16 12 15 4 -28 20 20 38 2 -21 16 -15 -39 22 -34 45 -48 -47 -47 || │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 28 33 -36 10 19 39 -7 43 7 19 40 37 7 15 28 -23 47 32 39 -17 43 -11 -16 48 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 8 25 35 -38 -10 4 27 1 -34 35 -1 39 -38 33 35 40 36 46 11 -28 46 1 11 -3 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -3 -23 48 -50 -12 35 9 49 23 -47 -2 29 -7 2 -37 29 22 7 -47 -37 15 -13 -23 -10 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -25 4 -28 -16 -27 -32 33 7 42 -18 15 -50 27 -22 -35 32 30 40 -9 -20 -32 24 39 -30 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || 41 44 9 24 48 -20 10 33 34 -15 -21 18 0 33 -14 -49 -48 -24 -33 17 -19 -20 39 44 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -44 -29 36 35 32 7 -3 41 12 36 -34 -47 -39 4 -35 13 -39 -40 -26 -49 22 -11 9 -8 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -24 5 -42 -39 -44 13 -15 16 -38 -8 -50 -8 -3 -22 -28 -30 43 5 41 -28 16 -6 36 35 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │ + || -13 14 8 19 16 -50 -46 -39 -12 -35 50 32 40 3 3 -31 -9 -3 25 -41 -2 -49 6 -13 | | │ │ │ + +--------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
    │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 || │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │
    │ │ │ │ │ │
    │ │ │

    Caveat

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -84,99 +84,99 @@ │ │ │ │ There are 2 components. We attempt to find points on each of these two │ │ │ │ components. We are successful. This indicates that the corresponding varieties │ │ │ │ are both rational. Also, if we can find one point, we can find as many as we │ │ │ │ want. │ │ │ │ i13 : netList randomPointsOnRationalVariety(compsJ_0, 10) │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 - │ │ │ │ -39 -47 39 -43 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 │ │ │ │ -22 -47 -47 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 - │ │ │ │ -16 48 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 │ │ │ │ -11 11 -3 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 │ │ │ │ --23 -10 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 │ │ │ │ --32 -9 39 -30 || │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 - │ │ │ │ -19 -33 39 44 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 - │ │ │ │ -26 9 -8 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 │ │ │ │ -41 36 35 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 - │ │ │ │ -2 25 6 -13 | | │ │ │ │ +--------------+ │ │ │ │ +o13 = || 47 5 40 -9 20 3 -12 -18 -3 -16 -15 48 21 34 -32 19 -38 2 -47 -18 -39 - │ │ │ │ +13 39 -43 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -21 -22 -24 -14 27 -16 12 15 4 -28 20 20 38 2 -21 16 -15 -39 22 -34 45 │ │ │ │ +-48 -47 -47 || │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || 28 33 -36 10 19 39 -7 43 7 19 40 37 7 15 28 -23 47 32 39 -17 43 -11 - │ │ │ │ +16 48 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || 8 25 35 -38 -10 4 27 1 -34 35 -1 39 -38 33 35 40 36 46 11 -28 46 1 11 │ │ │ │ +-3 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -3 -23 48 -50 -12 35 9 49 23 -47 -2 29 -7 2 -37 29 22 7 -47 -37 15 -13 │ │ │ │ +-23 -10 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -25 4 -28 -16 -27 -32 33 7 42 -18 15 -50 27 -22 -35 32 30 40 -9 -20 - │ │ │ │ +32 24 39 -30 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || 41 44 9 24 48 -20 10 33 34 -15 -21 18 0 33 -14 -49 -48 -24 -33 17 -19 │ │ │ │ +-20 39 44 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -44 -29 36 35 32 7 -3 41 12 36 -34 -47 -39 4 -35 13 -39 -40 -26 -49 22 │ │ │ │ +-11 9 -8 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -24 5 -42 -39 -44 13 -15 16 -38 -8 -50 -8 -3 -22 -28 -30 43 5 41 -28 │ │ │ │ +16 -6 36 35 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +--------------+ │ │ │ │ + || -13 14 8 19 16 -50 -46 -39 -12 -35 50 32 40 3 3 -31 -9 -3 25 -41 -2 - │ │ │ │ +49 6 -13 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ +--------------+ │ │ │ │ i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 │ │ │ │ -37 -47 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 - │ │ │ │ -48 30 -48 0 || │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 │ │ │ │ -40 -18 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 │ │ │ │ -3 13 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 - │ │ │ │ -18 30 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 │ │ │ │ -12 18 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 │ │ │ │ --37 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 │ │ │ │ -6 -28 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 - │ │ │ │ -33 26 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 │ │ │ │ --20 4 0 | | │ │ │ │ +---------------+ │ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 - │ │ │ │ +35 -47 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 - │ │ │ │ +48 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 - │ │ │ │ +22 -18 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 │ │ │ │ +0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 │ │ │ │ +30 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 │ │ │ │ +-18 18 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 - │ │ │ │ +39 19 20 -37 0 || │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 │ │ │ │ +-9 -28 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 │ │ │ │ +-28 26 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 - │ │ │ │ +13 4 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ +---------------+ │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_) -- find a random point on a variety │ │ │ │ that can be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_s_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_,_Z_Z_) -- find random points on a │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ @@ -350,90 +350,85 @@ │ │ │

    We can find random points on each component, since these components are rational.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11
│ │ │ +o12 = | -36 -30 42 21 -22 23 23 16 -7 -30 -11 -40 19 19 -41 -22 24 -36 -8 33
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 -29 24 -29 |
│ │ │ +      -29 -10 -29 -29 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39
│ │ │ +o13 = | 24 -7 -25 32 -28 13 -49 11 -17 -17 -2 -50 30 19 -41 -21 -38 -24 21 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0 -24 34 |
│ │ │ +      34 0 -16 -47 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2             2                            2               
│ │ │ -o14 = ideal (a  - 12b*c - 4c  - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c +
│ │ │ +              2             2                              2               
│ │ │ +o14 = ideal (a  - 7b*c + 21c  - 30a*d - 22b*d - 30c*d - 36d , a*b - 41b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      23c  + 19a*d + 38b*d + 20c*d - 19d , b  - 30b*c - 22c  - 29a*d - 29b*d
│ │ │ +      11c  - 22a*d - 40b*d + 23c*d + 42d , b  - 29b*c + 24c  - 10a*d - 36b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2                   2                              2
│ │ │ -      - 36c*d + 50d , a*c + 24b*c + 19c  - 29a*d + 11b*d - 10c*d - 34d )
│ │ │ +                   2                  2                              2
│ │ │ +      + 19c*d + 23d , a*c - 29b*c - 8c  - 29a*d + 33b*d + 19c*d + 16d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                           2               
│ │ │ -o15 = ideal (a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c -
│ │ │ +              2              2                             2               
│ │ │ +o15 = ideal (a  - 17b*c + 32c  - 17a*d - 28b*d - 7c*d + 24d , a*b - 41b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                           2   2              2                  
│ │ │ -      17c  + 3a*d - 2b*d - 8c*d - 29d , b  + 39b*c - 47c  + 19b*d - 49c*d -
│ │ │ +        2                              2   2              2                  
│ │ │ +      2c  - 21a*d - 50b*d + 13c*d - 25d , b  + 34b*c - 38c  - 24b*d + 30c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                   2                              2
│ │ │ -      47d , a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d )
│ │ │ +      49d , a*c - 16b*c + 21c  - 47a*d + 39b*d + 19c*d + 11d )
│ │ │  
│ │ │  o15 : Ideal of S
    │ │ │
    │ │ │ 
    i16 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o16 = {ideal (a + 24b + 19c + 36d, b  - 30b*c - 22c  - 40b*d + 10c*d - 17d ),
│ │ │ +                                   2              2                      2
│ │ │ +o16 = {ideal (a - 29b - 8c - 11d, b  - 29b*c + 24c  - 23b*d + 40c*d + 14d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 29d, b + 30d, a - 38d)}
│ │ │ +      ideal (c - 29d, b - 35d, a + 43d)}
│ │ │  
│ │ │  o16 : List
    │ │ │
    │ │ │ 
    i17 : decompose F2
│ │ │  
│ │ │ -                               2                              2   2          
│ │ │ -o17 = {ideal (a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d , b  + 39b*c -
│ │ │ +o17 = {ideal (b - 6c - 42d, a + 26c - 5d), ideal (b + 40c + 18d, a - 46c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                      2                   2                       
│ │ │ -      47c  + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c  + 3a*d - 2b*d - 8c*d -
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         2   2              2                           2
│ │ │ -      29d , a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d )}
│ │ │ +      19d)}
│ │ │  
│ │ │  o17 : List
    │ │ │
    │ │ │
    │ │ │

    Note, the general element of one component is a plane conic union a point. (The dimension of the locus of all such is: (choice of plane) + (choice of degree 2 in plane) + choice of point. This is 3 + 5 + 3 = 11.

    │ │ │ ├── html2text {} │ │ │ │ @@ -424,70 +424,65 @@ │ │ │ │ -----------------------------------------------------------+ │ │ │ │ This tells us that there are 2 components (at least over the given field). │ │ │ │ Their dimensions are 11, 8. │ │ │ │ We can find random points on each component, since these components are │ │ │ │ rational. │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11 │ │ │ │ +o12 = | -36 -30 42 21 -22 23 23 16 -7 -30 -11 -40 19 19 -41 -22 24 -36 -8 33 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -30 -29 24 -29 | │ │ │ │ + -29 -10 -29 -29 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o12 : Matrix kk <-- kk │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39 │ │ │ │ +o13 = | 24 -7 -25 32 -28 13 -49 11 -17 -17 -2 -50 30 19 -41 -21 -38 -24 21 39 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0 -24 34 | │ │ │ │ + 34 0 -16 -47 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o14 = ideal (a - 12b*c - 4c - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o14 = ideal (a - 7b*c + 21c - 30a*d - 22b*d - 30c*d - 36d , a*b - 41b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 │ │ │ │ - 23c + 19a*d + 38b*d + 20c*d - 19d , b - 30b*c - 22c - 29a*d - 29b*d │ │ │ │ + 11c - 22a*d - 40b*d + 23c*d + 42d , b - 29b*c + 24c - 10a*d - 36b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - - 36c*d + 50d , a*c + 24b*c + 19c - 29a*d + 11b*d - 10c*d - 34d ) │ │ │ │ + 2 2 2 │ │ │ │ + + 19c*d + 23d , a*c - 29b*c - 8c - 29a*d + 33b*d + 19c*d + 16d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = ideal (a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o15 = ideal (a - 17b*c + 32c - 17a*d - 28b*d - 7c*d + 24d , a*b - 41b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 17c + 3a*d - 2b*d - 8c*d - 29d , b + 39b*c - 47c + 19b*d - 49c*d - │ │ │ │ + 2 2 2 2 │ │ │ │ + 2c - 21a*d - 50b*d + 13c*d - 25d , b + 34b*c - 38c - 24b*d + 30c*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 │ │ │ │ - 47d , a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d ) │ │ │ │ + 49d , a*c - 16b*c + 21c - 47a*d + 39b*d + 19c*d + 11d ) │ │ │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = {ideal (a + 24b + 19c + 36d, b - 30b*c - 22c - 40b*d + 10c*d - 17d ), │ │ │ │ + 2 2 2 │ │ │ │ +o16 = {ideal (a - 29b - 8c - 11d, b - 29b*c + 24c - 23b*d + 40c*d + 14d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 29d, b + 30d, a - 38d)} │ │ │ │ + ideal (c - 29d, b - 35d, a + 43d)} │ │ │ │ │ │ │ │ o16 : List │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o17 = {ideal (a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d , b + 39b*c - │ │ │ │ +o17 = {ideal (b - 6c - 42d, a + 26c - 5d), ideal (b + 40c + 18d, a - 46c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 47c + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c + 3a*d - 2b*d - 8c*d - │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 29d , a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d )} │ │ │ │ + 19d)} │ │ │ │ │ │ │ │ o17 : List │ │ │ │ Note, the general element of one component is a plane conic union a point. (The │ │ │ │ dimension of the locus of all such is: (choice of plane) + (choice of degree 2 │ │ │ │ in plane) + choice of point. This is 3 + 5 + 3 = 11. │ │ │ │ The other component consists of two skew lines. This has dimension (choice of │ │ │ │ line) + (choice of line). This is 4 + 4 = 8. Also notice that the 2 skew lines │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ i5 : elapsedTime gb I2 │ │ │ - -- 3.27651s elapsed │ │ │ + -- 1.99657s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 2.21061s elapsed │ │ │ + -- 1.64224s 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
│ │ │ - -- 3.27651s elapsed
│ │ │ + -- 1.99657s 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)
│ │ │ - -- 2.21061s elapsed
│ │ │ + -- 1.64224s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,23 +38,23 @@ │ │ │ │ using a different weight vector and then graded reverse lexicographic we could │ │ │ │ substitute and compute directly, │ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ i5 : elapsedTime gb I2 │ │ │ │ - -- 3.27651s elapsed │ │ │ │ + -- 1.99657s 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) │ │ │ │ - -- 2.21061s elapsed │ │ │ │ + -- 1.64224s 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 +6,20 @@ │ │ │ o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}} │ │ │ 2 │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : hadamardPower(L,3) │ │ │ │ │ │ - 1 1 │ │ │ -o2 = {Point{1, 1, -}, Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, │ │ │ - 8 2 │ │ │ + 1 1 │ │ │ +o2 = {Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, -}, │ │ │ + 8 2 │ │ │ ------------------------------------------------------------------------ │ │ │ - 1 │ │ │ - Point{1, 0, 4}, Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, │ │ │ - 4 │ │ │ + 1 │ │ │ + Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1, 0, -}, Point{1, │ │ │ + 4 │ │ │ ------------------------------------------------------------------------ │ │ │ - Point{1, 0, 16}, Point{1, 0, 1}} │ │ │ + 8, 64}, Point{1, 4, 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{0, 2}, Point{2, 4}, Point{1, 0}} │ │ │ +o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out │ │ │ @@ -30,17 +30,17 @@ │ │ │ 2 2 │ │ │ + x*y - 6x*z ) │ │ │ │ │ │ o4 : Ideal of S │ │ │ │ │ │ i5 : X2 = hadamardPower(X,2) │ │ │ │ │ │ -o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, │ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0}, │ │ │ ------------------------------------------------------------------------ │ │ │ - Point{0, 1, 1}, Point{1, 1, 0}} │ │ │ + Point{1, 2, 0}, Point{1, 4, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : I2 == idealOfProjectivePoints(X2,S) │ │ │ │ │ │ o6 = true │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ @@ -84,23 +84,23 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1               1                                  
│ │ │ -o2 = {Point{1, 1, -}, Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1},
│ │ │ -                  8               2                                  
│ │ │ +                                                   1               1  
│ │ │ +o2 = {Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, -},
│ │ │ +                                                   8               2  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1
│ │ │ -     Point{1, 0, 4}, Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8},
│ │ │ -                                 4
│ │ │ +                                                                 1
│ │ │ +     Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1, 0, -}, Point{1,
│ │ │ +                                                                 4
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{1, 0, 16}, Point{1, 0, 1}}
│ │ │ +     8, 64}, Point{1, 4, 8}}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -21,23 +21,23 @@ │ │ │ │ 1 │ │ │ │ o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}} │ │ │ │ 2 │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : hadamardPower(L,3) │ │ │ │ │ │ │ │ - 1 1 │ │ │ │ -o2 = {Point{1, 1, -}, Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, │ │ │ │ - 8 2 │ │ │ │ + 1 1 │ │ │ │ +o2 = {Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, -}, │ │ │ │ + 8 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 1 │ │ │ │ - Point{1, 0, 4}, Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, │ │ │ │ - 4 │ │ │ │ + 1 │ │ │ │ + Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1, 0, -}, Point{1, │ │ │ │ + 4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - Point{1, 0, 16}, Point{1, 0, 1}} │ │ │ │ + 8, 64}, Point{1, 4, 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{0, 2}, Point{2, 4}, Point{1, 0}}
│ │ │ +o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │  
│ │ │  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{0, 2}, Point{2, 4}, Point{1, 0}} │ │ │ │ +o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}} │ │ │ │ │ │ │ │ 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- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/Hadamard.m2:345:0. │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_ideal__Of__Projective__Points.html │ │ │ @@ -116,17 +116,17 @@ │ │ │ o4 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : X2 = hadamardPower(X,2)
│ │ │  
│ │ │ -o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │ +     Point{1, 2, 0}, Point{1, 4, 1}}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,17 +39,17 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2       2
│ │ │ │       + x*y  - 6x*z )
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : X2 = hadamardPower(X,2)
│ │ │ │  
│ │ │ │ -o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │ │ +     Point{1, 2, 0}, Point{1, 4, 1}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** WWaayyss ttoo uussee iiddeeaallOOffPPrroojjeeccttiivveePPooiinnttss:: **********
│ │ │ │      * idealOfProjectivePoints(List,Ring)
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -42,19 +42,19 @@
│ │ │  i5 : w = {9,1,99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ - -- .000003647s elapsed
│ │ │ - -- .000002405s elapsed
│ │ │ - -- .000002054s elapsed
│ │ │ - -- .000003707s elapsed
│ │ │ - -- .000003126s elapsed
│ │ │ + -- .000002575s elapsed
│ │ │ + -- .000002829s elapsed
│ │ │ + -- .000002628s elapsed
│ │ │ + -- .000002546s elapsed
│ │ │ + -- .000002815s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[t_1..t_5];
│ │ │  
│ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004368s elapsed
│ │ │ + -- .000004446s 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)
│ │ │ - -- .000003907s elapsed
│ │ │ + -- .00000388s 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
│ │ │ @@ -132,19 +132,19 @@
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ - -- .000003647s elapsed
│ │ │ - -- .000002405s elapsed
│ │ │ - -- .000002054s elapsed
│ │ │ - -- .000003707s elapsed
│ │ │ - -- .000003126s elapsed
│ │ │ + -- .000002575s elapsed
│ │ │ + -- .000002829s elapsed
│ │ │ + -- .000002628s elapsed
│ │ │ + -- .000002546s elapsed
│ │ │ + -- .000002815s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,19 +54,19 @@
│ │ │ │                    1   6   1   6
│ │ │ │  i5 : w = {9,1,99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │ - -- .000003647s elapsed
│ │ │ │ - -- .000002405s elapsed
│ │ │ │ - -- .000002054s elapsed
│ │ │ │ - -- .000003707s elapsed
│ │ │ │ - -- .000003126s elapsed
│ │ │ │ + -- .000002575s elapsed
│ │ │ │ + -- .000002829s elapsed
│ │ │ │ + -- .000002628s elapsed
│ │ │ │ + -- .000002546s elapsed
│ │ │ │ + -- .000002815s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004368s elapsed
│ │ │ + -- .000004446s 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)
│ │ │ - -- .000003907s elapsed
│ │ │ + -- .00000388s 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
│ │ │ │ - -- .000004368s elapsed
│ │ │ │ + -- .000004446s 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)
│ │ │ │ - -- .000003907s elapsed
│ │ │ │ + -- .00000388s 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
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({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 
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  -
│ │ │ +          5   6      5 6    1 9      14      17      5   8    5 8    3 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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  
│ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T },
│ │ │ +         15      16      4   8      2 7    4 8      12      14      3   9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ -         5   8    5 8    3 9      15      16      4   8      2 7    4 8  
│ │ │ +      T T  - z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   +
│ │ │ +       3 9      15      17      3   6      3 6    5 7    1 8      12  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ), ({T , T }, - T T  -
│ │ │ -         12      14      3   9    3 9      15      17      3   6      3 6  
│ │ │ +      x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ +         14      1   7      1 7      13      5   9      2 8    5 9      15  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, -
│ │ │ -       5 7    1 8      12      14      1   7      1 7      13      5   9    
│ │ │ +      ({T , T  }, - T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ +         3   10      4 8    3 10      18      19      2   7      2 7    4 8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   - z*T   + x*T  ),
│ │ │ -       2 8    5 9      15      3   10      4 8    3 10      18      19  
│ │ │ +      y*T   + z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  },
│ │ │ +         12      14      4   8      4 8    3 10      18      19      2   10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, - T T  + T T   -
│ │ │ -         2   7      2 7    4 8      12      14      4   8      4 8    3 10  
│ │ │ +      - T T  - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, -
│ │ │ +         5 8    2 10      19      4   10      4 10      18      5   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ), ({T , T  }, -
│ │ │ -         18      19      2   10      5 8    2 10      19      4   10    
│ │ │ +      T T  - T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, -
│ │ │ +       5 8    2 10      19      3   8    3 8      15      17      1   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T , T }, T T  +
│ │ │ -       4 10      18      5   8      5 8    2 10      19      3   8    3 8  
│ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   +
│ │ │ +       3 6    5 7    1 8      12      14      4   9    4 9    5 10      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   - z*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ -         15      17      1   8      3 6    5 7    1 8      12      14      4 
│ │ │ +      z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  -
│ │ │ +         19      2   8      2 8    5 9      15      5   7      3 6    5 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, T T  - T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ -       9    4 9    5 10      17      19      2   8      2 8    5 9      15  
│ │ │ +      T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ),
│ │ │ +       1 8      12      14      4   9    2 6    3 8    4 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  +
│ │ │ -         5   7      3 6    5 7    1 8      12      14      4   9    2 6  
│ │ │ +      ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   -
│ │ │ +         3   10    5 6    3 10      18      20      5   6    5 6    3 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   +
│ │ │ -       3 8    4 9      14      17      3   10    5 6    3 10      18  
│ │ │ +      z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T },
│ │ │ +         18      20      4   10      5 7    4 10      12      20      1   6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  -
│ │ │ -         20      5   6    5 6    3 10      18      20      4   10      5 7  
│ │ │ +      - T T  - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ +         1 6    4 7      11      3   9    5 8    3 9      15      16      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + z*T   + z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T },
│ │ │ -       4 10      12      20      1   6      1 6    4 7      11      3   9  
│ │ │ +      T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, - T T   + y*T  ), ({T ,
│ │ │ +       6    4 6    3 7      11      13      5   10      5 10      18      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ),
│ │ │ -       5 8    3 9      15      16      4   6    4 6    3 7      11      13  
│ │ │ +      T }, T T  - T T   + x*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T ,
│ │ │ +       6    4 6    1 10      20      4   7      1 6    4 7      11      2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ -         5   10      5 10      18      4   6    4 6    1 10      20      4 
│ │ │ +      T }, T T  + T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  - T T   -
│ │ │ +       6    2 6    3 8    4 9      14      17      5   10    4 9    5 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14  
│ │ │ +      z*T   + z*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ +         17      19      3   8    2 6    3 8    4 9      14      17      3 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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  + y*T   - z*T  ), ({T , T }, - T T  - T T   + z*T   + z*T  ),
│ │ │ +       6    3 6      11      12      5   7      5 7    4 10      12      20  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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 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   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      ({T , T }, - T T  - 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  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14  
│ │ │ +      ({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 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  + z*T  )}
│ │ │ -         17      4   7      4 7      13
│ │ │ +      T }, T T  - z*T   + x*T  )}
│ │ │ +       7    3 7      11      12
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │  o16 = -1
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -215,82 +215,82 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({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 
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  -
│ │ │ +          5   6      5 6    1 9      14      17      5   8    5 8    3 9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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  
│ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T },
│ │ │ +         15      16      4   8      2 7    4 8      12      14      3   9  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ -         5   8    5 8    3 9      15      16      4   8      2 7    4 8  
│ │ │ +      T T  - z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   +
│ │ │ +       3 9      15      17      3   6      3 6    5 7    1 8      12  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ), ({T , T }, - T T  -
│ │ │ -         12      14      3   9    3 9      15      17      3   6      3 6  
│ │ │ +      x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ +         14      1   7      1 7      13      5   9      2 8    5 9      15  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, -
│ │ │ -       5 7    1 8      12      14      1   7      1 7      13      5   9    
│ │ │ +      ({T , T  }, - T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ +         3   10      4 8    3 10      18      19      2   7      2 7    4 8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   - z*T   + x*T  ),
│ │ │ -       2 8    5 9      15      3   10      4 8    3 10      18      19  
│ │ │ +      y*T   + z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  },
│ │ │ +         12      14      4   8      4 8    3 10      18      19      2   10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, - T T  + T T   -
│ │ │ -         2   7      2 7    4 8      12      14      4   8      4 8    3 10  
│ │ │ +      - T T  - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, -
│ │ │ +         5 8    2 10      19      4   10      4 10      18      5   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ), ({T , T  }, -
│ │ │ -         18      19      2   10      5 8    2 10      19      4   10    
│ │ │ +      T T  - T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, -
│ │ │ +       5 8    2 10      19      3   8    3 8      15      17      1   8    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T , T }, T T  +
│ │ │ -       4 10      18      5   8      5 8    2 10      19      3   8    3 8  
│ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   +
│ │ │ +       3 6    5 7    1 8      12      14      4   9    4 9    5 10      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   - z*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ -         15      17      1   8      3 6    5 7    1 8      12      14      4 
│ │ │ +      z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  -
│ │ │ +         19      2   8      2 8    5 9      15      5   7      3 6    5 7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, T T  - T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ -       9    4 9    5 10      17      19      2   8      2 8    5 9      15  
│ │ │ +      T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ),
│ │ │ +       1 8      12      14      4   9    2 6    3 8    4 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  +
│ │ │ -         5   7      3 6    5 7    1 8      12      14      4   9    2 6  
│ │ │ +      ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   -
│ │ │ +         3   10    5 6    3 10      18      20      5   6    5 6    3 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   +
│ │ │ -       3 8    4 9      14      17      3   10    5 6    3 10      18  
│ │ │ +      z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T },
│ │ │ +         18      20      4   10      5 7    4 10      12      20      1   6  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  -
│ │ │ -         20      5   6    5 6    3 10      18      20      4   10      5 7  
│ │ │ +      - T T  - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ +         1 6    4 7      11      3   9    5 8    3 9      15      16      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T   + z*T   + z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T },
│ │ │ -       4 10      12      20      1   6      1 6    4 7      11      3   9  
│ │ │ +      T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, - T T   + y*T  ), ({T ,
│ │ │ +       6    4 6    3 7      11      13      5   10      5 10      18      4 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T T  + T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ),
│ │ │ -       5 8    3 9      15      16      4   6    4 6    3 7      11      13  
│ │ │ +      T }, T T  - T T   + x*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T ,
│ │ │ +       6    4 6    1 10      20      4   7      1 6    4 7      11      2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ -         5   10      5 10      18      4   6    4 6    1 10      20      4 
│ │ │ +      T }, T T  + T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  - T T   -
│ │ │ +       6    2 6    3 8    4 9      14      17      5   10    4 9    5 10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14  
│ │ │ +      z*T   + z*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ +         17      19      3   8    2 6    3 8    4 9      14      17      3 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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  + y*T   - z*T  ), ({T , T }, - T T  - T T   + z*T   + z*T  ),
│ │ │ +       6    3 6      11      12      5   7      5 7    4 10      12      20  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      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 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   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      ({T , T }, - T T  - 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  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14  
│ │ │ +      ({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 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T  ), ({T , T }, - T T  + z*T  )}
│ │ │ -         17      4   7      4 7      13
│ │ │ +      T }, T T  - z*T   + x*T  )}
│ │ │ +       7    3 7      11      12
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -118,82 +118,82 @@
│ │ │ │  37   38   39   40   41   42   43   44
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │ │ -          1   10    4 6    1 10      20      5   9      5 9      16      3
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, T T  + T T  -
│ │ │ │ +          5   6      5 6    1 9      14      17      5   8    5 8    3 9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      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
│ │ │ │ +      z*T   + x*T  ), ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T },
│ │ │ │ +         15      16      4   8      2 7    4 8      12      14      3   9
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, T T  + T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ │ -         5   8    5 8    3 9      15      16      4   8      2 7    4 8
│ │ │ │ +      T T  - z*T   + y*T  ), ({T , T }, - T T  - T T  - T T  + z*T   +
│ │ │ │ +       3 9      15      17      3   6      3 6    5 7    1 8      12
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T   + z*T  ), ({T , T }, T T  - z*T   + y*T  ), ({T , T }, - T T  -
│ │ │ │ -         12      14      3   9    3 9      15      17      3   6      3 6
│ │ │ │ +      x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ │ +         14      1   7      1 7      13      5   9      2 8    5 9      15
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  - T T  + z*T   + x*T  ), ({T , T }, - T T  + x*T  ), ({T , T }, -
│ │ │ │ -       5 7    1 8      12      14      1   7      1 7      13      5   9
│ │ │ │ +      ({T , T  }, - T T  + T T   - z*T   + x*T  ), ({T , T }, - T T  - T T  +
│ │ │ │ +         3   10      4 8    3 10      18      19      2   7      2 7    4 8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  - T T  + y*T  ), ({T , T  }, - T T  + T T   - z*T   + x*T  ),
│ │ │ │ -       2 8    5 9      15      3   10      4 8    3 10      18      19
│ │ │ │ +      y*T   + z*T  ), ({T , T }, - T T  + T T   - z*T   + x*T  ), ({T , T  },
│ │ │ │ +         12      14      4   8      4 8    3 10      18      19      2   10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  + y*T   + z*T  ), ({T , T }, - T T  + T T   -
│ │ │ │ -         2   7      2 7    4 8      12      14      4   8      4 8    3 10
│ │ │ │ +      - T T  - T T   + y*T  ), ({T , T  }, - T T   + x*T  ), ({T , T }, -
│ │ │ │ +         5 8    2 10      19      4   10      4 10      18      5   8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   + x*T  ), ({T , T  }, - T T  - T T   + y*T  ), ({T , T  }, -
│ │ │ │ -         18      19      2   10      5 8    2 10      19      4   10
│ │ │ │ +      T T  - T T   + y*T  ), ({T , T }, T T  + x*T   - z*T  ), ({T , T }, -
│ │ │ │ +       5 8    2 10      19      3   8    3 8      15      17      1   8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T   + x*T  ), ({T , T }, - T T  - T T   + y*T  ), ({T , T }, T T  +
│ │ │ │ -       4 10      18      5   8      5 8    2 10      19      3   8    3 8
│ │ │ │ +      T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  - T T   - z*T   +
│ │ │ │ +       3 6    5 7    1 8      12      14      4   9    4 9    5 10      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T   - z*T  ), ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T ,
│ │ │ │ -         15      17      1   8      3 6    5 7    1 8      12      14      4
│ │ │ │ +      z*T  ), ({T , T }, - T T  - T T  + y*T  ), ({T , T }, - T T  - T T  -
│ │ │ │ +         19      2   8      2 8    5 9      15      5   7      3 6    5 7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, T T  - T T   - z*T   + z*T  ), ({T , T }, - T T  - T T  + y*T  ),
│ │ │ │ -       9    4 9    5 10      17      19      2   8      2 8    5 9      15
│ │ │ │ +      T T  + z*T   + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ),
│ │ │ │ +       1 8      12      14      4   9    2 6    3 8    4 9      14      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  - T T  + z*T   + x*T  ), ({T , T }, T T  +
│ │ │ │ -         5   7      3 6    5 7    1 8      12      14      4   9    2 6
│ │ │ │ +      ({T , T  }, T T  + T T   - z*T   + y*T  ), ({T , T }, T T  + T T   -
│ │ │ │ +         3   10    5 6    3 10      18      20      5   6    5 6    3 10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  + T T   - z*T   +
│ │ │ │ -       3 8    4 9      14      17      3   10    5 6    3 10      18
│ │ │ │ +      z*T   + y*T  ), ({T , T  }, - T T  - T T   + z*T   + z*T  ), ({T , T },
│ │ │ │ +         18      20      4   10      5 7    4 10      12      20      1   6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T  ), ({T , T }, T T  + T T   - z*T   + y*T  ), ({T , T  }, - T T  -
│ │ │ │ -         20      5   6    5 6    3 10      18      20      4   10      5 7
│ │ │ │ +      - T T  - T T  + x*T  ), ({T , T }, T T  + T T  - z*T   + x*T  ), ({T ,
│ │ │ │ +         1 6    4 7      11      3   9    5 8    3 9      15      16      4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T   + z*T   + z*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T , T },
│ │ │ │ -       4 10      12      20      1   6      1 6    4 7      11      3   9
│ │ │ │ +      T }, T T  + T T  - z*T   + y*T  ), ({T , T  }, - T T   + y*T  ), ({T ,
│ │ │ │ +       6    4 6    3 7      11      13      5   10      5 10      18      4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T T  + T T  - z*T   + x*T  ), ({T , T }, T T  + T T  - z*T   + y*T  ),
│ │ │ │ -       5 8    3 9      15      16      4   6    4 6    3 7      11      13
│ │ │ │ +      T }, T T  - T T   + x*T  ), ({T , T }, - T T  - T T  + x*T  ), ({T ,
│ │ │ │ +       6    4 6    1 10      20      4   7      1 6    4 7      11      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T , T  }, - T T   + y*T  ), ({T , T }, T T  - T T   + x*T  ), ({T ,
│ │ │ │ -         5   10      5 10      18      4   6    4 6    1 10      20      4
│ │ │ │ +      T }, T T  + T T  + T T  + y*T   - z*T  ), ({T , T  }, T T  - T T   -
│ │ │ │ +       6    2 6    3 8    4 9      14      17      5   10    4 9    5 10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T  + x*T  ), ({T , T }, T T  + T T  + T T  + y*T   -
│ │ │ │ -       7      1 6    4 7      11      2   6    2 6    3 8    4 9      14
│ │ │ │ +      z*T   + z*T  ), ({T , T }, T T  + T T  + T T  + y*T   - z*T  ), ({T ,
│ │ │ │ +         17      19      3   8    2 6    3 8    4 9      14      17      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      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  + y*T   - z*T  ), ({T , T }, - T T  - T T   + z*T   + z*T  ),
│ │ │ │ +       6    3 6      11      12      5   7      5 7    4 10      12      20
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      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 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   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ -       7      5 7    4 10      12      20      3   7    4 6    3 7      11
│ │ │ │ +      ({T , T }, - T T  - 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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T  ), ({T , T }, - T T  + y*T  ), ({T , T }, - T T  - T T  - z*T   +
│ │ │ │ -         13      2   9      2 9      16      1   9      5 6    1 9      14
│ │ │ │ +      ({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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T  ), ({T , T }, - T T  + z*T  )}
│ │ │ │ -         17      4   7      4 7      13
│ │ │ │ +      T }, T T  - z*T   + x*T  )}
│ │ │ │ +       7    3 7      11      12
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │  o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.out
│ │ │ @@ -10,35 +10,35 @@
│ │ │  
│ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : restriction(A,1)
│ │ │  
│ │ │  o4 = {x, z, x, x - z}
│ │ │  
│ │ │  o4 : Hyperplane Arrangement 
│ │ │  
│ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │  
│ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : prune image der(A,m)
│ │ │  
│ │ │        3
│ │ │  o9 = R
│ │ │ @@ -59,16 +59,16 @@
│ │ │  
│ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │  
│ │ │  i12 : A = arrangement "bracelet";
│ │ │  
│ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x , x  + x  + x , x  + x , x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         3   4   2    3    4   2    4   2   3    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
│ │ │  
│ │ │  i14 : C = restriction(A,0)
│ │ │  
│ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : restriction(A,1)
│ │ │ @@ -113,15 +113,15 @@
│ │ │  o4 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence.  See the paper of Abe, Terao and Wakefield for details.
    
│ │ │ @@ -132,15 +132,15 @@
│ │ │                
    i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : prune image der(A,m)
│ │ │ @@ -181,16 +181,16 @@
│ │ │                
    i12 : A = arrangement "bracelet";
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x , x  + x  + x , x  + x , x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         3   4   2    3    4   2    4   2   3    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : C = restriction(A,0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,36 +33,36 @@
│ │ │ │  i2 : A = arrangement {x,y,z,x-y,x-z}
│ │ │ │  
│ │ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │ │  
│ │ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : restriction(A,1)
│ │ │ │  
│ │ │ │  o4 = {x, z, x, x - z}
│ │ │ │  
│ │ │ │  o4 : Hyperplane Arrangement
│ │ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │ │  
│ │ │ │ -o5 = {x - z, z, x}
│ │ │ │ +o5 = {z, x, x - z}
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$,
│ │ │ │  where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the
│ │ │ │  Euler restriction is free as well, and the modules of _l_o_g_a_r_i_t_h_m_i_c_ _d_e_r_i_v_a_t_i_o_n_s
│ │ │ │  form a short exact sequence. See the paper of Abe, Terao and Wakefield for
│ │ │ │  details.
│ │ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │ │  
│ │ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : prune image der(A,m)
│ │ │ │  
│ │ │ │        3
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │ @@ -80,16 +80,16 @@
│ │ │ │  
│ │ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │ │  It may be the case that the Euler restriction is free, while the naive
│ │ │ │  restriction is not:
│ │ │ │  i12 : A = arrangement "bracelet";
│ │ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │ │  
│ │ │ │ -o13 = ({x , x , x  + x  + x , x  + x , x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ -         3   4   2    3    4   2    4   2   3    4
│ │ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ │ │  i14 : C = restriction(A,0)
│ │ │ │  
│ │ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./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 0.803476s (cpu); 0.400866s (thread); 0s (gc)
│ │ │ + -- used 0.705968s (cpu); 0.388203s (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 0.669248s (cpu); 0.369386s (thread); 0s (gc)
│ │ │ + -- used 0.733877s (cpu); 0.407391s (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 1.02394s (cpu); 0.472475s (thread); 0s (gc)
│ │ │ + -- used 0.842625s (cpu); 0.408987s (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 1.06285s (cpu); 0.537877s (thread); 0s (gc)
│ │ │ + -- used 0.879127s (cpu); 0.43333s (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 1.55504s (cpu); 0.731058s (thread); 0s (gc)
│ │ │ + -- used 1.85559s (cpu); 0.830133s (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 0.981816s (cpu); 0.466986s (thread); 0s (gc)
│ │ │ + -- used 1.12016s (cpu); 0.543152s (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 0.231485s (cpu); 0.0700508s (thread); 0s (gc)
│ │ │ + -- used 0.22591s (cpu); 0.0884232s (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 0.137094s (cpu); 0.0666161s (thread); 0s (gc)
│ │ │ + -- used 0.146667s (cpu); 0.072453s (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 0.0679401s (cpu); 0.067694s (thread); 0s (gc)
│ │ │ + -- used 0.0800267s (cpu); 0.0800652s (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 0.0479004s (cpu); 0.0470507s (thread); 0s (gc)
│ │ │ + -- used 0.054029s (cpu); 0.0561453s (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 0.156086s (cpu); 0.0812707s (thread); 0s (gc)
│ │ │ + -- used 0.177773s (cpu); 0.0926378s (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 0.0592825s (cpu); 0.0620712s (thread); 0s (gc)
│ │ │ + -- used 0.0680056s (cpu); 0.0696491s (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 0.467148s (cpu); 0.316433s (thread); 0s (gc)
│ │ │ + -- used 0.552828s (cpu); 0.398813s (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 0.512854s (cpu); 0.349394s (thread); 0s (gc)
│ │ │ + -- used 0.467981s (cpu); 0.322824s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -780,17 +780,17 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0944063 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.32417s (cpu); 0.240508s (thread); 0s (gc)
│ │ │ -  time .223987 sec  #fractions 6]
│ │ │ + [step 0:   time .111936 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.358624s (cpu); 0.282024s (thread); 0s (gc)
│ │ │ +  time .242594 sec  #fractions 6]
│ │ │  
│ │ │  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 0 sec #minors 4]
│ │ │ + [jacobian time .00101292 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .198954 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.459646s (cpu); 0.297601s (thread); 0s (gc)
│ │ │ -  time .256691 sec  #fractions 6]
│ │ │ + [step 0:   time .239657 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.502796s (cpu); 0.345973s (thread); 0s (gc)
│ │ │ +  time .259158 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -846,17 +846,17 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .111715 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.576437s (cpu); 0.405929s (thread); 0s (gc)
│ │ │ -  time .461045 sec  #fractions 6]
│ │ │ + [step 0:   time .131949 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.719266s (cpu); 0.5645s (thread); 0s (gc)
│ │ │ +  time .583337 sec  #fractions 6]
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -3,48 +3,48 @@
│ │ │  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 0 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00397269 seconds
│ │ │ -      idlizer1:  .00800146 seconds
│ │ │ -      idlizer2:  .00797529 seconds
│ │ │ -      minpres:   .0890644 seconds
│ │ │ -  time .117021 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00399742 seconds
│ │ │ +      idlizer1:  .007948 seconds
│ │ │ +      idlizer2:  .0114498 seconds
│ │ │ +      minpres:   .0795377 seconds
│ │ │ +  time .115107 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00400073 seconds
│ │ │ -      idlizer1:  .012073 seconds
│ │ │ -      idlizer2:  .0120406 seconds
│ │ │ -      minpres:   .103641 seconds
│ │ │ -  time .143737 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00399985 seconds
│ │ │ +      idlizer1:  .0162142 seconds
│ │ │ +      idlizer2:  .0121029 seconds
│ │ │ +      minpres:   .0839589 seconds
│ │ │ +  time .127971 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0120083 seconds
│ │ │ -      idlizer2:  .101167 seconds
│ │ │ -      minpres:   .0107394 seconds
│ │ │ -  time .143942 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .000017492 seconds
│ │ │ +      idlizer1:  .0118895 seconds
│ │ │ +      idlizer2:  .0859929 seconds
│ │ │ +      minpres:   .0120589 seconds
│ │ │ +  time .126127 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00378929 seconds
│ │ │ -      idlizer1:  .0118 seconds
│ │ │ -      idlizer2:  .0974523 seconds
│ │ │ -      minpres:   .0143427 seconds
│ │ │ -  time .140231 sec  #fractions 5]
│ │ │ +      radical (use minprimes) 0 seconds
│ │ │ +      idlizer1:  .0159406 seconds
│ │ │ +      idlizer2:  .0949109 seconds
│ │ │ +      minpres:   .0320606 seconds
│ │ │ +  time .169452 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00399997 seconds
│ │ │ -      idlizer1:  .00800158 seconds
│ │ │ -      idlizer2:  .10573 seconds
│ │ │ -      minpres:   .0128084 seconds
│ │ │ -  time .148113 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00402281 seconds
│ │ │ +      idlizer1:  .0199753 seconds
│ │ │ +      idlizer2:  .0981039 seconds
│ │ │ +      minpres:   .0120516 seconds
│ │ │ +  time .154149 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00387188 seconds
│ │ │ -      idlizer1:   -- used 0.715329s (cpu); 0.373506s (thread); 0s (gc)
│ │ │ -.0119473 seconds
│ │ │ -  time .0182781 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00399867 seconds
│ │ │ +      idlizer1:   -- used 0.716869s (cpu); 0.430815s (thread); 0s (gc)
│ │ │ +.01195 seconds
│ │ │ +  time .0200174 sec  #fractions 5]
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 2.04839s (cpu); 0.957564s (thread); 0s (gc)
│ │ │ + -- used 2.1599s (cpu); 0.977847s (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 1.09676s (cpu); 0.603839s (thread); 0s (gc)
│ │ │ + -- used 1.33304s (cpu); 0.648211s (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 0.803476s (cpu); 0.400866s (thread); 0s (gc)
│ │ │ + -- used 0.705968s (cpu); 0.388203s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -186,15 +186,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.669248s (cpu); 0.369386s (thread); 0s (gc)
│ │ │ + -- used 0.733877s (cpu); 0.407391s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -273,15 +273,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 1.02394s (cpu); 0.472475s (thread); 0s (gc)
│ │ │ + -- used 0.842625s (cpu); 0.408987s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -348,15 +348,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 1.06285s (cpu); 0.537877s (thread); 0s (gc)
│ │ │ + -- used 0.879127s (cpu); 0.43333s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -423,15 +423,15 @@ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.55504s (cpu); 0.731058s (thread); 0s (gc)
│ │ │ + -- used 1.85559s (cpu); 0.830133s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -498,15 +498,15 @@ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.981816s (cpu); 0.466986s (thread); 0s (gc)
│ │ │ + -- used 1.12016s (cpu); 0.543152s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -573,15 +573,15 @@ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.231485s (cpu); 0.0700508s (thread); 0s (gc)
│ │ │ + -- used 0.22591s (cpu); 0.0884232s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -643,15 +643,15 @@ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.137094s (cpu); 0.0666161s (thread); 0s (gc)
│ │ │ + -- used 0.146667s (cpu); 0.072453s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -695,15 +695,15 @@ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0679401s (cpu); 0.067694s (thread); 0s (gc)
│ │ │ + -- used 0.0800267s (cpu); 0.0800652s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -746,15 +746,15 @@ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0479004s (cpu); 0.0470507s (thread); 0s (gc)
│ │ │ + -- used 0.054029s (cpu); 0.0561453s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -798,15 +798,15 @@ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.156086s (cpu); 0.0812707s (thread); 0s (gc)
│ │ │ + -- used 0.177773s (cpu); 0.0926378s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -932,15 +932,15 @@ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0592825s (cpu); 0.0620712s (thread); 0s (gc)
│ │ │ + -- used 0.0680056s (cpu); 0.0696491s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1056,15 +1056,15 @@ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.467148s (cpu); 0.316433s (thread); 0s (gc)
│ │ │ + -- used 0.552828s (cpu); 0.398813s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1098,15 +1098,15 @@ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.512854s (cpu); 0.349394s (thread); 0s (gc)
│ │ │ + -- used 0.467981s (cpu); 0.322824s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1143,17 +1143,17 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0944063 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.32417s (cpu); 0.240508s (thread); 0s (gc)
│ │ │ -  time .223987 sec  #fractions 6]
│ │ │ + [step 0:   time .111936 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.358624s (cpu); 0.282024s (thread); 0s (gc)
│ │ │ +  time .242594 sec  #fractions 6]
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1187,20 +1187,20 @@ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time 0 sec #minors 4]
│ │ │ + [jacobian time .00101292 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .198954 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.459646s (cpu); 0.297601s (thread); 0s (gc)
│ │ │ -  time .256691 sec  #fractions 6]
│ │ │ + [step 0:   time .239657 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.502796s (cpu); 0.345973s (thread); 0s (gc)
│ │ │ +  time .259158 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1240,17 +1240,17 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │   [jacobian time 0 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .111715 sec  #fractions 6]
│ │ │ - [step 1:  -- used 0.576437s (cpu); 0.405929s (thread); 0s (gc)
│ │ │ -  time .461045 sec  #fractions 6]
│ │ │ + [step 0:   time .131949 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.719266s (cpu); 0.5645s (thread); 0s (gc)
│ │ │ +  time .583337 sec  #fractions 6]
│ │ │  
│ │ │  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 0.803476s (cpu); 0.400866s (thread); 0s (gc) │ │ │ │ + -- used 0.705968s (cpu); 0.388203s (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 0.669248s (cpu); 0.369386s (thread); 0s (gc) │ │ │ │ + -- used 0.733877s (cpu); 0.407391s (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 1.02394s (cpu); 0.472475s (thread); 0s (gc) │ │ │ │ + -- used 0.842625s (cpu); 0.408987s (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 1.06285s (cpu); 0.537877s (thread); 0s (gc) │ │ │ │ + -- used 0.879127s (cpu); 0.43333s (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 1.55504s (cpu); 0.731058s (thread); 0s (gc) │ │ │ │ + -- used 1.85559s (cpu); 0.830133s (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 0.981816s (cpu); 0.466986s (thread); 0s (gc) │ │ │ │ + -- used 1.12016s (cpu); 0.543152s (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 0.231485s (cpu); 0.0700508s (thread); 0s (gc) │ │ │ │ + -- used 0.22591s (cpu); 0.0884232s (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 0.137094s (cpu); 0.0666161s (thread); 0s (gc) │ │ │ │ + -- used 0.146667s (cpu); 0.072453s (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 0.0679401s (cpu); 0.067694s (thread); 0s (gc) │ │ │ │ + -- used 0.0800267s (cpu); 0.0800652s (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 0.0479004s (cpu); 0.0470507s (thread); 0s (gc) │ │ │ │ + -- used 0.054029s (cpu); 0.0561453s (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 0.156086s (cpu); 0.0812707s (thread); 0s (gc) │ │ │ │ + -- used 0.177773s (cpu); 0.0926378s (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 0.0592825s (cpu); 0.0620712s (thread); 0s (gc) │ │ │ │ + -- used 0.0680056s (cpu); 0.0696491s (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 0.467148s (cpu); 0.316433s (thread); 0s (gc) │ │ │ │ + -- used 0.552828s (cpu); 0.398813s (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 0.512854s (cpu); 0.349394s (thread); 0s (gc) │ │ │ │ + -- used 0.467981s (cpu); 0.322824s (thread); 0s (gc) │ │ │ │ │ │ │ │ o82 = R' │ │ │ │ │ │ │ │ o82 : QuotientRing │ │ │ │ i83 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -889,17 +889,17 @@ │ │ │ │ o85 = R │ │ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1) │ │ │ │ [jacobian time 0 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .0944063 sec #fractions 6] │ │ │ │ - [step 1: -- used 0.32417s (cpu); 0.240508s (thread); 0s (gc) │ │ │ │ - time .223987 sec #fractions 6] │ │ │ │ + [step 0: time .111936 sec #fractions 6] │ │ │ │ + [step 1: -- used 0.358624s (cpu); 0.282024s (thread); 0s (gc) │ │ │ │ + time .242594 sec #fractions 6] │ │ │ │ │ │ │ │ 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 0 sec #minors 4] │ │ │ │ + [jacobian time .00101292 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .198954 sec #fractions 6] │ │ │ │ - [step 1: -- used 0.459646s (cpu); 0.297601s (thread); 0s (gc) │ │ │ │ - time .256691 sec #fractions 6] │ │ │ │ + [step 0: time .239657 sec #fractions 6] │ │ │ │ + [step 1: -- used 0.502796s (cpu); 0.345973s (thread); 0s (gc) │ │ │ │ + time .259158 sec #fractions 6] │ │ │ │ │ │ │ │ o90 = R' │ │ │ │ │ │ │ │ o90 : QuotientRing │ │ │ │ i91 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -950,17 +950,17 @@ │ │ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, │ │ │ │ StartWithOneMinor}, Verbosity => 1) │ │ │ │ [jacobian time 0 sec #minors 1] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .111715 sec #fractions 6] │ │ │ │ - [step 1: -- used 0.576437s (cpu); 0.405929s (thread); 0s (gc) │ │ │ │ - time .461045 sec #fractions 6] │ │ │ │ + [step 0: time .131949 sec #fractions 6] │ │ │ │ + [step 1: -- used 0.719266s (cpu); 0.5645s (thread); 0s (gc) │ │ │ │ + time .583337 sec #fractions 6] │ │ │ │ │ │ │ │ o94 = R' │ │ │ │ │ │ │ │ o94 : QuotientRing │ │ │ │ i95 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 2 3 2 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ @@ -75,48 +75,48 @@ │ │ │ │ │ │ │ │ │ 
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │   [jacobian time 0 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00397269 seconds
│ │ │ -      idlizer1:  .00800146 seconds
│ │ │ -      idlizer2:  .00797529 seconds
│ │ │ -      minpres:   .0890644 seconds
│ │ │ -  time .117021 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00399742 seconds
│ │ │ +      idlizer1:  .007948 seconds
│ │ │ +      idlizer2:  .0114498 seconds
│ │ │ +      minpres:   .0795377 seconds
│ │ │ +  time .115107 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00400073 seconds
│ │ │ -      idlizer1:  .012073 seconds
│ │ │ -      idlizer2:  .0120406 seconds
│ │ │ -      minpres:   .103641 seconds
│ │ │ -  time .143737 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00399985 seconds
│ │ │ +      idlizer1:  .0162142 seconds
│ │ │ +      idlizer2:  .0121029 seconds
│ │ │ +      minpres:   .0839589 seconds
│ │ │ +  time .127971 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0120083 seconds
│ │ │ -      idlizer2:  .101167 seconds
│ │ │ -      minpres:   .0107394 seconds
│ │ │ -  time .143942 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .000017492 seconds
│ │ │ +      idlizer1:  .0118895 seconds
│ │ │ +      idlizer2:  .0859929 seconds
│ │ │ +      minpres:   .0120589 seconds
│ │ │ +  time .126127 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00378929 seconds
│ │ │ -      idlizer1:  .0118 seconds
│ │ │ -      idlizer2:  .0974523 seconds
│ │ │ -      minpres:   .0143427 seconds
│ │ │ -  time .140231 sec  #fractions 5]
│ │ │ +      radical (use minprimes) 0 seconds
│ │ │ +      idlizer1:  .0159406 seconds
│ │ │ +      idlizer2:  .0949109 seconds
│ │ │ +      minpres:   .0320606 seconds
│ │ │ +  time .169452 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00399997 seconds
│ │ │ -      idlizer1:  .00800158 seconds
│ │ │ -      idlizer2:  .10573 seconds
│ │ │ -      minpres:   .0128084 seconds
│ │ │ -  time .148113 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00402281 seconds
│ │ │ +      idlizer1:  .0199753 seconds
│ │ │ +      idlizer2:  .0981039 seconds
│ │ │ +      minpres:   .0120516 seconds
│ │ │ +  time .154149 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00387188 seconds
│ │ │ -      idlizer1:   -- used 0.715329s (cpu); 0.373506s (thread); 0s (gc)
│ │ │ -.0119473 seconds
│ │ │ -  time .0182781 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00399867 seconds
│ │ │ +      idlizer1:   -- used 0.716869s (cpu); 0.430815s (thread); 0s (gc)
│ │ │ +.01195 seconds
│ │ │ +  time .0200174 sec  #fractions 5]
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -16,48 +16,48 @@ │ │ │ │ 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 0 sec #minors 3] │ │ │ │ integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ [step 0: │ │ │ │ - radical (use minprimes) .00397269 seconds │ │ │ │ - idlizer1: .00800146 seconds │ │ │ │ - idlizer2: .00797529 seconds │ │ │ │ - minpres: .0890644 seconds │ │ │ │ - time .117021 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00399742 seconds │ │ │ │ + idlizer1: .007948 seconds │ │ │ │ + idlizer2: .0114498 seconds │ │ │ │ + minpres: .0795377 seconds │ │ │ │ + time .115107 sec #fractions 4] │ │ │ │ [step 1: │ │ │ │ - radical (use minprimes) .00400073 seconds │ │ │ │ - idlizer1: .012073 seconds │ │ │ │ - idlizer2: .0120406 seconds │ │ │ │ - minpres: .103641 seconds │ │ │ │ - time .143737 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00399985 seconds │ │ │ │ + idlizer1: .0162142 seconds │ │ │ │ + idlizer2: .0121029 seconds │ │ │ │ + minpres: .0839589 seconds │ │ │ │ + time .127971 sec #fractions 4] │ │ │ │ [step 2: │ │ │ │ - radical (use minprimes) 0 seconds │ │ │ │ - idlizer1: .0120083 seconds │ │ │ │ - idlizer2: .101167 seconds │ │ │ │ - minpres: .0107394 seconds │ │ │ │ - time .143942 sec #fractions 5] │ │ │ │ + radical (use minprimes) .000017492 seconds │ │ │ │ + idlizer1: .0118895 seconds │ │ │ │ + idlizer2: .0859929 seconds │ │ │ │ + minpres: .0120589 seconds │ │ │ │ + time .126127 sec #fractions 5] │ │ │ │ [step 3: │ │ │ │ - radical (use minprimes) .00378929 seconds │ │ │ │ - idlizer1: .0118 seconds │ │ │ │ - idlizer2: .0974523 seconds │ │ │ │ - minpres: .0143427 seconds │ │ │ │ - time .140231 sec #fractions 5] │ │ │ │ + radical (use minprimes) 0 seconds │ │ │ │ + idlizer1: .0159406 seconds │ │ │ │ + idlizer2: .0949109 seconds │ │ │ │ + minpres: .0320606 seconds │ │ │ │ + time .169452 sec #fractions 5] │ │ │ │ [step 4: │ │ │ │ - radical (use minprimes) .00399997 seconds │ │ │ │ - idlizer1: .00800158 seconds │ │ │ │ - idlizer2: .10573 seconds │ │ │ │ - minpres: .0128084 seconds │ │ │ │ - time .148113 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00402281 seconds │ │ │ │ + idlizer1: .0199753 seconds │ │ │ │ + idlizer2: .0981039 seconds │ │ │ │ + minpres: .0120516 seconds │ │ │ │ + time .154149 sec #fractions 5] │ │ │ │ [step 5: │ │ │ │ - radical (use minprimes) .00387188 seconds │ │ │ │ - idlizer1: -- used 0.715329s (cpu); 0.373506s (thread); 0s (gc) │ │ │ │ -.0119473 seconds │ │ │ │ - time .0182781 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00399867 seconds │ │ │ │ + idlizer1: -- used 0.716869s (cpu); 0.430815s (thread); 0s (gc) │ │ │ │ +.01195 seconds │ │ │ │ + time .0200174 sec #fractions 5] │ │ │ │ │ │ │ │ o2 = R' │ │ │ │ │ │ │ │ o2 : QuotientRing │ │ │ │ i3 : trim ideal R' │ │ │ │ │ │ │ │ 3 2 2 2 4 4 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ @@ -109,29 +109,29 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time integralClosure J
│ │ │ - -- used 2.04839s (cpu); 0.957564s (thread); 0s (gc)
│ │ │ + -- used 2.1599s (cpu); 0.977847s (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 1.09676s (cpu); 0.603839s (thread); 0s (gc)
│ │ │ + -- used 1.33304s (cpu); 0.648211s (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 2.04839s (cpu); 0.957564s (thread); 0s (gc)
│ │ │ │ + -- used 2.1599s (cpu); 0.977847s (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 1.09676s (cpu); 0.603839s (thread); 0s (gc)
│ │ │ │ + -- used 1.33304s (cpu); 0.648211s (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/_action.out
│ │ │ @@ -14,21 +14,17 @@
│ │ │       | 0 1 -1 1  |
│ │ │       | 1 0 -1 -1 |
│ │ │  
│ │ │  o2 : DiagonalAction
│ │ │  
│ │ │  i3 : S = R^T
│ │ │  
│ │ │ -o3 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o3 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o3 : RingOfInvariants
│ │ │  
│ │ │  i4 : action S
│ │ │  
│ │ │               * 2
│ │ │  o4 = R <- (QQ )  via
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_defining__Ideal.out
│ │ │ @@ -22,31 +22,21 @@
│ │ │       | 0 1 -1 1  |
│ │ │       | 1 0 -1 -1 |
│ │ │  
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : S = R^T
│ │ │  
│ │ │ -o4 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o4 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o4 : RingOfInvariants
│ │ │  
│ │ │  i5 : definingIdeal S
│ │ │  
│ │ │ -                             2        2            2        3        3      
│ │ │ -o5 = ideal (u  - u u , u  - u , u  - u u , u  - u u , u  - u , u  - u u , u 
│ │ │ -             5    8 9   6    9   3    8 9   1    8 9   4    9   2    8 9   7
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -        5
│ │ │ -     - u )
│ │ │ -        8
│ │ │ +o5 = ideal ()
│ │ │  
│ │ │  o5 : Ideal of QQ[u ..u ]
│ │ │ -                  1   9
│ │ │ +                  1   2
│ │ │  
│ │ │  i6 :
│ │ ├── ./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)
│ │ │ - -- .00378776s elapsed
│ │ │ + -- .00398263s 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);
│ │ │ - -- .000508261s elapsed
│ │ │ + -- .00080882s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Series_lp__Ring__Of__Invariants_rp.out
│ │ │ @@ -22,28 +22,23 @@
│ │ │       | 0 1 -1 1  |
│ │ │       | 1 0 -1 -1 |
│ │ │  
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : S = R^T
│ │ │  
│ │ │ -o4 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o4 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o4 : RingOfInvariants
│ │ │  
│ │ │  i5 : hilbertSeries S
│ │ │  
│ │ │ -          7    8    10    11    12    13    17     18     19     20     21     22     23    24    25    28     29     30     31     32     33     34     35     36     40     41     42     43     44     45     46     47    48    51    52     53     54     55     56     57     58    59    63    64    65    66    68    69    76
│ │ │ -     1 - T  - T  - T   - T   - T   - T   + T   + 2T   + 2T   + 2T   + 2T   + 2T   + 3T   + T   + T   - T   - 2T   - 4T   - 3T   - 3T   - 4T   - 3T   - 3T   - 2T   + 2T   + 3T   + 3T   + 4T   + 3T   + 3T   + 4T   + 2T   + T   - T   - T   - 3T   - 2T   - 2T   - 2T   - 2T   - 2T   - T   + T   + T   + T   + T   + T   + T   - T
│ │ │ -o5 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │ -                                                                                                                                     15       13       12       11       10       8       7       4       3
│ │ │ -                                                                                                                               (1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T )(1 - T )(1 - T )(1 - T )
│ │ │ +             1
│ │ │ +o5 = ----------------
│ │ │ +           4       3
│ │ │ +     (1 - T )(1 - T )
│ │ │  
│ │ │  o5 : Expression of class Divide
│ │ │  
│ │ │  i6 :
│ │ ├── ./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 0.832459s (cpu); 0.57933s (thread); 0s (gc)
│ │ │ + -- used 0.889397s (cpu); 0.635961s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.644773s (cpu); 0.39549s (thread); 0s (gc)
│ │ │ + -- used 0.647157s (cpu); 0.399192s (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 0.0237512s (cpu); 0.024733s (thread); 0s (gc)
│ │ │ + -- used 0.0279433s (cpu); 0.028013s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.1789s (cpu); 1.50352s (thread); 0s (gc)
│ │ │ + -- used 2.05812s (cpu); 1.42511s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .650845s elapsed
│ │ │ + -- .577943s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ - -- .58543s elapsed
│ │ │ + -- .485768s elapsed
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .734527s elapsed
│ │ │ + -- .570218s 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)
│ │ │ - -- .153156s elapsed
│ │ │ + -- .0905933s 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_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/_action.html
│ │ │ @@ -91,21 +91,17 @@
│ │ │  o2 : DiagonalAction
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : S = R^T
│ │ │  
│ │ │ -o3 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o3 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o3 : RingOfInvariants
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : action S
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,21 +29,17 @@
│ │ │ │  
│ │ │ │       | 0 1 -1 1  |
│ │ │ │       | 1 0 -1 -1 |
│ │ │ │  
│ │ │ │  o2 : DiagonalAction
│ │ │ │  i3 : S = R^T
│ │ │ │  
│ │ │ │ -o3 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5
│ │ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -              2
│ │ │ │ -     x x x , x x x ]
│ │ │ │ -      1 2 3   1 3 4
│ │ │ │ +o3 =             2
│ │ │ │ +     QQ[x x x , x x x ]
│ │ │ │ +         1 2 3   1 3 4
│ │ │ │  
│ │ │ │  o3 : RingOfInvariants
│ │ │ │  i4 : action S
│ │ │ │  
│ │ │ │               * 2
│ │ │ │  o4 = R <- (QQ )  via
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_defining__Ideal.html
│ │ │ @@ -107,39 +107,29 @@
│ │ │  o3 : DiagonalAction
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : S = R^T
│ │ │  
│ │ │ -o4 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o4 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o4 : RingOfInvariants
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : definingIdeal S
│ │ │  
│ │ │ -                             2        2            2        3        3      
│ │ │ -o5 = ideal (u  - u u , u  - u , u  - u u , u  - u u , u  - u , u  - u u , u 
│ │ │ -             5    8 9   6    9   3    8 9   1    8 9   4    9   2    8 9   7
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -        5
│ │ │ -     - u )
│ │ │ -        8
│ │ │ +o5 = ideal ()
│ │ │  
│ │ │  o5 : Ideal of QQ[u ..u ]
│ │ │ -                  1   9
    │ │ │ + 1 2 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use definingIdeal:

    │ │ │ ├── html2text {} │ │ │ │ @@ -39,35 +39,25 @@ │ │ │ │ │ │ │ │ | 0 1 -1 1 | │ │ │ │ | 1 0 -1 -1 | │ │ │ │ │ │ │ │ o3 : DiagonalAction │ │ │ │ i4 : S = R^T │ │ │ │ │ │ │ │ -o4 = 5 3 2 5 3 4 4 2 3 6 3 3 3 2 4 2 2 5 5 5 │ │ │ │ - QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x , │ │ │ │ - 1 2 3 4 1 2 3 4 1 2 3 4 1 3 4 1 2 3 4 1 3 4 1 2 3 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 2 │ │ │ │ - x x x , x x x ] │ │ │ │ - 1 2 3 1 3 4 │ │ │ │ +o4 = 2 │ │ │ │ + QQ[x x x , x x x ] │ │ │ │ + 1 2 3 1 3 4 │ │ │ │ │ │ │ │ o4 : RingOfInvariants │ │ │ │ i5 : definingIdeal S │ │ │ │ │ │ │ │ - 2 2 2 3 3 │ │ │ │ -o5 = ideal (u - u u , u - u , u - u u , u - u u , u - u , u - u u , u │ │ │ │ - 5 8 9 6 9 3 8 9 1 8 9 4 9 2 8 9 7 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 5 │ │ │ │ - - u ) │ │ │ │ - 8 │ │ │ │ +o5 = ideal () │ │ │ │ │ │ │ │ o5 : Ideal of QQ[u ..u ] │ │ │ │ - 1 9 │ │ │ │ + 1 2 │ │ │ │ ********** WWaayyss ttoo uussee ddeeffiinniinnggIIddeeaall:: ********** │ │ │ │ * definingIdeal(RingOfInvariants) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _d_e_f_i_n_i_n_g_I_d_e_a_l 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- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/InvariantRing/InvariantsDoc.m2:895:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ │ │ │ o4 = false │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00378776s elapsed
│ │ │ + -- .00398263s 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);
│ │ │ - -- .000508261s elapsed
    │ │ │ + -- .00080882s 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) │ │ │ │ - -- .00378776s elapsed │ │ │ │ + -- .00398263s 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); │ │ │ │ - -- .000508261s elapsed │ │ │ │ + -- .00080882s 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- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:185:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Series_lp__Ring__Of__Invariants_rp.html │ │ │ @@ -108,34 +108,29 @@ │ │ │ o3 : DiagonalAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : S = R^T
│ │ │  
│ │ │ -o4 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
│ │ │ -     QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x ,
│ │ │ -         1 2 3 4   1 2 3 4   1 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -              2
│ │ │ -     x x x , x x x ]
│ │ │ -      1 2 3   1 3 4
│ │ │ +o4 =             2
│ │ │ +     QQ[x x x , x x x ]
│ │ │ +         1 2 3   1 3 4
│ │ │  
│ │ │  o4 : RingOfInvariants
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : hilbertSeries S
│ │ │  
│ │ │ -          7    8    10    11    12    13    17     18     19     20     21     22     23    24    25    28     29     30     31     32     33     34     35     36     40     41     42     43     44     45     46     47    48    51    52     53     54     55     56     57     58    59    63    64    65    66    68    69    76
│ │ │ -     1 - T  - T  - T   - T   - T   - T   + T   + 2T   + 2T   + 2T   + 2T   + 2T   + 3T   + T   + T   - T   - 2T   - 4T   - 3T   - 3T   - 4T   - 3T   - 3T   - 2T   + 2T   + 3T   + 3T   + 4T   + 3T   + 3T   + 4T   + 2T   + T   - T   - T   - 3T   - 2T   - 2T   - 2T   - 2T   - 2T   - T   + T   + T   + T   + T   + T   + T   - T
│ │ │ -o5 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │ -                                                                                                                                     15       13       12       11       10       8       7       4       3
│ │ │ -                                                                                                                               (1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T )(1 - T )(1 - T )(1 - T )
│ │ │ +             1
│ │ │ +o5 = ----------------
│ │ │ +           4       3
│ │ │ +     (1 - T )(1 - T )
│ │ │  
│ │ │  o5 : Expression of class Divide
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,44 +38,25 @@ │ │ │ │ │ │ │ │ | 0 1 -1 1 | │ │ │ │ | 1 0 -1 -1 | │ │ │ │ │ │ │ │ o3 : DiagonalAction │ │ │ │ i4 : S = R^T │ │ │ │ │ │ │ │ -o4 = 5 3 2 5 3 4 4 2 3 6 3 3 3 2 4 2 2 5 5 5 │ │ │ │ - QQ[x x x x , x x x x , x x x x , x x x , x x x x , x x x , x x x , │ │ │ │ - 1 2 3 4 1 2 3 4 1 2 3 4 1 3 4 1 2 3 4 1 3 4 1 2 3 │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 2 │ │ │ │ - x x x , x x x ] │ │ │ │ - 1 2 3 1 3 4 │ │ │ │ +o4 = 2 │ │ │ │ + QQ[x x x , x x x ] │ │ │ │ + 1 2 3 1 3 4 │ │ │ │ │ │ │ │ o4 : RingOfInvariants │ │ │ │ i5 : hilbertSeries S │ │ │ │ │ │ │ │ - 7 8 10 11 12 13 17 18 19 20 21 │ │ │ │ -22 23 24 25 28 29 30 31 32 33 34 35 │ │ │ │ -36 40 41 42 43 44 45 46 47 48 51 52 │ │ │ │ -53 54 55 56 57 58 59 63 64 65 66 68 69 │ │ │ │ -76 │ │ │ │ - 1 - T - T - T - T - T - T + T + 2T + 2T + 2T + 2T + 2T │ │ │ │ -+ 3T + T + T - T - 2T - 4T - 3T - 3T - 4T - 3T - 3T - 2T │ │ │ │ -+ 2T + 3T + 3T + 4T + 3T + 3T + 4T + 2T + T - T - T - 3T │ │ │ │ -- 2T - 2T - 2T - 2T - 2T - T + T + T + T + T + T + T - │ │ │ │ -T │ │ │ │ -o5 = -------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ ----------- │ │ │ │ - │ │ │ │ -15 13 12 11 10 8 7 4 3 │ │ │ │ - │ │ │ │ -(1 - T )(1 - T )(1 - T )(1 - T )(1 - T )(1 - T )(1 - T )(1 - T )(1 - T ) │ │ │ │ + 1 │ │ │ │ +o5 = ---------------- │ │ │ │ + 4 3 │ │ │ │ + (1 - T )(1 - T ) │ │ │ │ │ │ │ │ o5 : Expression of class Divide │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _h_i_l_b_e_r_t_S_e_r_i_e_s_(_R_i_n_g_O_f_I_n_v_a_r_i_a_n_t_s_) -- Hilbert series of the invariant ring │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/InvariantRing/InvariantsDoc.m2:967:0. │ │ ├── ./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 0.832459s (cpu); 0.57933s (thread); 0s (gc)
│ │ │ + -- used 0.889397s (cpu); 0.635961s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.644773s (cpu); 0.39549s (thread); 0s (gc)
│ │ │ + -- used 0.647157s (cpu); 0.399192s (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 0.0237512s (cpu); 0.024733s (thread); 0s (gc)
│ │ │ + -- used 0.0279433s (cpu); 0.028013s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.1789s (cpu); 1.50352s (thread); 0s (gc)
│ │ │ + -- used 2.05812s (cpu); 1.42511s (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 0.832459s (cpu); 0.57933s (thread); 0s (gc)
│ │ │ │ + -- used 0.889397s (cpu); 0.635961s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2   2    2   3       3
│ │ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.644773s (cpu); 0.39549s (thread); 0s (gc)
│ │ │ │ + -- used 0.647157s (cpu); 0.399192s (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 0.0237512s (cpu); 0.024733s (thread); 0s (gc)
│ │ │ │ + -- used 0.0279433s (cpu); 0.028013s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            4    4
│ │ │ │  o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 2.1789s (cpu); 1.50352s (thread); 0s (gc)
│ │ │ │ + -- used 2.05812s (cpu); 1.42511s (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
│ │ │ - -- .650845s elapsed
│ │ │ + -- .577943s 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,DegreeBound=>4)
│ │ │ - -- .58543s elapsed
│ │ │ + -- .485768s elapsed
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,27 +33,27 @@
│ │ │ │  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
│ │ │ │ - -- .650845s elapsed
│ │ │ │ + -- .577943s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ │ - -- .58543s elapsed
│ │ │ │ + -- .485768s elapsed
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .734527s elapsed
│ │ │ + -- .570218s 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)
│ │ │ - -- .153156s elapsed
│ │ │ + -- .0905933s 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
│ │ │ │ - -- .734527s elapsed
│ │ │ │ + -- .570218s 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)
│ │ │ │ - -- .153156s elapsed
│ │ │ │ + -- .0905933s 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_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) │ │ │ - -- 1.3382s elapsed │ │ │ + -- 1.59457s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isomorphism(T1, T2) │ │ │ - -- .00002146s elapsed │ │ │ + -- .000022665s 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)
│ │ │ - -- 1.3382s elapsed
│ │ │ + -- 1.59457s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .00002146s elapsed
│ │ │ + -- .000022665s 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)
│ │ │ │ - -- 1.3382s elapsed
│ │ │ │ + -- 1.59457s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .00002146s elapsed
│ │ │ │ + -- .000022665s 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/M2-138204-0/0.json
│ │ │ +o9 = /tmp/M2-250019-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-138204-0/0.json
│ │ │ +o10 = /tmp/M2-250019-0/0.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/M2-138204-0/0.json
    │ │ │ +o9 = /tmp/M2-250019-0/0.json │ │ │
    │ │ │ 
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-138204-0/0.json
│ │ │ +o10 = /tmp/M2-250019-0/0.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/M2-138204-0/0.json
│ │ │ │ +o9 = /tmp/M2-250019-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-138204-0/0.json
│ │ │ │ +o10 = /tmp/M2-250019-0/0.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)
│ │ │ - -- .00217575s elapsed
│ │ │ + -- .00233183s 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
│ │ │ - -- .302008s elapsed
│ │ │ + -- .258072s 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)
│ │ │ - -- .00903519s elapsed
│ │ │ + -- .0114217s 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)
│ │ │ - -- .00225613s elapsed
│ │ │ + -- .00302204s 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)
│ │ │ - -- .00206473s elapsed
│ │ │ + -- .00276935s 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)
│ │ │ - -- .0123258s elapsed
│ │ │ + -- .0150422s 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)
│ │ │ - -- .000672949s elapsed
│ │ │ + -- .000843218s 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)
│ │ │ │ - -- .00217575s elapsed
│ │ │ │ + -- .00233183s 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
│ │ │ │ - -- .302008s elapsed
│ │ │ │ + -- .258072s 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)
│ │ │ - -- .00217575s elapsed
│ │ │ + -- .00233183s 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
│ │ │ - -- .302008s elapsed
│ │ │ + -- .258072s 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)
│ │ │ - -- .00903519s elapsed
│ │ │ + -- .0114217s 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)
│ │ │ - -- .00225613s elapsed
│ │ │ + -- .00302204s 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)
│ │ │ - -- .00206473s elapsed
│ │ │ + -- .00276935s 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)
│ │ │ - -- .0123258s elapsed
│ │ │ + -- .0150422s 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)
│ │ │ - -- .000672949s elapsed
│ │ │ + -- .000843218s 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)
│ │ │ │ - -- .00903519s elapsed
│ │ │ │ + -- .0114217s 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)
│ │ │ │ - -- .00225613s elapsed
│ │ │ │ + -- .00302204s 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)
│ │ │ │ - -- .00206473s elapsed
│ │ │ │ + -- .00276935s 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)
│ │ │ │ - -- .0123258s elapsed
│ │ │ │ + -- .0150422s 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)
│ │ │ │ - -- .000672949s elapsed
│ │ │ │ + -- .000843218s 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
│ │ │ - -- .0259654s elapsed
│ │ │ + -- .0302623s 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)
│ │ │ - -- .00236066s elapsed
│ │ │ - -- .00683903s elapsed
│ │ │ - -- .0232518s elapsed
│ │ │ - -- .0102286s elapsed
│ │ │ - -- .00377167s elapsed
│ │ │ + -- .00265302s elapsed
│ │ │ + -- .00739593s elapsed
│ │ │ + -- .0272381s elapsed
│ │ │ + -- .0126229s elapsed
│ │ │ + -- .00417434s 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)
│ │ │ - -- .00246423s elapsed
│ │ │ - -- .00627075s elapsed
│ │ │ - -- .0234075s elapsed
│ │ │ - -- .0110125s elapsed
│ │ │ - -- .00359968s elapsed
│ │ │ - -- .553045s elapsed
│ │ │ + -- .00293913s elapsed
│ │ │ + -- .00787492s elapsed
│ │ │ + -- .0283482s elapsed
│ │ │ + -- .0123755s elapsed
│ │ │ + -- .00412053s elapsed
│ │ │ + -- .44406s 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
│ │ │ - -- .340732s elapsed
│ │ │ + -- .272496s 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);
│ │ │ - -- .00470885s elapsed
│ │ │ - -- .0171843s elapsed
│ │ │ - -- .0951582s elapsed
│ │ │ - -- 1.0674s elapsed
│ │ │ - -- .499851s elapsed
│ │ │ - -- .0393332s elapsed
│ │ │ - -- .00669755s elapsed
│ │ │ - -- 6.34327s elapsed
│ │ │ + -- .00522795s elapsed
│ │ │ + -- .0216107s elapsed
│ │ │ + -- .108808s elapsed
│ │ │ + -- 1.05801s elapsed
│ │ │ + -- .455304s elapsed
│ │ │ + -- .0410704s elapsed
│ │ │ + -- .00788978s elapsed
│ │ │ + -- 6.06507s 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)
│ │ │ - -- .00254853s elapsed
│ │ │ - -- .00629121s elapsed
│ │ │ - -- .0240282s elapsed
│ │ │ - -- .0113519s elapsed
│ │ │ - -- .00368606s elapsed
│ │ │ + -- .00300216s elapsed
│ │ │ + -- .00771659s elapsed
│ │ │ + -- .0291543s elapsed
│ │ │ + -- .0114388s elapsed
│ │ │ + -- .00410029s 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
│ │ │ - -- .247359s elapsed
│ │ │ + -- .236928s 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);
│ │ │ - -- .007547s elapsed
│ │ │ - -- .0336731s elapsed
│ │ │ - -- .11684s elapsed
│ │ │ - -- 1.10155s elapsed
│ │ │ - -- .453186s elapsed
│ │ │ - -- .0395934s elapsed
│ │ │ - -- .00659858s elapsed
│ │ │ - -- 6.96887s elapsed
│ │ │ + -- .00713205s elapsed
│ │ │ + -- .0215352s elapsed
│ │ │ + -- .135614s elapsed
│ │ │ + -- 1.05811s elapsed
│ │ │ + -- .427268s elapsed
│ │ │ + -- .0430608s elapsed
│ │ │ + -- .00774745s elapsed
│ │ │ + -- 6.04023s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,42 +3,42 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .00741432s elapsed
│ │ │ - -- .0131122s elapsed
│ │ │ - -- .000340948s elapsed
│ │ │ - -- .000173654s elapsed
│ │ │ - -- .000145061s elapsed
│ │ │ - -- .000136617s elapsed
│ │ │ - -- .00012314s elapsed
│ │ │ - -- .000131686s elapsed
│ │ │ - -- .000164106s elapsed
│ │ │ - -- .000168546s elapsed
│ │ │ - -- .000141084s elapsed
│ │ │ - -- .000132988s elapsed
│ │ │ - -- .000289191s elapsed
│ │ │ - -- .000131376s elapsed
│ │ │ - -- .00012862s elapsed
│ │ │ - -- .000128541s elapsed
│ │ │ - -- .0001334s elapsed
│ │ │ - -- .000151062s elapsed
│ │ │ - -- .000143969s elapsed
│ │ │ - -- .00014432s elapsed
│ │ │ - -- .000151594s elapsed
│ │ │ - -- .000151393s elapsed
│ │ │ - -- .000147766s elapsed
│ │ │ - -- .00013296s elapsed
│ │ │ - -- .000130113s elapsed
│ │ │ - -- .000136976s elapsed
│ │ │ - -- .000136727s elapsed
│ │ │ - -- .000137126s elapsed
│ │ │ + -- .0092257s elapsed
│ │ │ + -- .0126772s elapsed
│ │ │ + -- .000248361s elapsed
│ │ │ + -- .000193208s elapsed
│ │ │ + -- .000191645s elapsed
│ │ │ + -- .000154399s elapsed
│ │ │ + -- .000184512s elapsed
│ │ │ + -- .000208194s elapsed
│ │ │ + -- .000178275s elapsed
│ │ │ + -- .000212982s elapsed
│ │ │ + -- .000186587s elapsed
│ │ │ + -- .000167579s elapsed
│ │ │ + -- .000235189s elapsed
│ │ │ + -- .000199103s elapsed
│ │ │ + -- .00015487s elapsed
│ │ │ + -- .000166657s elapsed
│ │ │ + -- .000168004s elapsed
│ │ │ + -- .000181172s elapsed
│ │ │ + -- .000181567s elapsed
│ │ │ + -- .000182704s elapsed
│ │ │ + -- .000192597s elapsed
│ │ │ + -- .000187697s elapsed
│ │ │ + -- .000183881s elapsed
│ │ │ + -- .000176545s elapsed
│ │ │ + -- .000165162s elapsed
│ │ │ + -- .000172027s elapsed
│ │ │ + -- .000183194s elapsed
│ │ │ + -- .000140133s elapsed
│ │ │  (number Of blocks, 26)
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  2
│ │ │   2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .280556s elapsed
│ │ │ - -- .274042s elapsed
│ │ │ - -- .207278s elapsed
│ │ │ - -- .343307s elapsed
│ │ │ - -- .344206s elapsed
│ │ │ - -- .319911s elapsed
│ │ │ + -- .18348s elapsed
│ │ │ + -- .213709s elapsed
│ │ │ + -- .145051s elapsed
│ │ │ + -- .259669s elapsed
│ │ │ + -- .18622s elapsed
│ │ │ + -- .180466s 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)
│ │ │ - -- .00258639s elapsed
│ │ │ - -- .0300927s elapsed
│ │ │ - -- .0243393s elapsed
│ │ │ - -- .00941187s elapsed
│ │ │ - -- .00336785s elapsed
│ │ │ + -- .00268486s elapsed
│ │ │ + -- .00711347s elapsed
│ │ │ + -- .0272806s elapsed
│ │ │ + -- .0108124s elapsed
│ │ │ + -- .00373279s 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)
│ │ │ - -- .252648s elapsed
│ │ │ + -- .240002s 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)
│ │ │ - -- .0028042s elapsed
│ │ │ - -- .00610624s elapsed
│ │ │ - -- .0231543s elapsed
│ │ │ - -- .00939143s elapsed
│ │ │ - -- .00362288s elapsed
│ │ │ + -- .00285302s elapsed
│ │ │ + -- .00703046s elapsed
│ │ │ + -- .0294182s elapsed
│ │ │ + -- .0105433s elapsed
│ │ │ + -- .0050064s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,33 +1,33 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0531025s elapsed
│ │ │ - -- .000039765s elapsed
│ │ │ - -- .000091661s elapsed
│ │ │ - -- .000068136s elapsed
│ │ │ - -- .000086272s elapsed
│ │ │ - -- .000098233s elapsed
│ │ │ - -- .000094527s elapsed
│ │ │ - -- .000029445s elapsed
│ │ │ - -- .000075421s elapsed
│ │ │ - -- .000095759s elapsed
│ │ │ - -- .000202318s elapsed
│ │ │ - -- .000092814s elapsed
│ │ │ - -- .000104546s elapsed
│ │ │ - -- .000080951s elapsed
│ │ │ - -- .000075392s elapsed
│ │ │ - -- .00014968s elapsed
│ │ │ - -- .000024105s elapsed
│ │ │ - -- .000066555s elapsed
│ │ │ - -- .000023724s elapsed
│ │ │ + -- .0203277s elapsed
│ │ │ + -- .000058955s elapsed
│ │ │ + -- .000085572s elapsed
│ │ │ + -- .000081125s elapsed
│ │ │ + -- .00008802s elapsed
│ │ │ + -- .0001447s elapsed
│ │ │ + -- .000146772s elapsed
│ │ │ + -- .000051573s elapsed
│ │ │ + -- .000091195s elapsed
│ │ │ + -- .000098701s elapsed
│ │ │ + -- .000110253s elapsed
│ │ │ + -- .000106543s elapsed
│ │ │ + -- .000121952s elapsed
│ │ │ + -- .000107866s elapsed
│ │ │ + -- .000075993s elapsed
│ │ │ + -- .000201568s elapsed
│ │ │ + -- .000052734s elapsed
│ │ │ + -- .000143843s elapsed
│ │ │ + -- .000030082s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0259654s elapsed
│ │ │ + -- .0302623s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -141,19 +141,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00236066s elapsed
│ │ │ - -- .00683903s elapsed
│ │ │ - -- .0232518s elapsed
│ │ │ - -- .0102286s elapsed
│ │ │ - -- .00377167s elapsed
│ │ │ + -- .00265302s elapsed
│ │ │ + -- .00739593s elapsed
│ │ │ + -- .0272381s elapsed
│ │ │ + -- .0126229s elapsed
│ │ │ + -- .00417434s 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
│ │ │ │ - -- .0259654s elapsed
│ │ │ │ + -- .0302623s 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)
│ │ │ │ - -- .00236066s elapsed
│ │ │ │ - -- .00683903s elapsed
│ │ │ │ - -- .0232518s elapsed
│ │ │ │ - -- .0102286s elapsed
│ │ │ │ - -- .00377167s elapsed
│ │ │ │ + -- .00265302s elapsed
│ │ │ │ + -- .00739593s elapsed
│ │ │ │ + -- .0272381s elapsed
│ │ │ │ + -- .0126229s elapsed
│ │ │ │ + -- .00417434s 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)
│ │ │ - -- .00246423s elapsed
│ │ │ - -- .00627075s elapsed
│ │ │ - -- .0234075s elapsed
│ │ │ - -- .0110125s elapsed
│ │ │ - -- .00359968s elapsed
│ │ │ - -- .553045s elapsed
│ │ │ + -- .00293913s elapsed
│ │ │ + -- .00787492s elapsed
│ │ │ + -- .0283482s elapsed
│ │ │ + -- .0123755s elapsed
│ │ │ + -- .00412053s elapsed
│ │ │ + -- .44406s 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
│ │ │ - -- .340732s elapsed
│ │ │ + -- .272496s 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);
│ │ │ - -- .00470885s elapsed
│ │ │ - -- .0171843s elapsed
│ │ │ - -- .0951582s elapsed
│ │ │ - -- 1.0674s elapsed
│ │ │ - -- .499851s elapsed
│ │ │ - -- .0393332s elapsed
│ │ │ - -- .00669755s elapsed
│ │ │ - -- 6.34327s elapsed
    
│ │ │ + -- .00522795s elapsed
│ │ │ + -- .0216107s elapsed
│ │ │ + -- .108808s elapsed
│ │ │ + -- 1.05801s elapsed
│ │ │ + -- .455304s elapsed
│ │ │ + -- .0410704s elapsed
│ │ │ + -- .00788978s elapsed
│ │ │ + -- 6.06507s 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)
│ │ │ │ - -- .00246423s elapsed
│ │ │ │ - -- .00627075s elapsed
│ │ │ │ - -- .0234075s elapsed
│ │ │ │ - -- .0110125s elapsed
│ │ │ │ - -- .00359968s elapsed
│ │ │ │ - -- .553045s elapsed
│ │ │ │ + -- .00293913s elapsed
│ │ │ │ + -- .00787492s elapsed
│ │ │ │ + -- .0283482s elapsed
│ │ │ │ + -- .0123755s elapsed
│ │ │ │ + -- .00412053s elapsed
│ │ │ │ + -- .44406s 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
│ │ │ │ - -- .340732s elapsed
│ │ │ │ + -- .272496s 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);
│ │ │ │ - -- .00470885s elapsed
│ │ │ │ - -- .0171843s elapsed
│ │ │ │ - -- .0951582s elapsed
│ │ │ │ - -- 1.0674s elapsed
│ │ │ │ - -- .499851s elapsed
│ │ │ │ - -- .0393332s elapsed
│ │ │ │ - -- .00669755s elapsed
│ │ │ │ - -- 6.34327s elapsed
│ │ │ │ + -- .00522795s elapsed
│ │ │ │ + -- .0216107s elapsed
│ │ │ │ + -- .108808s elapsed
│ │ │ │ + -- 1.05801s elapsed
│ │ │ │ + -- .455304s elapsed
│ │ │ │ + -- .0410704s elapsed
│ │ │ │ + -- .00788978s elapsed
│ │ │ │ + -- 6.06507s 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)
│ │ │ - -- .00254853s elapsed
│ │ │ - -- .00629121s elapsed
│ │ │ - -- .0240282s elapsed
│ │ │ - -- .0113519s elapsed
│ │ │ - -- .00368606s elapsed
│ │ │ + -- .00300216s elapsed
│ │ │ + -- .00771659s elapsed
│ │ │ + -- .0291543s elapsed
│ │ │ + -- .0114388s elapsed
│ │ │ + -- .00410029s 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
│ │ │ - -- .247359s elapsed
│ │ │ + -- .236928s 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);
│ │ │ - -- .007547s elapsed
│ │ │ - -- .0336731s elapsed
│ │ │ - -- .11684s elapsed
│ │ │ - -- 1.10155s elapsed
│ │ │ - -- .453186s elapsed
│ │ │ - -- .0395934s elapsed
│ │ │ - -- .00659858s elapsed
│ │ │ - -- 6.96887s elapsed
    
│ │ │ + -- .00713205s elapsed
│ │ │ + -- .0215352s elapsed
│ │ │ + -- .135614s elapsed
│ │ │ + -- 1.05811s elapsed
│ │ │ + -- .427268s elapsed
│ │ │ + -- .0430608s elapsed
│ │ │ + -- .00774745s elapsed
│ │ │ + -- 6.04023s 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)
│ │ │ │ - -- .00254853s elapsed
│ │ │ │ - -- .00629121s elapsed
│ │ │ │ - -- .0240282s elapsed
│ │ │ │ - -- .0113519s elapsed
│ │ │ │ - -- .00368606s elapsed
│ │ │ │ + -- .00300216s elapsed
│ │ │ │ + -- .00771659s elapsed
│ │ │ │ + -- .0291543s elapsed
│ │ │ │ + -- .0114388s elapsed
│ │ │ │ + -- .00410029s 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
│ │ │ │ - -- .247359s elapsed
│ │ │ │ + -- .236928s 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);
│ │ │ │ - -- .007547s elapsed
│ │ │ │ - -- .0336731s elapsed
│ │ │ │ - -- .11684s elapsed
│ │ │ │ - -- 1.10155s elapsed
│ │ │ │ - -- .453186s elapsed
│ │ │ │ - -- .0395934s elapsed
│ │ │ │ - -- .00659858s elapsed
│ │ │ │ - -- 6.96887s elapsed
│ │ │ │ + -- .00713205s elapsed
│ │ │ │ + -- .0215352s elapsed
│ │ │ │ + -- .135614s elapsed
│ │ │ │ + -- 1.05811s elapsed
│ │ │ │ + -- .427268s elapsed
│ │ │ │ + -- .0430608s elapsed
│ │ │ │ + -- .00774745s elapsed
│ │ │ │ + -- 6.04023s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -80,42 +80,42 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00741432s elapsed
│ │ │ - -- .0131122s elapsed
│ │ │ - -- .000340948s elapsed
│ │ │ - -- .000173654s elapsed
│ │ │ - -- .000145061s elapsed
│ │ │ - -- .000136617s elapsed
│ │ │ - -- .00012314s elapsed
│ │ │ - -- .000131686s elapsed
│ │ │ - -- .000164106s elapsed
│ │ │ - -- .000168546s elapsed
│ │ │ - -- .000141084s elapsed
│ │ │ - -- .000132988s elapsed
│ │ │ - -- .000289191s elapsed
│ │ │ - -- .000131376s elapsed
│ │ │ - -- .00012862s elapsed
│ │ │ - -- .000128541s elapsed
│ │ │ - -- .0001334s elapsed
│ │ │ - -- .000151062s elapsed
│ │ │ - -- .000143969s elapsed
│ │ │ - -- .00014432s elapsed
│ │ │ - -- .000151594s elapsed
│ │ │ - -- .000151393s elapsed
│ │ │ - -- .000147766s elapsed
│ │ │ - -- .00013296s elapsed
│ │ │ - -- .000130113s elapsed
│ │ │ - -- .000136976s elapsed
│ │ │ - -- .000136727s elapsed
│ │ │ - -- .000137126s elapsed
│ │ │ + -- .0092257s elapsed
│ │ │ + -- .0126772s elapsed
│ │ │ + -- .000248361s elapsed
│ │ │ + -- .000193208s elapsed
│ │ │ + -- .000191645s elapsed
│ │ │ + -- .000154399s elapsed
│ │ │ + -- .000184512s elapsed
│ │ │ + -- .000208194s elapsed
│ │ │ + -- .000178275s elapsed
│ │ │ + -- .000212982s elapsed
│ │ │ + -- .000186587s elapsed
│ │ │ + -- .000167579s elapsed
│ │ │ + -- .000235189s elapsed
│ │ │ + -- .000199103s elapsed
│ │ │ + -- .00015487s elapsed
│ │ │ + -- .000166657s elapsed
│ │ │ + -- .000168004s elapsed
│ │ │ + -- .000181172s elapsed
│ │ │ + -- .000181567s elapsed
│ │ │ + -- .000182704s elapsed
│ │ │ + -- .000192597s elapsed
│ │ │ + -- .000187697s elapsed
│ │ │ + -- .000183881s elapsed
│ │ │ + -- .000176545s elapsed
│ │ │ + -- .000165162s elapsed
│ │ │ + -- .000172027s elapsed
│ │ │ + -- .000183194s elapsed
│ │ │ + -- .000140133s elapsed
│ │ │  (number Of blocks, 26)
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  1
│ │ │  2
│ │ │   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,42 +19,42 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .00741432s elapsed
│ │ │ │ - -- .0131122s elapsed
│ │ │ │ - -- .000340948s elapsed
│ │ │ │ - -- .000173654s elapsed
│ │ │ │ - -- .000145061s elapsed
│ │ │ │ - -- .000136617s elapsed
│ │ │ │ - -- .00012314s elapsed
│ │ │ │ - -- .000131686s elapsed
│ │ │ │ - -- .000164106s elapsed
│ │ │ │ - -- .000168546s elapsed
│ │ │ │ - -- .000141084s elapsed
│ │ │ │ - -- .000132988s elapsed
│ │ │ │ - -- .000289191s elapsed
│ │ │ │ - -- .000131376s elapsed
│ │ │ │ - -- .00012862s elapsed
│ │ │ │ - -- .000128541s elapsed
│ │ │ │ - -- .0001334s elapsed
│ │ │ │ - -- .000151062s elapsed
│ │ │ │ - -- .000143969s elapsed
│ │ │ │ - -- .00014432s elapsed
│ │ │ │ - -- .000151594s elapsed
│ │ │ │ - -- .000151393s elapsed
│ │ │ │ - -- .000147766s elapsed
│ │ │ │ - -- .00013296s elapsed
│ │ │ │ - -- .000130113s elapsed
│ │ │ │ - -- .000136976s elapsed
│ │ │ │ - -- .000136727s elapsed
│ │ │ │ - -- .000137126s elapsed
│ │ │ │ + -- .0092257s elapsed
│ │ │ │ + -- .0126772s elapsed
│ │ │ │ + -- .000248361s elapsed
│ │ │ │ + -- .000193208s elapsed
│ │ │ │ + -- .000191645s elapsed
│ │ │ │ + -- .000154399s elapsed
│ │ │ │ + -- .000184512s elapsed
│ │ │ │ + -- .000208194s elapsed
│ │ │ │ + -- .000178275s elapsed
│ │ │ │ + -- .000212982s elapsed
│ │ │ │ + -- .000186587s elapsed
│ │ │ │ + -- .000167579s elapsed
│ │ │ │ + -- .000235189s elapsed
│ │ │ │ + -- .000199103s elapsed
│ │ │ │ + -- .00015487s elapsed
│ │ │ │ + -- .000166657s elapsed
│ │ │ │ + -- .000168004s elapsed
│ │ │ │ + -- .000181172s elapsed
│ │ │ │ + -- .000181567s elapsed
│ │ │ │ + -- .000182704s elapsed
│ │ │ │ + -- .000192597s elapsed
│ │ │ │ + -- .000187697s elapsed
│ │ │ │ + -- .000183881s elapsed
│ │ │ │ + -- .000176545s elapsed
│ │ │ │ + -- .000165162s elapsed
│ │ │ │ + -- .000172027s elapsed
│ │ │ │ + -- .000183194s elapsed
│ │ │ │ + -- .000140133s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │  1
│ │ │ │  1
│ │ │ │  1
│ │ │ │  1
│ │ │ │  2
│ │ │ │   2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -85,20 +85,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .280556s elapsed
│ │ │ - -- .274042s elapsed
│ │ │ - -- .207278s elapsed
│ │ │ - -- .343307s elapsed
│ │ │ - -- .344206s elapsed
│ │ │ - -- .319911s elapsed
│ │ │ + -- .18348s elapsed
│ │ │ + -- .213709s elapsed
│ │ │ + -- .145051s elapsed
│ │ │ + -- .259669s elapsed
│ │ │ + -- .18622s elapsed
│ │ │ + -- .180466s 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
│ │ │ │ - -- .280556s elapsed
│ │ │ │ - -- .274042s elapsed
│ │ │ │ - -- .207278s elapsed
│ │ │ │ - -- .343307s elapsed
│ │ │ │ - -- .344206s elapsed
│ │ │ │ - -- .319911s elapsed
│ │ │ │ + -- .18348s elapsed
│ │ │ │ + -- .213709s elapsed
│ │ │ │ + -- .145051s elapsed
│ │ │ │ + -- .259669s elapsed
│ │ │ │ + -- .18622s elapsed
│ │ │ │ + -- .180466s 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)
│ │ │ - -- .00258639s elapsed
│ │ │ - -- .0300927s elapsed
│ │ │ - -- .0243393s elapsed
│ │ │ - -- .00941187s elapsed
│ │ │ - -- .00336785s elapsed
│ │ │ + -- .00268486s elapsed
│ │ │ + -- .00711347s elapsed
│ │ │ + -- .0272806s elapsed
│ │ │ + -- .0108124s elapsed
│ │ │ + -- .00373279s 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)
│ │ │ - -- .252648s elapsed
│ │ │ + -- .240002s 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)
│ │ │ - -- .0028042s elapsed
│ │ │ - -- .00610624s elapsed
│ │ │ - -- .0231543s elapsed
│ │ │ - -- .00939143s elapsed
│ │ │ - -- .00362288s elapsed
│ │ │ + -- .00285302s elapsed
│ │ │ + -- .00703046s elapsed
│ │ │ + -- .0294182s elapsed
│ │ │ + -- .0105433s elapsed
│ │ │ + -- .0050064s 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)
│ │ │ │ - -- .00258639s elapsed
│ │ │ │ - -- .0300927s elapsed
│ │ │ │ - -- .0243393s elapsed
│ │ │ │ - -- .00941187s elapsed
│ │ │ │ - -- .00336785s elapsed
│ │ │ │ + -- .00268486s elapsed
│ │ │ │ + -- .00711347s elapsed
│ │ │ │ + -- .0272806s elapsed
│ │ │ │ + -- .0108124s elapsed
│ │ │ │ + -- .00373279s 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)
│ │ │ │ - -- .252648s elapsed
│ │ │ │ + -- .240002s 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)
│ │ │ │ - -- .0028042s elapsed
│ │ │ │ - -- .00610624s elapsed
│ │ │ │ - -- .0231543s elapsed
│ │ │ │ - -- .00939143s elapsed
│ │ │ │ - -- .00362288s elapsed
│ │ │ │ + -- .00285302s elapsed
│ │ │ │ + -- .00703046s elapsed
│ │ │ │ + -- .0294182s elapsed
│ │ │ │ + -- .0105433s elapsed
│ │ │ │ + -- .0050064s 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
│ │ │ @@ -78,33 +78,33 @@
│ │ │  
│ │ │  o1 = 4
    │ │ │
    │ │ │ 
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0531025s elapsed
│ │ │ - -- .000039765s elapsed
│ │ │ - -- .000091661s elapsed
│ │ │ - -- .000068136s elapsed
│ │ │ - -- .000086272s elapsed
│ │ │ - -- .000098233s elapsed
│ │ │ - -- .000094527s elapsed
│ │ │ - -- .000029445s elapsed
│ │ │ - -- .000075421s elapsed
│ │ │ - -- .000095759s elapsed
│ │ │ - -- .000202318s elapsed
│ │ │ - -- .000092814s elapsed
│ │ │ - -- .000104546s elapsed
│ │ │ - -- .000080951s elapsed
│ │ │ - -- .000075392s elapsed
│ │ │ - -- .00014968s elapsed
│ │ │ - -- .000024105s elapsed
│ │ │ - -- .000066555s elapsed
│ │ │ - -- .000023724s elapsed
│ │ │ + -- .0203277s elapsed
│ │ │ + -- .000058955s elapsed
│ │ │ + -- .000085572s elapsed
│ │ │ + -- .000081125s elapsed
│ │ │ + -- .00008802s elapsed
│ │ │ + -- .0001447s elapsed
│ │ │ + -- .000146772s elapsed
│ │ │ + -- .000051573s elapsed
│ │ │ + -- .000091195s elapsed
│ │ │ + -- .000098701s elapsed
│ │ │ + -- .000110253s elapsed
│ │ │ + -- .000106543s elapsed
│ │ │ + -- .000121952s elapsed
│ │ │ + -- .000107866s elapsed
│ │ │ + -- .000075993s elapsed
│ │ │ + -- .000201568s elapsed
│ │ │ + -- .000052734s elapsed
│ │ │ + -- .000143843s elapsed
│ │ │ + -- .000030082s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,33 +19,33 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0531025s elapsed
│ │ │ │ - -- .000039765s elapsed
│ │ │ │ - -- .000091661s elapsed
│ │ │ │ - -- .000068136s elapsed
│ │ │ │ - -- .000086272s elapsed
│ │ │ │ - -- .000098233s elapsed
│ │ │ │ - -- .000094527s elapsed
│ │ │ │ - -- .000029445s elapsed
│ │ │ │ - -- .000075421s elapsed
│ │ │ │ - -- .000095759s elapsed
│ │ │ │ - -- .000202318s elapsed
│ │ │ │ - -- .000092814s elapsed
│ │ │ │ - -- .000104546s elapsed
│ │ │ │ - -- .000080951s elapsed
│ │ │ │ - -- .000075392s elapsed
│ │ │ │ - -- .00014968s elapsed
│ │ │ │ - -- .000024105s elapsed
│ │ │ │ - -- .000066555s elapsed
│ │ │ │ - -- .000023724s elapsed
│ │ │ │ + -- .0203277s elapsed
│ │ │ │ + -- .000058955s elapsed
│ │ │ │ + -- .000085572s elapsed
│ │ │ │ + -- .000081125s elapsed
│ │ │ │ + -- .00008802s elapsed
│ │ │ │ + -- .0001447s elapsed
│ │ │ │ + -- .000146772s elapsed
│ │ │ │ + -- .000051573s elapsed
│ │ │ │ + -- .000091195s elapsed
│ │ │ │ + -- .000098701s elapsed
│ │ │ │ + -- .000110253s elapsed
│ │ │ │ + -- .000106543s elapsed
│ │ │ │ + -- .000121952s elapsed
│ │ │ │ + -- .000107866s elapsed
│ │ │ │ + -- .000075993s elapsed
│ │ │ │ + -- .000201568s elapsed
│ │ │ │ + -- .000052734s elapsed
│ │ │ │ + -- .000143843s elapsed
│ │ │ │ + -- .000030082s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ ├── ./usr/share/doc/Macaulay2/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 0.00799269s (cpu); 0.00932259s (thread); 0s (gc)
│ │ │ + -- used 0.00799581s (cpu); 0.0100649s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0247692s (cpu); 0.028421s (thread); 0s (gc)
│ │ │ + -- used 0.0283456s (cpu); 0.0300074s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107589s (cpu); 0.108842s (thread); 0s (gc)
│ │ │ + -- used 0.114081s (cpu); 0.115989s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0105352s (cpu); 0.0122315s (thread); 0s (gc)
│ │ │ + -- used 0.00986948s (cpu); 0.0128073s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0467487s (cpu); 0.0477321s (thread); 0s (gc)
│ │ │ + -- used 0.0608397s (cpu); 0.0632076s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0588381s (cpu); 0.0597089s (thread); 0s (gc)
│ │ │ + -- used 0.0613066s (cpu); 0.0627295s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.362082s (cpu); 0.364588s (thread); 0s (gc)
│ │ │ + -- used 0.331431s (cpu); 0.3337s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11273s (cpu); 0.114812s (thread); 0s (gc)
│ │ │ + -- used 0.148516s (cpu); 0.150758s (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 0.00799269s (cpu); 0.00932259s (thread); 0s (gc)
│ │ │ + -- used 0.00799581s (cpu); 0.0100649s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0247692s (cpu); 0.028421s (thread); 0s (gc)
│ │ │ + -- used 0.0283456s (cpu); 0.0300074s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107589s (cpu); 0.108842s (thread); 0s (gc)
│ │ │ + -- used 0.114081s (cpu); 0.115989s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0105352s (cpu); 0.0122315s (thread); 0s (gc)
│ │ │ + -- used 0.00986948s (cpu); 0.0128073s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0467487s (cpu); 0.0477321s (thread); 0s (gc)
│ │ │ + -- used 0.0608397s (cpu); 0.0632076s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0588381s (cpu); 0.0597089s (thread); 0s (gc)
│ │ │ + -- used 0.0613066s (cpu); 0.0627295s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.362082s (cpu); 0.364588s (thread); 0s (gc)
│ │ │ + -- used 0.331431s (cpu); 0.3337s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11273s (cpu); 0.114812s (thread); 0s (gc)
│ │ │ + -- used 0.148516s (cpu); 0.150758s (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 0.00799269s (cpu); 0.00932259s (thread); 0s (gc) │ │ │ │ + -- used 0.00799581s (cpu); 0.0100649s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o3 : Matrix ZZ <-- ZZ │ │ │ │ i4 : time LLL(m, Strategy=>CohenEngine); │ │ │ │ - -- used 0.0247692s (cpu); 0.028421s (thread); 0s (gc) │ │ │ │ + -- used 0.0283456s (cpu); 0.0300074s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : time LLL(m, Strategy=>CohenTopLevel); │ │ │ │ - -- used 0.107589s (cpu); 0.108842s (thread); 0s (gc) │ │ │ │ + -- used 0.114081s (cpu); 0.115989s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o5 : Matrix ZZ <-- ZZ │ │ │ │ i6 : time LLL(m, Strategy=>{Givens,RealFP}); │ │ │ │ - -- used 0.0105352s (cpu); 0.0122315s (thread); 0s (gc) │ │ │ │ + -- used 0.00986948s (cpu); 0.0128073s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o6 : Matrix ZZ <-- ZZ │ │ │ │ i7 : time LLL(m, Strategy=>{Givens,RealQP}); │ │ │ │ - -- used 0.0467487s (cpu); 0.0477321s (thread); 0s (gc) │ │ │ │ + -- used 0.0608397s (cpu); 0.0632076s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o7 : Matrix ZZ <-- ZZ │ │ │ │ i8 : time LLL(m, Strategy=>{Givens,RealXD}); │ │ │ │ - -- used 0.0588381s (cpu); 0.0597089s (thread); 0s (gc) │ │ │ │ + -- used 0.0613066s (cpu); 0.0627295s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o8 : Matrix ZZ <-- ZZ │ │ │ │ i9 : time LLL(m, Strategy=>{Givens,RealRR}); │ │ │ │ - -- used 0.362082s (cpu); 0.364588s (thread); 0s (gc) │ │ │ │ + -- used 0.331431s (cpu); 0.3337s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o9 : Matrix ZZ <-- ZZ │ │ │ │ i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP}); │ │ │ │ - -- used 0.11273s (cpu); 0.114812s (thread); 0s (gc) │ │ │ │ + -- used 0.148516s (cpu); 0.150758s (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 0.727086s (cpu); 0.51562s (thread); 0s (gc) │ │ │ + -- used 1.0863s (cpu); 0.537947s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.337625s (cpu); 0.255335s (thread); 0s (gc) │ │ │ + -- used 0.663042s (cpu); 0.309423s (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 0.727086s (cpu); 0.51562s (thread); 0s (gc)
    │ │ │ + -- used 1.0863s (cpu); 0.537947s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.337625s (cpu); 0.255335s (thread); 0s (gc)
    │ │ │ + -- used 0.663042s (cpu); 0.309423s (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 0.727086s (cpu); 0.51562s (thread); 0s (gc) │ │ │ │ + -- used 1.0863s (cpu); 0.537947s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.337625s (cpu); 0.255335s (thread); 0s (gc) │ │ │ │ + -- used 0.663042s (cpu); 0.309423s (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) │ │ │ - -- .133995s elapsed │ │ │ + -- .0970661s 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}}) │ │ │ - -- .022496s elapsed │ │ │ + -- .015139s 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) │ │ │ - -- 3.45673s elapsed │ │ │ + -- 2.73874s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .0418014s elapsed │ │ │ + -- .0301798s 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)
│ │ │ - -- .133995s elapsed
│ │ │ + -- .0970661s 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}})
│ │ │ - -- .022496s elapsed
│ │ │ + -- .015139s 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) │ │ │ │ - -- .133995s elapsed │ │ │ │ + -- .0970661s 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}}) │ │ │ │ - -- .022496s elapsed │ │ │ │ + -- .015139s 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)
│ │ │ - -- 3.45673s elapsed
│ │ │ + -- 2.73874s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0418014s elapsed
│ │ │ + -- .0301798s 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) │ │ │ │ - -- 3.45673s elapsed │ │ │ │ + -- 2.73874s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .0418014s elapsed │ │ │ │ + -- .0301798s 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) │ │ │ - -- .183201s elapsed │ │ │ + -- .207308s 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 │ │ │ - -- .0124109s elapsed │ │ │ + -- .0153582s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .282257s elapsed │ │ │ + -- .261206s 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)
│ │ │ - -- .183201s elapsed
│ │ │ + -- .207308s 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
│ │ │ - -- .0124109s elapsed
│ │ │ + -- .0153582s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .282257s elapsed
│ │ │ + -- .261206s 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) │ │ │ │ - -- .183201s elapsed │ │ │ │ + -- .207308s 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 │ │ │ │ - -- .0124109s elapsed │ │ │ │ + -- .0153582s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .282257s elapsed │ │ │ │ + -- .261206s 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 │ │ │ -Mon Aug 25 12:29:34 UTC 2025 │ │ │ +Sat Oct 4 04:41:44 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/M2-11848-0/0.dbm │ │ │ +o1 = /tmp/M2-13438-0/0.dbm │ │ │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ -o2 = /tmp/M2-11848-0/0.dbm │ │ │ +o2 = /tmp/M2-13438-0/0.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" => 42493534762 } │ │ │ +o1 = HashTable{"bytesAlloc" => 42596709034 } │ │ │ "GC_free_space_divisor" => 3 │ │ │ "GC_LARGE_ALLOC_WARN_INTERVAL" => 1 │ │ │ "gcCpuTimeSecs" => 0 │ │ │ - "heapSize" => 207200256 │ │ │ - "numGCs" => 793 │ │ │ - "numGCThreads" => 6 │ │ │ + "heapSize" => 224821248 │ │ │ + "numGCs" => 782 │ │ │ + "numGCThreads" => 16 │ │ │ │ │ │ o1 : HashTable │ │ │ │ │ │ i2 : s#"heapSize" │ │ │ │ │ │ -o2 = 207200256 │ │ │ +o2 = 224821248 │ │ │ │ │ │ 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 0.00985475s (cpu); 0.00985013s (thread); 0s (gc) │ │ │ + -- used 0.00599737s (cpu); 0.00599416s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of R │ │ │ │ │ │ i9 : time K = truncate(8, I, MinimalGenerators => true); │ │ │ - -- used 0.0817304s (cpu); 0.0817367s (thread); 0s (gc) │ │ │ + -- used 0.057939s (cpu); 0.0579513s (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 0.0304881s (cpu); 0.0304864s (thread); 0s (gc) │ │ │ + -- used 0.0472629s (cpu); 0.0472646s (thread); 0s (gc) │ │ │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ - -- used 0.0321648s (cpu); 0.0321706s (thread); 0s (gc) │ │ │ + -- used 0.0458996s (cpu); 0.0459103s (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 0.00194988s (cpu); 0.00190997s (thread); 0s (gc) │ │ │ + -- used 0.00173968s (cpu); 0.00173148s (thread); 0s (gc) │ │ │ │ │ │ 3 6 15 18 6 │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ @@ -14,10 +14,10 @@ │ │ │ │ │ │ i4 : peek read f │ │ │ │ │ │ o4 = "hi there" │ │ │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ -o5 = false │ │ │ +o5 = true │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1330379359420 │ │ │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ -o1 = .0002894243684497773 │ │ │ +o1 = .0003340345282869781 │ │ │ │ │ │ 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' │ │ │ - -- .00525587s elapsed │ │ │ + -- .00403624s elapsed │ │ │ │ │ │ o4 = 3 │ │ │ │ │ │ i5 : elapsedTime pdim' M │ │ │ - -- .000001342s elapsed │ │ │ + -- .000001518s 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 = 722293 │ │ │ +o5 = 1106263 │ │ │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ o6 = 0 │ │ │ │ │ │ i7 : t │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ i8 : n │ │ │ │ │ │ -o8 = 1464540 │ │ │ +o8 = 2232431 │ │ │ │ │ │ i9 : isReady t │ │ │ │ │ │ o9 = false │ │ │ │ │ │ i10 : cancelTask t │ │ │ │ │ │ @@ -53,22 +53,22 @@ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ i13 : n │ │ │ │ │ │ -o13 = 1464775 │ │ │ +o13 = 2232776 │ │ │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ o14 = 0 │ │ │ │ │ │ i15 : n │ │ │ │ │ │ -o15 = 1464775 │ │ │ +o15 = 2232776 │ │ │ │ │ │ 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/M2-10670-0/0 │ │ │ +o1 = /tmp/M2-11037-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10670-0/0 │ │ │ +o2 = /tmp/M2-11037-0/0 │ │ │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ -o3 = /tmp/M2-10670-0/0/ │ │ │ +o3 = /tmp/M2-11037-0/0/ │ │ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ -o4 = /tmp/M2-10670-0/0/ │ │ │ +o4 = /tmp/M2-11037-0/0/ │ │ │ │ │ │ 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") -- .15275s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .113525s elapsed │ │ │ │ │ │ i3 : check FirstPackage │ │ │ - -- capturing check(0, "FirstPackage") -- .149336s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .149216s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .11506s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .114414s elapsed │ │ │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ - -- capturing check(1, "FirstPackage") -- .150392s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .113545s elapsed │ │ │ │ │ │ i5 : check "FirstPackage" │ │ │ - -- capturing check(0, "FirstPackage") -- .152341s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .230863s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .11866s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .11878s 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") -- .234646s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .112802s 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") -- .228259s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .228697s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .112478s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .110078s 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") -- .229285s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .110789s elapsed │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ @@ -1,26 +1,26 @@ │ │ │ -- -*- M2-comint -*- hash: 10986518019608335719 │ │ │ │ │ │ i1 : run "uname -a" │ │ │ -Linux sbuild 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-08-02) x86_64 GNU/Linux │ │ │ +Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) x86_64 GNU/Linux │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close │ │ │ ba │ │ │ ad │ │ │ │ │ │ o2 = !grep a │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : peek get "!uname -a" │ │ │ │ │ │ -o3 = "Linux sbuild 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 │ │ │ - (2025-08-02) x86_64 GNU/Linux\n" │ │ │ +o3 = "Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ + 6.12.48-1 (2025-09-20) x86_64 GNU/Linux\n" │ │ │ │ │ │ i4 : f = openInOut "!egrep '^in'" │ │ │ │ │ │ o4 = !egrep '^in' │ │ │ │ │ │ o4 : File │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ @@ -126,15 +126,15 @@ │ │ │ │ │ │ ZZ │ │ │ o23 : Ideal of ----[x..z, w] │ │ │ 1277 │ │ │ │ │ │ i24 : gb I │ │ │ │ │ │ - -- registering gb 5 at 0x7fcf2c0f3380 │ │ │ + -- registering gb 5 at 0x7f00d2f29380 │ │ │ │ │ │ -- [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 0.279998s (cpu); 0.283653s (thread); 0s (gc) │ │ │ + -- used 0.195656s (cpu); 0.196903s (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 0.00799689s (cpu); 0.00497973s (thread); 0s (gc) │ │ │ + -- used 0.00399011s (cpu); 0.00308441s (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/M2-11392-0/0/ │ │ │ +o1 = /tmp/M2-12502-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11392-0/1/ │ │ │ +o2 = /tmp/M2-12502-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11392-0/0/a/ │ │ │ +o3 = /tmp/M2-12502-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11392-0/0/b/ │ │ │ +o4 = /tmp/M2-12502-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11392-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12502-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11392-0/0/a/f │ │ │ +o6 = /tmp/M2-12502-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11392-0/0/a/g │ │ │ +o7 = /tmp/M2-12502-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11392-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12502-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : stack findFiles src │ │ │ │ │ │ -o9 = /tmp/M2-11392-0/0/ │ │ │ - /tmp/M2-11392-0/0/b/ │ │ │ - /tmp/M2-11392-0/0/b/c/ │ │ │ - /tmp/M2-11392-0/0/b/c/g │ │ │ - /tmp/M2-11392-0/0/a/ │ │ │ - /tmp/M2-11392-0/0/a/g │ │ │ - /tmp/M2-11392-0/0/a/f │ │ │ +o9 = /tmp/M2-12502-0/0/ │ │ │ + /tmp/M2-12502-0/0/a/ │ │ │ + /tmp/M2-12502-0/0/a/g │ │ │ + /tmp/M2-12502-0/0/a/f │ │ │ + /tmp/M2-12502-0/0/b/ │ │ │ + /tmp/M2-12502-0/0/b/c/ │ │ │ + /tmp/M2-12502-0/0/b/c/g │ │ │ │ │ │ i10 : copyDirectory(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11392-0/0/b/c/g -> /tmp/M2-11392-0/1/b/c/g │ │ │ - -- copying: /tmp/M2-11392-0/0/a/g -> /tmp/M2-11392-0/1/a/g │ │ │ - -- copying: /tmp/M2-11392-0/0/a/f -> /tmp/M2-11392-0/1/a/f │ │ │ + -- copying: /tmp/M2-12502-0/0/a/g -> /tmp/M2-12502-0/1/a/g │ │ │ + -- copying: /tmp/M2-12502-0/0/a/f -> /tmp/M2-12502-0/1/a/f │ │ │ + -- copying: /tmp/M2-12502-0/0/b/c/g -> /tmp/M2-12502-0/1/b/c/g │ │ │ │ │ │ i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11392-0/0/b/c/g not newer than /tmp/M2-11392-0/1/b/c/g │ │ │ - -- skipping: /tmp/M2-11392-0/0/a/g not newer than /tmp/M2-11392-0/1/a/g │ │ │ - -- skipping: /tmp/M2-11392-0/0/a/f not newer than /tmp/M2-11392-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12502-0/0/a/g not newer than /tmp/M2-12502-0/1/a/g │ │ │ + -- skipping: /tmp/M2-12502-0/0/a/f not newer than /tmp/M2-12502-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12502-0/0/b/c/g not newer than /tmp/M2-12502-0/1/b/c/g │ │ │ │ │ │ i12 : stack findFiles dst │ │ │ │ │ │ -o12 = /tmp/M2-11392-0/1/ │ │ │ - /tmp/M2-11392-0/1/a/ │ │ │ - /tmp/M2-11392-0/1/a/f │ │ │ - /tmp/M2-11392-0/1/a/g │ │ │ - /tmp/M2-11392-0/1/b/ │ │ │ - /tmp/M2-11392-0/1/b/c/ │ │ │ - /tmp/M2-11392-0/1/b/c/g │ │ │ +o12 = /tmp/M2-12502-0/1/ │ │ │ + /tmp/M2-12502-0/1/a/ │ │ │ + /tmp/M2-12502-0/1/a/g │ │ │ + /tmp/M2-12502-0/1/a/f │ │ │ + /tmp/M2-12502-0/1/b/ │ │ │ + /tmp/M2-12502-0/1/b/c/ │ │ │ + /tmp/M2-12502-0/1/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/M2-11177-0/0 │ │ │ +o1 = /tmp/M2-12067-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-11177-0/1 │ │ │ +o2 = /tmp/M2-12067-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11177-0/0 │ │ │ +o3 = /tmp/M2-12067-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : copyFile(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11177-0/0 -> /tmp/M2-11177-0/1 │ │ │ + -- copying: /tmp/M2-12067-0/0 -> /tmp/M2-12067-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1 │ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1 │ │ │ │ │ │ i7 : src << "ho there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11177-0/0 │ │ │ +o7 = /tmp/M2-12067-0/0 │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1 │ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1 │ │ │ │ │ │ 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 = 365.096465098 │ │ │ +o1 = 294.147154702 │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ │ │ i2 : for i from 0 to 1000000 do 223131321321*324234324324; │ │ │ │ │ │ i3 : t2 = cpuTime() │ │ │ │ │ │ -o3 = 367.292409687 │ │ │ +o3 = 295.012007467 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : t2-t1 │ │ │ │ │ │ -o4 = 2.195944589000021 │ │ │ +o4 = .8648527650000233 │ │ │ │ │ │ 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 = 1756125053 │ │ │ +o1 = 1759552954 │ │ │ │ │ │ i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 ) │ │ │ │ │ │ -o2 = 55.64938758977737 │ │ │ +o2 = 55.75801344819361 │ │ │ │ │ │ o2 : RR (of precision 53) │ │ │ │ │ │ i3 : 12 * (oo - floor oo) │ │ │ │ │ │ -o3 = 7.792651077328429 │ │ │ +o3 = 9.096161378323302 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : run "date" │ │ │ -Mon Aug 25 12:30:53 UTC 2025 │ │ │ +Sat Oct 4 04:42:34 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 │ │ │ - -- 1.00011s elapsed │ │ │ + -- 1.00022s 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 │ │ │ - -- 1.00017 seconds │ │ │ + -- 1.00013 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{1.00017, 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 0.528053s (cpu); 0.289697s (thread); 0s (gc) │ │ │ + -- used 0.131402s (cpu); 0.131402s (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 0.796872s (cpu); 0.31766s (thread); 0s (gc) │ │ │ + -- used 0.344631s (cpu); 0.174895s (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 0.0493859s (cpu); 0.0493947s (thread); 0s (gc) │ │ │ + -- used 0.0321462s (cpu); 0.0321482s (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 0.447427s (cpu); 0.216815s (thread); 0s (gc) │ │ │ + -- used 0.105021s (cpu); 0.105027s (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 0.00185985s (cpu); 0.00186044s (thread); 0s (gc) │ │ │ + -- used 0.00157095s (cpu); 0.00156769s (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 0.321427s (cpu); 0.109696s (thread); 0s (gc) │ │ │ + -- used 0.0386393s (cpu); 0.0386488s (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/M2-10398-0/91-rundir/ │ │ │ + source directory => /tmp/M2-10518-0/91-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/M2-10765-0/0 │ │ │ +o1 = /tmp/M2-11232-0/0 │ │ │ │ │ │ i2 : fileExists fn │ │ │ │ │ │ o2 = false │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10765-0/0 │ │ │ +o3 = /tmp/M2-11232-0/0 │ │ │ │ │ │ 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/M2-12338-0/0 │ │ │ +o1 = /tmp/M2-14438-0/0 │ │ │ │ │ │ o1 : File │ │ │ │ │ │ i2 : fileLength f │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : close f │ │ │ │ │ │ -o3 = /tmp/M2-12338-0/0 │ │ │ +o3 = /tmp/M2-14438-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : filename = toString f │ │ │ │ │ │ -o4 = /tmp/M2-12338-0/0 │ │ │ +o4 = /tmp/M2-14438-0/0 │ │ │ │ │ │ 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/M2-11582-0/0 │ │ │ +o1 = /tmp/M2-12892-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-11582-0/0 │ │ │ +o2 = /tmp/M2-12892-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fileMode f │ │ │ │ │ │ o3 = 420 │ │ │ │ │ │ i4 : close f │ │ │ │ │ │ -o4 = /tmp/M2-11582-0/0 │ │ │ +o4 = /tmp/M2-12892-0/0 │ │ │ │ │ │ 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/M2-11196-0/0 │ │ │ +o1 = /tmp/M2-12106-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-11196-0/0 │ │ │ +o2 = /tmp/M2-12106-0/0 │ │ │ │ │ │ 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/M2-11061-0/0 │ │ │ +o1 = /tmp/M2-11831-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-11061-0/0 │ │ │ +o2 = /tmp/M2-11831-0/0 │ │ │ │ │ │ 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/M2-11061-0/0 │ │ │ +o6 = /tmp/M2-11831-0/0 │ │ │ │ │ │ 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/M2-12184-0/0 │ │ │ +o1 = /tmp/M2-14124-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12184-0/0 │ │ │ +o2 = /tmp/M2-14124-0/0 │ │ │ │ │ │ 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 = 68 │ │ │ +o1 = 46 │ │ │ │ │ │ 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 0x7f500f309a80 │ │ │ + -- registering gb 0 at 0x7fc81f326a80 │ │ │ │ │ │ -- [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 = Mon Aug 25 12:30:02 UTC 2025 │ │ │ +o4 = Sat Oct 4 04:42:02 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/M2-10587-0/0 │ │ │ +o2 = /tmp/M2-10874-0/0 │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10587-0/0 │ │ │ +o3 = /tmp/M2-10874-0/0 │ │ │ │ │ │ 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) │ │ │ - -- 4.27565s elapsed │ │ │ + -- 5.29813s 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 │ │ │ - -- .0554294s elapsed │ │ │ + -- .059371s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isPseudoprime m3 │ │ │ - -- .000132769s elapsed │ │ │ + -- .000157637s 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/M2-12376-0/0 │ │ │ +o1 = /tmp/M2-14516-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12376-0/0 │ │ │ +o2 = /tmp/M2-14516-0/0 │ │ │ │ │ │ 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/M2-10929-0/0 │ │ │ +o1 = /tmp/M2-11559-0/0 │ │ │ │ │ │ i2 : makeDirectory (dir|"/a/b/c") │ │ │ │ │ │ -o2 = /tmp/M2-10929-0/0/a/b/c │ │ │ +o2 = /tmp/M2-11559-0/0/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 1.61151s (cpu); 1.13816s (thread); 0s (gc) │ │ │ + -- used 0.842547s (cpu); 0.640172s (thread); 0s (gc) │ │ │ │ │ │ o2 = 514229 │ │ │ │ │ │ i3 : fib = memoize fib │ │ │ │ │ │ o3 = fib │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ │ │ i4 : time fib 28 │ │ │ - -- used 9.631e-05s (cpu); 9.588e-05s (thread); 0s (gc) │ │ │ + -- used 6.8982e-05s (cpu); 6.5623e-05s (thread); 0s (gc) │ │ │ │ │ │ o4 = 514229 │ │ │ │ │ │ i5 : time fib 28 │ │ │ - -- used 5.581e-06s (cpu); 5.109e-06s (thread); 0s (gc) │ │ │ + -- used 3.27e-06s (cpu); 3.035e-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 │ │ │ @@ -4,32 +4,32 @@ │ │ │ │ │ │ o1 = {0 => (==, BettiTally, BettiTally) } │ │ │ {1 => (++, BettiTally, BettiTally) } │ │ │ {2 => (**, BettiTally, BettiTally) } │ │ │ {3 => (SPACE, BettiTally, Array) } │ │ │ {4 => (SPACE, BettiTally, ZZ) } │ │ │ {5 => (lift, BettiTally, ZZ) } │ │ │ - {6 => (*, ZZ, BettiTally) } │ │ │ - {7 => (*, QQ, BettiTally) } │ │ │ + {6 => (*, QQ, BettiTally) } │ │ │ + {7 => (*, ZZ, BettiTally) } │ │ │ {8 => (multigraded, BettiTally) } │ │ │ {9 => (net, BettiTally) } │ │ │ {10 => (texMath, BettiTally) } │ │ │ {11 => (betti, BettiTally) } │ │ │ {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 => (codim, BettiTally) } │ │ │ - {21 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ - {22 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ - {23 => (dual, BettiTally) } │ │ │ + {20 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ + {21 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ + {22 => (dual, BettiTally) } │ │ │ + {23 => (codim, BettiTally) } │ │ │ {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)} │ │ │ {25 => (^, Ring, BettiTally) } │ │ │ │ │ │ o1 : NumberedVerticalList │ │ │ │ │ │ i2 : methods resolution │ │ │ │ │ │ @@ -60,20 +60,20 @@ │ │ │ {1 => (++, Module, GradedModule)} │ │ │ {2 => (++, Module, Module) } │ │ │ │ │ │ o4 : NumberedVerticalList │ │ │ │ │ │ i5 : methods( Matrix, Matrix ) │ │ │ │ │ │ -o5 = {0 => (+, Matrix, Matrix) } │ │ │ - {1 => (diff, Matrix, Matrix) } │ │ │ - {2 => (-, Matrix, Matrix) } │ │ │ - {3 => (contract', Matrix, Matrix) } │ │ │ - {4 => (diff', Matrix, Matrix) } │ │ │ - {5 => (contract, Matrix, Matrix) } │ │ │ +o5 = {0 => (contract', Matrix, Matrix) } │ │ │ + {1 => (+, Matrix, Matrix) } │ │ │ + {2 => (contract, Matrix, Matrix) } │ │ │ + {3 => (diff', Matrix, Matrix) } │ │ │ + {4 => (-, Matrix, Matrix) } │ │ │ + {5 => (diff, Matrix, Matrix) } │ │ │ {6 => (markedGB, Matrix, Matrix) } │ │ │ {7 => (Hom, Matrix, Matrix) } │ │ │ {8 => (==, Matrix, Matrix) } │ │ │ {9 => (*, Matrix, Matrix) } │ │ │ {10 => (|, Matrix, Matrix) } │ │ │ {11 => (||, Matrix, Matrix) } │ │ │ {12 => (subquotient, Matrix, Matrix) } │ │ │ @@ -84,21 +84,21 @@ │ │ │ {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 => (tensor, Matrix, Matrix) } │ │ │ - {29 => (intersect, Matrix, Matrix) } │ │ │ - {30 => (intersect, Matrix, Matrix, Matrix, Matrix) } │ │ │ + {28 => (intersect, Matrix, Matrix) } │ │ │ + {29 => (intersect, Matrix, Matrix, Matrix, Matrix) } │ │ │ + {30 => (tensor, Matrix, Matrix) } │ │ │ {31 => (pullback, 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) } │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i2 : S = ring I │ │ │ │ │ │ o2 = S │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ │ │ i3 : elapsedTime C = minimalBetti I │ │ │ - -- 2.06644s elapsed │ │ │ + -- 2.20206s 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) │ │ │ - -- .754668s elapsed │ │ │ + -- .906559s 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) │ │ │ - -- .0311935s elapsed │ │ │ + -- .0386584s 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) │ │ │ - -- 1.22721s elapsed │ │ │ + -- 1.5132s 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/M2-10948-0/0/ │ │ │ +o1 = /tmp/M2-11598-0/0/ │ │ │ │ │ │ i2 : mkdir p │ │ │ │ │ │ i3 : isDirectory p │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : (fn = p | "foo") << "hi there" << close │ │ │ │ │ │ -o4 = /tmp/M2-10948-0/0/foo │ │ │ +o4 = /tmp/M2-11598-0/0/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/M2-10822-0/0 │ │ │ +o1 = /tmp/M2-11349-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10822-0/1 │ │ │ +o2 = /tmp/M2-11349-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10822-0/0 │ │ │ +o3 = /tmp/M2-11349-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : moveFile(src,dst,Verbose=>true) │ │ │ ---moving: /tmp/M2-10822-0/0 -> /tmp/M2-10822-0/1 │ │ │ +--moving: /tmp/M2-11349-0/0 -> /tmp/M2-11349-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : bak = moveFile(dst,Verbose=>true) │ │ │ ---backup file created: /tmp/M2-10822-0/1.bak │ │ │ +--backup file created: /tmp/M2-11349-0/1.bak │ │ │ │ │ │ -o6 = /tmp/M2-10822-0/1.bak │ │ │ +o6 = /tmp/M2-11349-0/1.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 │ │ │ - -- .500143s elapsed │ │ │ + -- .500127s elapsed │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ @@ -5,26 +5,26 @@ │ │ │ o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : L = random toList (1..10000); │ │ │ │ │ │ i3 : elapsedTime apply(1..100, n -> sort L); │ │ │ - -- .701868s elapsed │ │ │ + -- .698851s elapsed │ │ │ │ │ │ i4 : elapsedTime parallelApply(1..100, n -> sort L); │ │ │ - -- .292671s elapsed │ │ │ + -- .102976s 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) │ │ ├── ./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 │ │ │ - -- 2.20012s elapsed │ │ │ + -- 2.16146s 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) │ │ │ - -- 1.79946s elapsed │ │ │ + -- 2.15376s elapsed │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ o6 = total: 1 35 140 385 819 1080 819 385 140 35 1 │ │ │ 0: 1 . . . . . . . . . . │ │ │ 1: . 35 140 189 84 . . . . . . │ │ │ 2: . . . 196 735 1080 735 196 . . . │ │ │ 3: . . . . . . 84 189 140 35 . │ │ │ @@ -105,15 +105,15 @@ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : numTBBThreads = 1 │ │ │ │ │ │ o8 = 1 │ │ │ │ │ │ i9 : elapsedTime minimalBetti(I) │ │ │ - -- 2.58082s elapsed │ │ │ + -- 2.20319s 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) │ │ │ - -- 2.04482s elapsed │ │ │ + -- 4.81958s 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) │ │ │ - -- 1.96083s elapsed │ │ │ + -- 2.44598s 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"); │ │ │ - -- 4.75265s elapsed │ │ │ + -- 3.70385s 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"); │ │ │ - -- 6.95852s elapsed │ │ │ + -- 8.07595s 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"); │ │ │ - -- 4.85792s elapsed │ │ │ + -- 3.05118s 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); │ │ │ - -- 1.1079s elapsed │ │ │ + -- .909785s 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); │ │ │ - -- 1.14666s elapsed │ │ │ + -- 1.41368s 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 0.00262431s (cpu); 1.0961e-05s (thread); 0s (gc) │ │ │ + -- used 0.00183579s (cpu); 1.1938e-05s (thread); 0s (gc) │ │ │ │ │ │ 3 6 9 │ │ │ o28 = 1 - 3T + 3T - T │ │ │ │ │ │ o28 : ZZ[T] │ │ │ │ │ │ i29 : time gens gb I; │ │ │ │ │ │ - -- registering gb 19 at 0x7f6037679540 │ │ │ + -- registering gb 19 at 0x7f5ed9ee2540 │ │ │ │ │ │ -- [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 0.00936292s (cpu); 0.0130541s (thread); 0s (gc) │ │ │ + -- -- used 0.0101486s (cpu); 0.0136747s (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 0x7f6037679380 │ │ │ + -- registering gb 20 at 0x7f5ed9ee2380 │ │ │ │ │ │ -- [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 0x7f60376791c0 │ │ │ + -- registering gb 21 at 0x7f5ed9ee21c0 │ │ │ │ │ │ -- [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 0.00799987s (cpu); 0.00562168s (thread); 0s (gc) │ │ │ + -- -- used 0.00798882s (cpu); 0.00553421s (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 0x7f6037679000 │ │ │ + -- registering gb 22 at 0x7f5ed9ee2000 │ │ │ │ │ │ -- [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 0.0479977s (cpu); 0.0471216s (thread); 0s (gc) │ │ │ + -- -- used 0.051934s (cpu); 0.0530726s (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/M2-11139-0/0 │ │ │ +o3 = /tmp/M2-11989-0/0 │ │ │ │ │ │ i4 : fn << "hi there" << endl << close │ │ │ │ │ │ -o4 = /tmp/M2-11139-0/0 │ │ │ +o4 = /tmp/M2-11989-0/0 │ │ │ │ │ │ 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/M2-11139-0/0 │ │ │ +o9 = /tmp/M2-11989-0/0 │ │ │ │ │ │ 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/M2-11139-0/0 │ │ │ +o11 = /tmp/M2-11989-0/0 │ │ │ │ │ │ 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 = 10398 │ │ │ +o1 = 10518 │ │ │ │ │ │ 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 94.58 ../../m2/matrix1.m2:279:4-282:58 │ │ │ - 1 92.01 ../../m2/matrix1.m2:281:22-281:43 │ │ │ - 1 45.11 ../../m2/matrix1.m2:193:25-193:52 │ │ │ - 1 31.76 ../../m2/matrix1.m2:114:5-156:72 │ │ │ - 1 30.58 ../../m2/matrix1.m2:140:10-155:16 │ │ │ - 1 22.63 ../../m2/matrix1.m2:181:4-181:42 │ │ │ - 1 21.4 ../../m2/matrix1.m2:45:10-49:22 │ │ │ - 1 21.26 ../../m2/set.m2:122:5-122:61 │ │ │ - 1 3.49 ../../m2/matrix1.m2:112:5-112:29 │ │ │ - 1 2.76 ../../m2/matrix1.m2:141:13-141:78 │ │ │ - 1 2.11 ../../m2/matrix1.m2:96:5-109:11 │ │ │ - 1 1.57 ../../m2/matrix1.m2:281:7-281:16 │ │ │ - 1 1.5 ../../m2/matrix1.m2:147:20-147:64 │ │ │ - 1 1.26 ../../m2/matrix1.m2:276:4-277:73 │ │ │ - 1 1.15 ../../m2/matrix1.m2:111:5-111:91 │ │ │ - 1 1.09 ../../m2/matrix1.m2:98:10-98:46 │ │ │ - 1 1.08 ../../m2/matrix1.m2:182:4-184:74 │ │ │ - 1 .76 ../../m2/modules.m2:278:4-278:52 │ │ │ - 20 .6 ../../m2/matrix1.m2:191:14-192:67 │ │ │ - 20 .5 ../../m2/matrix1.m2:47:43-47:71 │ │ │ - 1 .0035s elapsed total │ │ │ + 1 97.29 ../../m2/matrix1.m2:279:4-282:58 │ │ │ + 1 94.92 ../../m2/matrix1.m2:281:22-281:43 │ │ │ + 1 48.2 ../../m2/matrix1.m2:193:25-193:52 │ │ │ + 1 35.92 ../../m2/matrix1.m2:114:5-156:72 │ │ │ + 1 34.27 ../../m2/matrix1.m2:140:10-155:16 │ │ │ + 1 25.83 ../../m2/matrix1.m2:45:10-49:22 │ │ │ + 1 21.83 ../../m2/matrix1.m2:181:4-181:42 │ │ │ + 1 20.52 ../../m2/set.m2:122:5-122:61 │ │ │ + 1 3.1 ../../m2/matrix1.m2:112:5-112:29 │ │ │ + 1 2.16 ../../m2/matrix1.m2:141:13-141:78 │ │ │ + 1 2 ../../m2/matrix1.m2:96:5-109:11 │ │ │ + 1 1.46 ../../m2/matrix1.m2:147:20-147:64 │ │ │ + 1 1.38 ../../m2/matrix1.m2:281:7-281:16 │ │ │ + 1 1.18 ../../m2/matrix1.m2:276:4-277:73 │ │ │ + 1 1.03 ../../m2/matrix1.m2:182:4-184:74 │ │ │ + 1 1.02 ../../m2/matrix1.m2:98:10-98:46 │ │ │ + 1 1.01 ../../m2/matrix1.m2:111:5-111:91 │ │ │ + 1 .63 ../../m2/modules.m2:278:4-278:52 │ │ │ + 20 .49 ../../m2/matrix1.m2:191:14-192:67 │ │ │ + 20 .36 ../../m2/matrix1.m2:47:43-47:71 │ │ │ + 1 .0030s 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 0.153074s (cpu); 0.12278s (thread); 0s (gc) │ │ │ + -- used 0.239174s (cpu); 0.118246s (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 0.604592s (cpu); 0.306923s (thread); 0s (gc) │ │ │ + -- used 2.33961s (cpu); 0.40782s (thread); 0s (gc) │ │ │ │ │ │ o10 = ideal (x - 27x , x - 16x , x - 9x , x + 44x , x - 52x ) │ │ │ 4 5 3 5 2 5 1 5 0 5 │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2] │ │ │ @@ -58,12 +58,12 @@ │ │ │ │ │ │ i14 : I=ideal random(n,R); │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ │ │ i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c); │ │ │ #select(degs,d->d==1))),f->f>0)) │ │ │ - -- used 3.962s (cpu); 2.28788s (thread); 0s (gc) │ │ │ + -- used 3.59943s (cpu); 1.82917s (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/M2-11772-0/0 │ │ │ +o1 = /tmp/M2-13282-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-11772-0/0 │ │ │ +o2 = /tmp/M2-13282-0/0 │ │ │ │ │ │ i3 : (fn = dir | "/" | "foo") << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11772-0/0/foo │ │ │ +o3 = /tmp/M2-13282-0/0/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/M2-11314-0/0 │ │ │ +o1 = /tmp/M2-12344-0/0 │ │ │ │ │ │ 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/M2-11314-0/0 │ │ │ +o2 = /tmp/M2-12344-0/0 │ │ │ │ │ │ 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/M2-11314-0/0 │ │ │ +o7 = /tmp/M2-12344-0/0 │ │ │ │ │ │ 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/M2-12013-0/0 │ │ │ +o1 = /tmp/M2-13763-0/0 │ │ │ │ │ │ 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/M2-10398-0/86-rundir/ │ │ │ +o1 = /tmp/M2-10518-0/86-rundir/ │ │ │ │ │ │ i2 : p = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-12032-0/0 │ │ │ +o2 = /tmp/M2-13802-0/0 │ │ │ │ │ │ i3 : q = temporaryFileName() │ │ │ │ │ │ -o3 = /tmp/M2-12032-0/1 │ │ │ +o3 = /tmp/M2-13802-0/1 │ │ │ │ │ │ i4 : symlinkFile(p,q) │ │ │ │ │ │ i5 : p << close │ │ │ │ │ │ -o5 = /tmp/M2-12032-0/0 │ │ │ +o5 = /tmp/M2-13802-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : readlink q │ │ │ │ │ │ -o6 = /tmp/M2-12032-0/0 │ │ │ +o6 = /tmp/M2-13802-0/0 │ │ │ │ │ │ i7 : realpath q │ │ │ │ │ │ -o7 = /tmp/M2-12032-0/0 │ │ │ +o7 = /tmp/M2-13802-0/0 │ │ │ │ │ │ i8 : removeFile p │ │ │ │ │ │ i9 : removeFile q │ │ │ │ │ │ i10 : realpath "" │ │ │ │ │ │ -o10 = /tmp/M2-10398-0/86-rundir/ │ │ │ +o10 = /tmp/M2-10518-0/86-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)[5]: -- finalizing sequence #6 -- │ │ │ ---finalization: (2)[2]: -- finalizing sequence #3 -- │ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 -- │ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 -- │ │ │ ---finalization: (5)[0]: -- finalizing sequence #1 -- │ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 -- │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 -- │ │ │ ---finalization: (8)[3]: -- finalizing sequence #4 -- │ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 -- │ │ │ +--finalization: (2)[4]: -- finalizing sequence #5 -- │ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 -- │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 -- │ │ │ +--finalization: (5)[2]: -- finalizing sequence #3 -- │ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 -- │ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 -- │ │ │ +--finalization: (8)[6]: -- finalizing sequence #7 -- │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out │ │ │ @@ -1,19 +1,19 @@ │ │ │ -- -*- M2-comint -*- hash: 8532980310097060089 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10986-0/0 │ │ │ +o1 = /tmp/M2-11676-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10986-0/0 │ │ │ +o2 = /tmp/M2-11676-0/0 │ │ │ │ │ │ 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/M2-10490-0/0 │ │ │ +o1 = /tmp/M2-10677-0/0 │ │ │ │ │ │ i2 : rootPath | fn │ │ │ │ │ │ -o2 = /tmp/M2-10490-0/0 │ │ │ +o2 = /tmp/M2-10677-0/0 │ │ │ │ │ │ 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/M2-11715-0/0 │ │ │ +o1 = /tmp/M2-13165-0/0 │ │ │ │ │ │ i2 : rootURI | fn │ │ │ │ │ │ -o2 = file:///tmp/M2-11715-0/0 │ │ │ +o2 = file:///tmp/M2-13165-0/0 │ │ │ │ │ │ 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/M2-11563-0/0 │ │ │ +o5 = /tmp/M2-12853-0/0 │ │ │ │ │ │ i6 : f << toString (p,m,M) << close │ │ │ │ │ │ -o6 = /tmp/M2-11563-0/0 │ │ │ +o6 = /tmp/M2-12853-0/0 │ │ │ │ │ │ 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 = 1424720 │ │ │ +o1 = 1524720 │ │ │ │ │ │ i2 : serialNumber foo │ │ │ │ │ │ -o2 = 1424722 │ │ │ +o2 = 1524722 │ │ │ │ │ │ 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 0.000248716s (cpu); 0.000241994s (thread); 0s (gc) │ │ │ + -- used 0.000201675s (cpu); 0.000193846s (thread); 0s (gc) │ │ │ │ │ │ i31 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.00015953s (cpu); 0.000159609s (thread); 0s (gc) │ │ │ + -- used 0.000112283s (cpu); 0.000112223s (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 0.241135s (cpu); 0.24107s (thread); 0s (gc) │ │ │ + -- used 0.139317s (cpu); 0.139324s (thread); 0s (gc) │ │ │ │ │ │ i39 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.240169s (cpu); 0.240175s (thread); 0s (gc) │ │ │ + -- used 0.337356s (cpu); 0.173462s (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/M2-11354-0/0/ │ │ │ +o1 = /tmp/M2-12424-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11354-0/1/ │ │ │ +o2 = /tmp/M2-12424-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11354-0/0/a/ │ │ │ +o3 = /tmp/M2-12424-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11354-0/0/b/ │ │ │ +o4 = /tmp/M2-12424-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11354-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12424-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11354-0/0/a/f │ │ │ +o6 = /tmp/M2-12424-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11354-0/0/a/g │ │ │ +o7 = /tmp/M2-12424-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11354-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12424-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : symlinkDirectory(src,dst,Verbose=>true) │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11354-0/1/b/c/g │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11354-0/1/b/c/g │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11411-0/0 │ │ │ +o1 = /tmp/M2-12541-0/0 │ │ │ │ │ │ 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/M2-12357-0/0.tex │ │ │ +o1 = /tmp/M2-14477-0/0.tex │ │ │ │ │ │ i2 : temporaryFileName () | ".html" │ │ │ │ │ │ -o2 = /tmp/M2-12357-0/1.html │ │ │ +o2 = /tmp/M2-14477-0/1.html │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ @@ -1,8 +1,8 @@ │ │ │ -- -*- M2-comint -*- hash: 1332435500723 │ │ │ │ │ │ i1 : time 3^30 │ │ │ - -- used 1.8946e-05s (cpu); 1.074e-05s (thread); 0s (gc) │ │ │ + -- used 1.9179e-05s (cpu); 5.057e-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 │ │ │ - -- .000014046 seconds │ │ │ + -- .000015327 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{.000014046, 205891132094649} │ │ │ +o2 = Time{.000015327, 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" => 3.2.1 │ │ │ "mysql version" => not present │ │ │ "normaliz version" => 3.10.5 │ │ │ "ntl version" => 11.5.1 │ │ │ - "operating system release" => 6.1.0-38-amd64 │ │ │ + "operating system release" => 6.12.48+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 Divisor EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieTypes ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds 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 Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases │ │ │ "pointer size" => 8 │ │ │ "python version" => 3.13.7 │ │ │ "readline version" => 8.2 │ │ │ "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
│ │ │ -Mon Aug 25 12:29:34 UTC 2025
│ │ │ +Sat Oct  4 04:41:44 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 │ │ │ │ -Mon Aug 25 12:29:34 UTC 2025 │ │ │ │ +Sat Oct 4 04:41:44 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/M2-11848-0/0.dbm
    │ │ │ +o1 = /tmp/M2-13438-0/0.dbm │ │ │
    │ │ │ 
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11848-0/0.dbm
│ │ │ +o2 = /tmp/M2-13438-0/0.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/M2-11848-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-13438-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11848-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-13438-0/0.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" => 42493534762        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 42596709034        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 207200256
│ │ │ -               "numGCs" => 793
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 224821248
│ │ │ +               "numGCs" => 782
│ │ │ +               "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 = 207200256
    
│ │ │ +o2 = 224821248
│ │ │              
    
│ │ │          
    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" => 42493534762        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 42596709034        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 207200256
│ │ │ │ -               "numGCs" => 793
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 224821248
│ │ │ │ +               "numGCs" => 782
│ │ │ │ +               "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 = 207200256
│ │ │ │ +o2 = 224821248
│ │ │ │  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 0.00985475s (cpu); 0.00985013s (thread); 0s (gc)
│ │ │ + -- used 0.00599737s (cpu); 0.00599416s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    │ │ │
    │ │ │ 
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0817304s (cpu); 0.0817367s (thread); 0s (gc)
│ │ │ + -- used 0.057939s (cpu); 0.0579513s (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 0.00985475s (cpu); 0.00985013s (thread); 0s (gc)
│ │ │ │ + -- used 0.00599737s (cpu); 0.00599416s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0817304s (cpu); 0.0817367s (thread); 0s (gc)
│ │ │ │ + -- used 0.057939s (cpu); 0.0579513s (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 0.0304881s (cpu); 0.0304864s (thread); 0s (gc)
    │ │ │ + -- used 0.0472629s (cpu); 0.0472646s (thread); 0s (gc) │ │ │
    │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0321648s (cpu); 0.0321706s (thread); 0s (gc)
    │ │ │ + -- used 0.0458996s (cpu); 0.0459103s (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 0.0304881s (cpu); 0.0304864s (thread); 0s (gc) │ │ │ │ + -- used 0.0472629s (cpu); 0.0472646s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0321648s (cpu); 0.0321706s (thread); 0s (gc) │ │ │ │ + -- used 0.0458996s (cpu); 0.0459103s (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 0.00194988s (cpu); 0.00190997s (thread); 0s (gc)
│ │ │ + -- used 0.00173968s (cpu); 0.00173148s (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 0.00194988s (cpu); 0.00190997s (thread); 0s (gc) │ │ │ │ + -- used 0.00173968s (cpu); 0.00173148s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 6 15 18 6 │ │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ o4 = "hi there" │ │ │
    │ │ │ 
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    │ │ │ +o5 = true │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,13 +23,13 @@ │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : peek read f │ │ │ │ │ │ │ │ o4 = "hi there" │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ │ │ -o5 = false │ │ │ │ +o5 = true │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0. │ │ ├── ./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 = .0002894243684497773
│ │ │ +o1 = .0003340345282869781
│ │ │  
│ │ │  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 = .0002894243684497773 │ │ │ │ +o1 = .0003340345282869781 │ │ │ │ │ │ │ │ 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'
│ │ │ - -- .00525587s elapsed
│ │ │ + -- .00403624s elapsed
│ │ │  
│ │ │  o4 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001342s elapsed
│ │ │ + -- .000001518s 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'
│ │ │ │ - -- .00525587s elapsed
│ │ │ │ + -- .00403624s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001342s elapsed
│ │ │ │ + -- .000001518s 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 = 722293
    │ │ │ +o5 = 1106263 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ @@ -127,15 +127,15 @@ │ │ │ o7 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 1464540
    │ │ │ +o8 = 2232431 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ @@ -163,29 +163,29 @@ │ │ │ o12 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 1464775
    │ │ │ +o13 = 2232776 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 1464775
    │ │ │ +o15 = 2232776 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,26 +28,26 @@ │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 722293 │ │ │ │ +o5 = 1106263 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 1464540 │ │ │ │ +o8 = 2232431 │ │ │ │ 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 = 1464775 │ │ │ │ +o13 = 2232776 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 1464775 │ │ │ │ +o15 = 2232776 │ │ │ │ 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/M2-10670-0/0
    │ │ │ +o1 = /tmp/M2-11037-0/0 │ │ │
    │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10670-0/0
    │ │ │ +o2 = /tmp/M2-11037-0/0 │ │ │
    │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10670-0/0/
    │ │ │ +o3 = /tmp/M2-11037-0/0/ │ │ │
    │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10670-0/0/
    │ │ │ +o4 = /tmp/M2-11037-0/0/ │ │ │
    │ │ │
    │ │ │

    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/M2-10670-0/0 │ │ │ │ +o1 = /tmp/M2-11037-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-10670-0/0 │ │ │ │ +o2 = /tmp/M2-11037-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-10670-0/0/ │ │ │ │ +o3 = /tmp/M2-11037-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-10670-0/0/ │ │ │ │ +o4 = /tmp/M2-11037-0/0/ │ │ │ │ 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")        -- .15275s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .113525s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .149336s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .149216s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .11506s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .114414s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Alternatively, if the package is installed somewhere accessible, one can do the following.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150392s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .113545s elapsed │ │ │
    │ │ │ 
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152341s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .230863s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .11866s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .11878s 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")        -- .234646s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .112802s 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")        -- .228259s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .228697s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .112478s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .110078s 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")        -- .229285s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .110789s 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") -- .15275s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .113525s elapsed │ │ │ │ i3 : check FirstPackage │ │ │ │ - -- capturing check(0, "FirstPackage") -- .149336s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .149216s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .11506s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .114414s elapsed │ │ │ │ Alternatively, if the package is installed somewhere accessible, one can do the │ │ │ │ following. │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ │ - -- capturing check(1, "FirstPackage") -- .150392s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .113545s elapsed │ │ │ │ i5 : check "FirstPackage" │ │ │ │ - -- capturing check(0, "FirstPackage") -- .152341s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .230863s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .11866s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .11878s 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") -- .234646s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .112802s 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") -- .228259s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .228697s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .112478s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .110078s 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") -- .229285s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .110789s 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 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-08-02) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1 (2025-09-20) 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 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1
│ │ │ -     (2025-08-02) x86_64 GNU/Linux\n"
    │ │ │ +o3 = "Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ + 6.12.48-1 (2025-09-20) 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 egrep and use it to locate the symbol names in Macaulay2 that begin with in. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 │ │ │ │ - -- .0937701s elapsed │ │ │ │ + -- .050861s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : elapsedTime monjMult I │ │ │ │ - -- .196612s elapsed │ │ │ │ + -- .153266s elapsed │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ │ - -- .152482s elapsed │ │ │ │ + -- .164015s 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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : f = openInOut "!egrep '^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 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1 (2025-08-
│ │ │ │ -02) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.48-1
│ │ │ │ +(2025-09-20) 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 6.1.0-38-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.1.147-1
│ │ │ │ -     (2025-08-02) x86_64 GNU/Linux\n"
│ │ │ │ +o3 = "Linux sbuild 6.12.48+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +     6.12.48-1 (2025-09-20) 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 egrep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!egrep '^in'"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -269,15 +269,15 @@
│ │ │                 1277
    │ │ │
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7fcf2c0f3380
│ │ │ +   -- registering gb 5 at 0x7f00d2f29380
│ │ │  
│ │ │     -- [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 0.279998s (cpu); 0.283653s (thread); 0s (gc)
│ │ │ + -- used 0.195656s (cpu); 0.196903s (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 0.00799689s (cpu); 0.00497973s (thread); 0s (gc)
│ │ │ + -- used 0.00399011s (cpu); 0.00308441s (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 0x7fcf2c0f3380
│ │ │ │ +   -- registering gb 5 at 0x7f00d2f29380
│ │ │ │  
│ │ │ │     -- [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 0.279998s (cpu); 0.283653s (thread); 0s (gc)
│ │ │ │ + -- used 0.195656s (cpu); 0.196903s (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 0.00799689s (cpu); 0.00497973s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399011s (cpu); 0.00308441s (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
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11392-0/0/
    
│ │ │ +o1 = /tmp/M2-12502-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11392-0/1/
    
│ │ │ +o2 = /tmp/M2-12502-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11392-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12502-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11392-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12502-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11392-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12502-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11392-0/0/a/f
│ │ │ +o6 = /tmp/M2-12502-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11392-0/0/a/g
│ │ │ +o7 = /tmp/M2-12502-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11392-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12502-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11392-0/0/
│ │ │ -     /tmp/M2-11392-0/0/b/
│ │ │ -     /tmp/M2-11392-0/0/b/c/
│ │ │ -     /tmp/M2-11392-0/0/b/c/g
│ │ │ -     /tmp/M2-11392-0/0/a/
│ │ │ -     /tmp/M2-11392-0/0/a/g
│ │ │ -     /tmp/M2-11392-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12502-0/0/
│ │ │ +     /tmp/M2-12502-0/0/a/
│ │ │ +     /tmp/M2-12502-0/0/a/g
│ │ │ +     /tmp/M2-12502-0/0/a/f
│ │ │ +     /tmp/M2-12502-0/0/b/
│ │ │ +     /tmp/M2-12502-0/0/b/c/
│ │ │ +     /tmp/M2-12502-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11392-0/0/b/c/g -> /tmp/M2-11392-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11392-0/0/a/g -> /tmp/M2-11392-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11392-0/0/a/f -> /tmp/M2-11392-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12502-0/0/a/g -> /tmp/M2-12502-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12502-0/0/a/f -> /tmp/M2-12502-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12502-0/0/b/c/g -> /tmp/M2-12502-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11392-0/0/b/c/g not newer than /tmp/M2-11392-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11392-0/0/a/g not newer than /tmp/M2-11392-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11392-0/0/a/f not newer than /tmp/M2-11392-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12502-0/0/a/g not newer than /tmp/M2-12502-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12502-0/0/a/f not newer than /tmp/M2-12502-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12502-0/0/b/c/g not newer than /tmp/M2-12502-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11392-0/1/
│ │ │ -      /tmp/M2-11392-0/1/a/
│ │ │ -      /tmp/M2-11392-0/1/a/f
│ │ │ -      /tmp/M2-11392-0/1/a/g
│ │ │ -      /tmp/M2-11392-0/1/b/
│ │ │ -      /tmp/M2-11392-0/1/b/c/
│ │ │ -      /tmp/M2-11392-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12502-0/1/
│ │ │ +      /tmp/M2-12502-0/1/a/
│ │ │ +      /tmp/M2-12502-0/1/a/g
│ │ │ +      /tmp/M2-12502-0/1/a/f
│ │ │ +      /tmp/M2-12502-0/1/b/
│ │ │ +      /tmp/M2-12502-0/1/b/c/
│ │ │ +      /tmp/M2-12502-0/1/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/M2-11392-0/0/
│ │ │ │ +o1 = /tmp/M2-12502-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11392-0/1/
│ │ │ │ +o2 = /tmp/M2-12502-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11392-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12502-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11392-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12502-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11392-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12502-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11392-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12502-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11392-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12502-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11392-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12502-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11392-0/0/
│ │ │ │ -     /tmp/M2-11392-0/0/b/
│ │ │ │ -     /tmp/M2-11392-0/0/b/c/
│ │ │ │ -     /tmp/M2-11392-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11392-0/0/a/
│ │ │ │ -     /tmp/M2-11392-0/0/a/g
│ │ │ │ -     /tmp/M2-11392-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12502-0/0/
│ │ │ │ +     /tmp/M2-12502-0/0/a/
│ │ │ │ +     /tmp/M2-12502-0/0/a/g
│ │ │ │ +     /tmp/M2-12502-0/0/a/f
│ │ │ │ +     /tmp/M2-12502-0/0/b/
│ │ │ │ +     /tmp/M2-12502-0/0/b/c/
│ │ │ │ +     /tmp/M2-12502-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11392-0/0/b/c/g -> /tmp/M2-11392-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11392-0/0/a/g -> /tmp/M2-11392-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11392-0/0/a/f -> /tmp/M2-11392-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12502-0/0/a/g -> /tmp/M2-12502-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12502-0/0/a/f -> /tmp/M2-12502-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12502-0/0/b/c/g -> /tmp/M2-12502-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11392-0/0/b/c/g not newer than /tmp/M2-11392-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11392-0/0/a/g not newer than /tmp/M2-11392-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11392-0/0/a/f not newer than /tmp/M2-11392-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12502-0/0/a/g not newer than /tmp/M2-12502-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12502-0/0/a/f not newer than /tmp/M2-12502-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12502-0/0/b/c/g not newer than /tmp/M2-12502-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11392-0/1/
│ │ │ │ -      /tmp/M2-11392-0/1/a/
│ │ │ │ -      /tmp/M2-11392-0/1/a/f
│ │ │ │ -      /tmp/M2-11392-0/1/a/g
│ │ │ │ -      /tmp/M2-11392-0/1/b/
│ │ │ │ -      /tmp/M2-11392-0/1/b/c/
│ │ │ │ -      /tmp/M2-11392-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12502-0/1/
│ │ │ │ +      /tmp/M2-12502-0/1/a/
│ │ │ │ +      /tmp/M2-12502-0/1/a/g
│ │ │ │ +      /tmp/M2-12502-0/1/a/f
│ │ │ │ +      /tmp/M2-12502-0/1/b/
│ │ │ │ +      /tmp/M2-12502-0/1/b/c/
│ │ │ │ +      /tmp/M2-12502-0/1/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/M2-11177-0/0
    
│ │ │ +o1 = /tmp/M2-12067-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11177-0/1
    
│ │ │ +o2 = /tmp/M2-12067-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11177-0/0
│ │ │ +o3 = /tmp/M2-12067-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11177-0/0 -> /tmp/M2-11177-0/1
    
│ │ │ + -- copying: /tmp/M2-12067-0/0 -> /tmp/M2-12067-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1
    
│ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11177-0/0
│ │ │ +o7 = /tmp/M2-12067-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1
    
│ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1
│ │ │              
    
│ │ │                
    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/M2-11177-0/0
│ │ │ │ +o1 = /tmp/M2-12067-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11177-0/1
│ │ │ │ +o2 = /tmp/M2-12067-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11177-0/0
│ │ │ │ +o3 = /tmp/M2-12067-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11177-0/0 -> /tmp/M2-11177-0/1
│ │ │ │ + -- copying: /tmp/M2-12067-0/0 -> /tmp/M2-12067-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1
│ │ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11177-0/0
│ │ │ │ +o7 = /tmp/M2-12067-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11177-0/0 not newer than /tmp/M2-11177-0/1
│ │ │ │ + -- skipping: /tmp/M2-12067-0/0 not newer than /tmp/M2-12067-0/1
│ │ │ │  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 = 365.096465098
│ │ │ +o1 = 294.147154702
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 367.292409687
│ │ │ +o3 = 295.012007467
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 2.195944589000021
│ │ │ +o4 = .8648527650000233
│ │ │  
│ │ │  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 = 365.096465098
│ │ │ │ +o1 = 294.147154702
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 367.292409687
│ │ │ │ +o3 = 295.012007467
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 2.195944589000021
│ │ │ │ +o4 = .8648527650000233
│ │ │ │  
│ │ │ │  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 = 1756125053
    
│ │ │ +o1 = 1759552954
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.64938758977737
│ │ │ +o2 = 55.75801344819361
│ │ │  
│ │ │  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 = 7.792651077328429
│ │ │ +o3 = 9.096161378323302
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Mon Aug 25 12:30:53 UTC 2025
│ │ │ +Sat Oct  4 04:42:34 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 = 1756125053
│ │ │ │ +o1 = 1759552954
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 55.64938758977737
│ │ │ │ +o2 = 55.75801344819361
│ │ │ │  
│ │ │ │  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 = 7.792651077328429
│ │ │ │ +o3 = 9.096161378323302
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Mon Aug 25 12:30:53 UTC 2025
│ │ │ │ +Sat Oct  4 04:42:34 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-
│ │ │ │  1.25.06+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
│ │ │ - -- 1.00011s elapsed
│ │ │ + -- 1.00022s 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
│ │ │ │ - -- 1.00011s elapsed
│ │ │ │ + -- 1.00022s 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
│ │ │ -     -- 1.00017 seconds
│ │ │ +     -- 1.00013 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00017, 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
│ │ │ │ -     -- 1.00017 seconds
│ │ │ │ +     -- 1.00013 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00017, 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 0.528053s (cpu); 0.289697s (thread); 0s (gc)
│ │ │ + -- used 0.131402s (cpu); 0.131402s (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 0.796872s (cpu); 0.31766s (thread); 0s (gc)
│ │ │ + -- used 0.344631s (cpu); 0.174895s (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 0.0493859s (cpu); 0.0493947s (thread); 0s (gc)
│ │ │ + -- used 0.0321462s (cpu); 0.0321482s (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 0.447427s (cpu); 0.216815s (thread); 0s (gc)
│ │ │ + -- used 0.105021s (cpu); 0.105027s (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 0.00185985s (cpu); 0.00186044s (thread); 0s (gc)
│ │ │ + -- used 0.00157095s (cpu); 0.00156769s (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 0.321427s (cpu); 0.109696s (thread); 0s (gc)
│ │ │ + -- used 0.0386393s (cpu); 0.0386488s (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 0.528053s (cpu); 0.289697s (thread); 0s (gc)
│ │ │ │ + -- used 0.131402s (cpu); 0.131402s (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 0.796872s (cpu); 0.31766s (thread); 0s (gc)
│ │ │ │ + -- used 0.344631s (cpu); 0.174895s (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 0.0493859s (cpu); 0.0493947s (thread); 0s (gc)
│ │ │ │ + -- used 0.0321462s (cpu); 0.0321482s (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 0.447427s (cpu); 0.216815s (thread); 0s (gc)
│ │ │ │ + -- used 0.105021s (cpu); 0.105027s (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 0.00185985s (cpu); 0.00186044s (thread); 0s (gc)
│ │ │ │ + -- used 0.00157095s (cpu); 0.00156769s (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 0.321427s (cpu); 0.109696s (thread); 0s (gc)
│ │ │ │ + -- used 0.0386393s (cpu); 0.0386488s (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/M2-10398-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10518-0/91-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/M2-10398-0/91-rundir/
│ │ │ │ +             source directory => /tmp/M2-10518-0/91-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
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -510,27 +510,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10765-0/0
    
│ │ │ +o1 = /tmp/M2-11232-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10765-0/0
│ │ │ +o3 = /tmp/M2-11232-0/0
│ │ │  
│ │ │  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/M2-10765-0/0
│ │ │ │ +o1 = /tmp/M2-11232-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10765-0/0
│ │ │ │ +o3 = /tmp/M2-11232-0/0
│ │ │ │  
│ │ │ │  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.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12338-0/0
│ │ │ +o1 = /tmp/M2-14438-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12338-0/0
│ │ │ +o3 = /tmp/M2-14438-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12338-0/0
    
│ │ │ +o4 = /tmp/M2-14438-0/0
│ │ │              
    
│ │ │                
    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/M2-12338-0/0
│ │ │ │ +o1 = /tmp/M2-14438-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12338-0/0
│ │ │ │ +o3 = /tmp/M2-14438-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12338-0/0
│ │ │ │ +o4 = /tmp/M2-14438-0/0
│ │ │ │  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
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -425,15 +425,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11582-0/0
    
│ │ │ +o1 = /tmp/M2-12892-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11582-0/0
│ │ │ +o2 = /tmp/M2-12892-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11582-0/0
│ │ │ +o4 = /tmp/M2-12892-0/0
│ │ │  
│ │ │  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/M2-11582-0/0
│ │ │ │ +o1 = /tmp/M2-12892-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11582-0/0
│ │ │ │ +o2 = /tmp/M2-12892-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11582-0/0
│ │ │ │ +o4 = /tmp/M2-12892-0/0
│ │ │ │  
│ │ │ │  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
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │ @@ -188,20 +188,20 @@
│ │ │              
│ │ │            
│ │ │            
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11196-0/0
    
│ │ │ +o1 = /tmp/M2-12106-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11196-0/0
│ │ │ +o2 = /tmp/M2-12106-0/0
│ │ │  
│ │ │  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/M2-11196-0/0
│ │ │ │ +o1 = /tmp/M2-12106-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11196-0/0
│ │ │ │ +o2 = /tmp/M2-12106-0/0
│ │ │ │  
│ │ │ │  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/M2-11061-0/0
    
│ │ │ +o1 = /tmp/M2-11831-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11061-0/0
│ │ │ +o2 = /tmp/M2-11831-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -108,15 +108,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11061-0/0
│ │ │ +o6 = /tmp/M2-11831-0/0
│ │ │  
│ │ │  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/M2-11061-0/0
│ │ │ │ +o1 = /tmp/M2-11831-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11061-0/0
│ │ │ │ +o2 = /tmp/M2-11831-0/0
│ │ │ │  
│ │ │ │  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/M2-11061-0/0
│ │ │ │ +o6 = /tmp/M2-11831-0/0
│ │ │ │  
│ │ │ │  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/M2-12184-0/0
    
│ │ │ +o1 = /tmp/M2-14124-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12184-0/0
│ │ │ +o2 = /tmp/M2-14124-0/0
│ │ │  
│ │ │  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/M2-12184-0/0
│ │ │ │ +o1 = /tmp/M2-14124-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12184-0/0
│ │ │ │ +o2 = /tmp/M2-14124-0/0
│ │ │ │  
│ │ │ │  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 = 68
    
│ │ │ +o1 = 46
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 68
│ │ │ │ +o1 = 46
│ │ │ │  ********** 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 0x7f500f309a80
│ │ │ +   -- registering gb 0 at 0x7fc81f326a80
│ │ │  
│ │ │     -- [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 0x7f500f309a80
│ │ │ │ +   -- registering gb 0 at 0x7fc81f326a80
│ │ │ │  
│ │ │ │     -- [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 = Mon Aug 25 12:30:02 UTC 2025
    
│ │ │ +o4 = Sat Oct  4 04:42:02 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 = Mon Aug 25 12:30:02 UTC 2025
│ │ │ │ +o4 = Sat Oct  4 04:42:02 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/M2-10587-0/0
    
│ │ │ +o2 = /tmp/M2-10874-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10587-0/0
│ │ │ +o3 = /tmp/M2-10874-0/0
│ │ │  
│ │ │  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/M2-10587-0/0
│ │ │ │ +o2 = /tmp/M2-10874-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10587-0/0
│ │ │ │ +o3 = /tmp/M2-10874-0/0
│ │ │ │  
│ │ │ │  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)
│ │ │ - -- 4.27565s elapsed
│ │ │ + -- 5.29813s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0554294s elapsed
│ │ │ + -- .059371s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000132769s elapsed
│ │ │ + -- .000157637s 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)
│ │ │ │ - -- 4.27565s elapsed
│ │ │ │ + -- 5.29813s 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
│ │ │ │ - -- .0554294s elapsed
│ │ │ │ + -- .059371s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000132769s elapsed
│ │ │ │ + -- .000157637s 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/M2-12376-0/0
    
│ │ │ +o1 = /tmp/M2-14516-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12376-0/0
│ │ │ +o2 = /tmp/M2-14516-0/0
│ │ │  
│ │ │  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/M2-12376-0/0
│ │ │ │ +o1 = /tmp/M2-14516-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12376-0/0
│ │ │ │ +o2 = /tmp/M2-14516-0/0
│ │ │ │  
│ │ │ │  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/M2-10929-0/0
│ │ │ │ +o1 = /tmp/M2-11559-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10929-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11559-0/0/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/M2-10929-0/0
    
│ │ │ +o1 = /tmp/M2-11559-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10929-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11559-0/0/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-
│ │ │ │  1.25.06+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 1.61151s (cpu); 1.13816s (thread); 0s (gc)
│ │ │ + -- used 0.842547s (cpu); 0.640172s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -78,23 +78,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 9.631e-05s (cpu); 9.588e-05s (thread); 0s (gc)
│ │ │ + -- used 6.8982e-05s (cpu); 6.5623e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 5.581e-06s (cpu); 5.109e-06s (thread); 0s (gc)
│ │ │ + -- used 3.27e-06s (cpu); 3.035e-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 1.61151s (cpu); 1.13816s (thread); 0s (gc)
│ │ │ │ + -- used 0.842547s (cpu); 0.640172s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 9.631e-05s (cpu); 9.588e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.8982e-05s (cpu); 6.5623e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 5.581e-06s (cpu); 5.109e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.27e-06s (cpu); 3.035e-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
│ │ │ @@ -76,32 +76,32 @@
│ │ │  
│ │ │  o1 = {0 => (==, BettiTally, BettiTally)                           }
│ │ │       {1 => (++, BettiTally, BettiTally)                           }
│ │ │       {2 => (**, BettiTally, BettiTally)                           }
│ │ │       {3 => (SPACE, BettiTally, Array)                             }
│ │ │       {4 => (SPACE, BettiTally, ZZ)                                }
│ │ │       {5 => (lift, BettiTally, ZZ)                                 }
│ │ │ -     {6 => (*, ZZ, BettiTally)                                    }
│ │ │ -     {7 => (*, QQ, BettiTally)                                    }
│ │ │ +     {6 => (*, QQ, BettiTally)                                    }
│ │ │ +     {7 => (*, ZZ, BettiTally)                                    }
│ │ │       {8 => (multigraded, BettiTally)                              }
│ │ │       {9 => (net, BettiTally)                                      }
│ │ │       {10 => (texMath, BettiTally)                                 }
│ │ │       {11 => (betti, BettiTally)                                   }
│ │ │       {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 => (codim, BettiTally)                                   }
│ │ │ -     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {22 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {23 => (dual, BettiTally)                                    }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {22 => (dual, BettiTally)                                    }
│ │ │ +     {23 => (codim, BettiTally)                                   }
│ │ │       {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ -     {2 => (-, Matrix, Matrix)                                   }
│ │ │ -     {3 => (contract', Matrix, Matrix)                           }
│ │ │ -     {4 => (diff', Matrix, Matrix)                               }
│ │ │ -     {5 => (contract, Matrix, Matrix)                            }
│ │ │ +o5 = {0 => (contract', Matrix, Matrix)                           }
│ │ │ +     {1 => (+, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (diff', Matrix, Matrix)                               }
│ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ +     {5 => (diff, Matrix, Matrix)                                }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -212,21 +212,21 @@
│ │ │       {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 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {28 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │       {31 => (pullback, 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)           }
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,32 +16,32 @@
│ │ │ │  
│ │ │ │  o1 = {0 => (==, BettiTally, BettiTally)                           }
│ │ │ │       {1 => (++, BettiTally, BettiTally)                           }
│ │ │ │       {2 => (**, BettiTally, BettiTally)                           }
│ │ │ │       {3 => (SPACE, BettiTally, Array)                             }
│ │ │ │       {4 => (SPACE, BettiTally, ZZ)                                }
│ │ │ │       {5 => (lift, BettiTally, ZZ)                                 }
│ │ │ │ -     {6 => (*, ZZ, BettiTally)                                    }
│ │ │ │ -     {7 => (*, QQ, BettiTally)                                    }
│ │ │ │ +     {6 => (*, QQ, BettiTally)                                    }
│ │ │ │ +     {7 => (*, ZZ, BettiTally)                                    }
│ │ │ │       {8 => (multigraded, BettiTally)                              }
│ │ │ │       {9 => (net, BettiTally)                                      }
│ │ │ │       {10 => (texMath, BettiTally)                                 }
│ │ │ │       {11 => (betti, BettiTally)                                   }
│ │ │ │       {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 => (codim, BettiTally)                                   }
│ │ │ │ -     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {22 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {23 => (dual, BettiTally)                                    }
│ │ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {22 => (dual, BettiTally)                                    }
│ │ │ │ +     {23 => (codim, BettiTally)                                   }
│ │ │ │       {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {2 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {3 => (contract', Matrix, Matrix)                           }
│ │ │ │ -     {4 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {5 => (contract, Matrix, Matrix)                            }
│ │ │ │ +o5 = {0 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {1 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {3 => (diff', Matrix, Matrix)                               }
│ │ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {5 => (diff, Matrix, Matrix)                                }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -109,21 +109,21 @@
│ │ │ │       {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 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {28 => (intersect, Matrix, Matrix)                          }
│ │ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │       {31 => (pullback, 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)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 2.06644s elapsed
│ │ │ + -- 2.20206s 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)
│ │ │ - -- .754668s elapsed
│ │ │ + -- .906559s 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)
│ │ │ - -- .0311935s elapsed
│ │ │ + -- .0386584s 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)
│ │ │ - -- 1.22721s elapsed
│ │ │ + -- 1.5132s 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
│ │ │ │ - -- 2.06644s elapsed
│ │ │ │ + -- 2.20206s 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)
│ │ │ │ - -- .754668s elapsed
│ │ │ │ + -- .906559s 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)
│ │ │ │ - -- .0311935s elapsed
│ │ │ │ + -- .0386584s 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)
│ │ │ │ - -- 1.22721s elapsed
│ │ │ │ + -- 1.5132s 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/M2-10948-0/0/
    
│ │ │ +o1 = /tmp/M2-11598-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10948-0/0/foo
│ │ │ +o4 = /tmp/M2-11598-0/0/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/M2-10948-0/0/
│ │ │ │ +o1 = /tmp/M2-11598-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10948-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11598-0/0/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/M2-10822-0/0
│ │ │ │ +o1 = /tmp/M2-11349-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10822-0/1
│ │ │ │ +o2 = /tmp/M2-11349-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10822-0/0
│ │ │ │ +o3 = /tmp/M2-11349-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10822-0/0 -> /tmp/M2-10822-0/1
│ │ │ │ +--moving: /tmp/M2-11349-0/0 -> /tmp/M2-11349-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10822-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11349-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10822-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11349-0/1.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/M2-10822-0/0
    
│ │ │ +o1 = /tmp/M2-11349-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10822-0/1
    
│ │ │ +o2 = /tmp/M2-11349-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10822-0/0
│ │ │ +o3 = /tmp/M2-11349-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10822-0/0 -> /tmp/M2-10822-0/1
    
│ │ │ +--moving: /tmp/M2-11349-0/0 -> /tmp/M2-11349-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10822-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11349-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10822-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11349-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500143s elapsed
│ │ │ + -- .500127s 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
│ │ │ │ - -- .500143s elapsed
│ │ │ │ + -- .500127s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │              
    		
│ │ │                
    i2 : L = random toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .701868s elapsed
    
│ │ │ + -- .698851s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .292671s elapsed
    
│ │ │ + -- .102976s 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.
    
│ │ │          
    
│ │ │          
│ │ │ ├── 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);
│ │ │ │ - -- .701868s elapsed
│ │ │ │ + -- .698851s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .292671s elapsed
│ │ │ │ + -- .102976s 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)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.20012s elapsed
│ │ │ + -- 2.16146s 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)
│ │ │ - -- 1.79946s elapsed
│ │ │ + -- 2.15376s 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)
│ │ │ - -- 2.58082s elapsed
│ │ │ + -- 2.20319s 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)
│ │ │ - -- 2.04482s elapsed
│ │ │ + -- 4.81958s 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)
│ │ │ - -- 1.96083s elapsed
│ │ │ + -- 2.44598s 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");
│ │ │ - -- 4.75265s elapsed
│ │ │ + -- 3.70385s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -322,15 +322,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 6.95852s elapsed
│ │ │ + -- 8.07595s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -345,15 +345,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.85792s elapsed
│ │ │ + -- 3.05118s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -396,15 +396,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.1079s elapsed
│ │ │ + -- .909785s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.14666s elapsed
│ │ │ + -- 1.41368s 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
│ │ │ │ - -- 2.20012s elapsed
│ │ │ │ + -- 2.16146s 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)
│ │ │ │ - -- 1.79946s elapsed
│ │ │ │ + -- 2.15376s 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)
│ │ │ │ - -- 2.58082s elapsed
│ │ │ │ + -- 2.20319s 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)
│ │ │ │ - -- 2.04482s elapsed
│ │ │ │ + -- 4.81958s 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)
│ │ │ │ - -- 1.96083s elapsed
│ │ │ │ + -- 2.44598s 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");
│ │ │ │ - -- 4.75265s elapsed
│ │ │ │ + -- 3.70385s 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");
│ │ │ │ - -- 6.95852s elapsed
│ │ │ │ + -- 8.07595s 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");
│ │ │ │ - -- 4.85792s elapsed
│ │ │ │ + -- 3.05118s 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);
│ │ │ │ - -- 1.1079s elapsed
│ │ │ │ + -- .909785s 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);
│ │ │ │ - -- 1.14666s elapsed
│ │ │ │ + -- 1.41368s 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 0.00262431s (cpu); 1.0961e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00183579s (cpu); 1.1938e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f6037679540
│ │ │ +   -- registering gb 19 at 0x7f5ed9ee2540
│ │ │  
│ │ │     -- [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 0.00936292s (cpu); 0.0130541s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0101486s (cpu); 0.0136747s (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 0x7f6037679380
│ │ │ +   -- registering gb 20 at 0x7f5ed9ee2380
│ │ │  
│ │ │     -- [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 0x7f60376791c0
│ │ │ +   -- registering gb 21 at 0x7f5ed9ee21c0
│ │ │  
│ │ │     -- [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 0.00799987s (cpu); 0.00562168s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00798882s (cpu); 0.00553421s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7f6037679000
│ │ │ +   -- registering gb 22 at 0x7f5ed9ee2000
│ │ │  
│ │ │     -- [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 0.0479977s (cpu); 0.0471216s (thread); 0s (gc)
│ │ │ +   --  -- used 0.051934s (cpu); 0.0530726s (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 0.00262431s (cpu); 1.0961e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00183579s (cpu); 1.1938e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7f6037679540
│ │ │ │ +   -- registering gb 19 at 0x7f5ed9ee2540
│ │ │ │  
│ │ │ │     -- [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 0.00936292s (cpu); 0.0130541s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0101486s (cpu); 0.0136747s (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 0x7f6037679380
│ │ │ │ +   -- registering gb 20 at 0x7f5ed9ee2380
│ │ │ │  
│ │ │ │     -- [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 0x7f60376791c0
│ │ │ │ +   -- registering gb 21 at 0x7f5ed9ee21c0
│ │ │ │  
│ │ │ │     -- [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 0.00799987s (cpu); 0.00562168s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00798882s (cpu); 0.00553421s (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 0x7f6037679000
│ │ │ │ +   -- registering gb 22 at 0x7f5ed9ee2000
│ │ │ │  
│ │ │ │     -- [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 0.0479977s (cpu); 0.0471216s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.051934s (cpu); 0.0530726s (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/M2-11139-0/0
    
│ │ │ +o3 = /tmp/M2-11989-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11139-0/0
│ │ │ +o4 = /tmp/M2-11989-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -151,15 +151,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11139-0/0
│ │ │ +o9 = /tmp/M2-11989-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -169,15 +169,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11139-0/0
│ │ │ +o11 = /tmp/M2-11989-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11139-0/0
│ │ │ │ +o3 = /tmp/M2-11989-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11139-0/0
│ │ │ │ +o4 = /tmp/M2-11989-0/0
│ │ │ │  
│ │ │ │  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/M2-11139-0/0
│ │ │ │ +o9 = /tmp/M2-11989-0/0
│ │ │ │  
│ │ │ │  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/M2-11139-0/0
│ │ │ │ +o11 = /tmp/M2-11989-0/0
│ │ │ │  
│ │ │ │  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 = 10398
    
│ │ │ +o1 = 10518
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 10398
│ │ │ │ +o1 = 10518
│ │ │ │  ********** 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     94.58   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.01   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     45.11   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     31.76   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     30.58   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     22.63   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     21.4    ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     21.26   ../../m2/set.m2:122:5-122:61     
│ │ │ -     1     3.49    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.76    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.11    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.57    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.5     ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.26    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.15    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     1.08    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .76     ../../m2/modules.m2:278:4-278:52 
│ │ │ -     20    .6      ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .5      ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0035s  elapsed total                    
    
│ │ │ +     1     97.29   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     94.92   ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     48.2    ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     35.92   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     34.27   ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     25.83   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     21.83   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     20.52   ../../m2/set.m2:122:5-122:61     
│ │ │ +     1     3.1     ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.16    ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     2       ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     1.46    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.38    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.18    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.03    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     1.02    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     1     1.01    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     .63     ../../m2/modules.m2:278:4-278:52 
│ │ │ +     20    .49     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .36     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     .0030s  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     94.58   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ -     1     92.01   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ -     1     45.11   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ -     1     31.76   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ -     1     30.58   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ -     1     22.63   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ -     1     21.4    ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ -     1     21.26   ../../m2/set.m2:122:5-122:61
│ │ │ │ -     1     3.49    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ -     1     2.76    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ -     1     2.11    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ -     1     1.57    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ -     1     1.5     ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ -     1     1.26    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ -     1     1.15    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ -     1     1.09    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ -     1     1.08    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ -     1     .76     ../../m2/modules.m2:278:4-278:52
│ │ │ │ -     20    .6      ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ -     20    .5      ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ -     1     .0035s  elapsed total
│ │ │ │ +     1     97.29   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ +     1     94.92   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ +     1     48.2    ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ +     1     35.92   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ +     1     34.27   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ +     1     25.83   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ +     1     21.83   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ +     1     20.52   ../../m2/set.m2:122:5-122:61
│ │ │ │ +     1     3.1     ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ +     1     2.16    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ +     1     2       ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ +     1     1.46    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     1.38    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ +     1     1.18    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ +     1     1.03    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ +     1     1.02    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ +     1     1.01    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ +     1     .63     ../../m2/modules.m2:278:4-278:52
│ │ │ │ +     20    .49     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ +     20    .36     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ +     1     .0030s  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 0.153074s (cpu); 0.12278s (thread); 0s (gc)
│ │ │ + -- used 0.239174s (cpu); 0.118246s (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 0.604592s (cpu); 0.306923s (thread); 0s (gc)
│ │ │ + -- used 2.33961s (cpu); 0.40782s (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 3.962s (cpu); 2.28788s (thread); 0s (gc)
│ │ │ + -- used 3.59943s (cpu); 1.82917s (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 0.153074s (cpu); 0.12278s (thread); 0s (gc)
│ │ │ │ + -- used 0.239174s (cpu); 0.118246s (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 0.604592s (cpu); 0.306923s (thread); 0s (gc)
│ │ │ │ + -- used 2.33961s (cpu); 0.40782s (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 3.962s (cpu); 2.28788s (thread); 0s (gc)
│ │ │ │ + -- used 3.59943s (cpu); 1.82917s (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/M2-11772-0/0
    
│ │ │ +o1 = /tmp/M2-13282-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11772-0/0
    
│ │ │ +o2 = /tmp/M2-13282-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11772-0/0/foo
│ │ │ +o3 = /tmp/M2-13282-0/0/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/M2-11772-0/0
│ │ │ │ +o1 = /tmp/M2-13282-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11772-0/0
│ │ │ │ +o2 = /tmp/M2-13282-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11772-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13282-0/0/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/M2-11314-0/0
    
│ │ │ +o1 = /tmp/M2-12344-0/0
│ │ │              
    
│ │ │                
    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/M2-11314-0/0
│ │ │ +o2 = /tmp/M2-12344-0/0
│ │ │  
│ │ │  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/M2-11314-0/0
│ │ │ +o7 = /tmp/M2-12344-0/0
│ │ │  
│ │ │  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/M2-11314-0/0
│ │ │ │ +o1 = /tmp/M2-12344-0/0
│ │ │ │  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/M2-11314-0/0
│ │ │ │ +o2 = /tmp/M2-12344-0/0
│ │ │ │  
│ │ │ │  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/M2-11314-0/0
│ │ │ │ +o7 = /tmp/M2-12344-0/0
│ │ │ │  
│ │ │ │  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/M2-12013-0/0
│ │ │ │ +o1 = /tmp/M2-13763-0/0
│ │ │ │  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/M2-12013-0/0
    
│ │ │ +o1 = /tmp/M2-13763-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10398-0/86-rundir/
    
│ │ │ +o1 = /tmp/M2-10518-0/86-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12032-0/0
    
│ │ │ +o2 = /tmp/M2-13802-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12032-0/1
    
│ │ │ +o3 = /tmp/M2-13802-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12032-0/0
│ │ │ +o5 = /tmp/M2-13802-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12032-0/0
    
│ │ │ +o6 = /tmp/M2-13802-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12032-0/0
    
│ │ │ +o7 = /tmp/M2-13802-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10398-0/86-rundir/
    
│ │ │ +o10 = /tmp/M2-10518-0/86-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/M2-10398-0/86-rundir/
│ │ │ │ +o1 = /tmp/M2-10518-0/86-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12032-0/0
│ │ │ │ +o2 = /tmp/M2-13802-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12032-0/1
│ │ │ │ +o3 = /tmp/M2-13802-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12032-0/0
│ │ │ │ +o5 = /tmp/M2-13802-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12032-0/0
│ │ │ │ +o6 = /tmp/M2-13802-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12032-0/0
│ │ │ │ +o7 = /tmp/M2-13802-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10398-0/86-rundir/
│ │ │ │ +o10 = /tmp/M2-10518-0/86-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)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (2)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (5)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (8)[3]: -- finalizing sequence #4 --
    
│ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (2)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (8)[6]: -- finalizing sequence #7 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (2)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (5)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (8)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (2)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (5)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (8)[6]: -- finalizing sequence #7 --
│ │ │ │  ********** 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/M2-10986-0/0
    
│ │ │ +o1 = /tmp/M2-11676-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10986-0/0
    
│ │ │ +o2 = /tmp/M2-11676-0/0
│ │ │              
    
│ │ │                
    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/M2-10986-0/0
│ │ │ │ +o1 = /tmp/M2-11676-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10986-0/0
│ │ │ │ +o2 = /tmp/M2-11676-0/0
│ │ │ │  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/M2-10490-0/0
    
│ │ │ +o1 = /tmp/M2-10677-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10490-0/0
    
│ │ │ +o2 = /tmp/M2-10677-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    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/M2-10490-0/0
│ │ │ │ +o1 = /tmp/M2-10677-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10490-0/0
│ │ │ │ +o2 = /tmp/M2-10677-0/0
│ │ │ │  ********** 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-
│ │ │ │  1.25.06+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/M2-11715-0/0
      
│ │ │ +o1 = /tmp/M2-13165-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11715-0/0
      
│ │ │ +o2 = file:///tmp/M2-13165-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    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/M2-11715-0/0
│ │ │ │ +o1 = /tmp/M2-13165-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11715-0/0
│ │ │ │ +o2 = file:///tmp/M2-13165-0/0
│ │ │ │  ********** 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-
│ │ │ │  1.25.06+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/M2-11563-0/0
    
│ │ │ +o5 = /tmp/M2-12853-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11563-0/0
│ │ │ +o6 = /tmp/M2-12853-0/0
│ │ │  
│ │ │  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/M2-11563-0/0
│ │ │ │ +o5 = /tmp/M2-12853-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11563-0/0
│ │ │ │ +o6 = /tmp/M2-12853-0/0
│ │ │ │  
│ │ │ │  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 = 1424720
    
│ │ │ +o1 = 1524720
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1424722
    
│ │ │ +o2 = 1524722
│ │ │              
    
│ │ │                
    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 = 1424720
│ │ │ │ +o1 = 1524720
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1424722
│ │ │ │ +o2 = 1524722
│ │ │ │  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 0.000248716s (cpu); 0.000241994s (thread); 0s (gc)
    
│ │ │ + -- used 0.000201675s (cpu); 0.000193846s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.00015953s (cpu); 0.000159609s (thread); 0s (gc)
    
│ │ │ + -- used 0.000112283s (cpu); 0.000112223s (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 0.241135s (cpu); 0.24107s (thread); 0s (gc)
    
│ │ │ + -- used 0.139317s (cpu); 0.139324s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.240169s (cpu); 0.240175s (thread); 0s (gc)
    
│ │ │ + -- used 0.337356s (cpu); 0.173462s (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 0.000248716s (cpu); 0.000241994s (thread); 0s (gc)
│ │ │ │ + -- used 0.000201675s (cpu); 0.000193846s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.00015953s (cpu); 0.000159609s (thread); 0s (gc)
│ │ │ │ + -- used 0.000112283s (cpu); 0.000112223s (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 0.241135s (cpu); 0.24107s (thread); 0s (gc)
│ │ │ │ + -- used 0.139317s (cpu); 0.139324s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.240169s (cpu); 0.240175s (thread); 0s (gc)
│ │ │ │ + -- used 0.337356s (cpu); 0.173462s (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/M2-11354-0/0/
    
│ │ │ +o1 = /tmp/M2-12424-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11354-0/1/
    
│ │ │ +o2 = /tmp/M2-12424-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11354-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12424-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11354-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12424-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11354-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12424-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11354-0/0/a/f
│ │ │ +o6 = /tmp/M2-12424-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11354-0/0/a/g
│ │ │ +o7 = /tmp/M2-12424-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11354-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12424-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11354-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11354-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11354-0/0/
│ │ │ │ +o1 = /tmp/M2-12424-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11354-0/1/
│ │ │ │ +o2 = /tmp/M2-12424-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11354-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12424-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11354-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12424-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11354-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12424-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11354-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12424-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11354-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12424-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11354-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12424-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11354-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11354-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11354-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11354-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12424-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12424-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12424-0/1/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/M2-11411-0/0
│ │ │ │ +o1 = /tmp/M2-12541-0/0
│ │ │ │  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/M2-11411-0/0
    
│ │ │ +o1 = /tmp/M2-12541-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12357-0/0.tex
    
│ │ │ +o1 = /tmp/M2-14477-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12357-0/1.html
    
│ │ │ +o2 = /tmp/M2-14477-0/1.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/M2-12357-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14477-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12357-0/1.html
│ │ │ │ +o2 = /tmp/M2-14477-0/1.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 1.8946e-05s (cpu); 1.074e-05s (thread); 0s (gc)
│ │ │ + -- used 1.9179e-05s (cpu); 5.057e-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 1.8946e-05s (cpu); 1.074e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.9179e-05s (cpu); 5.057e-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
│ │ │ -     -- .000014046 seconds
│ │ │ +     -- .000015327 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000014046, 205891132094649}
    
│ │ │ +o2 = Time{.000015327, 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
│ │ │ │ -     -- .000014046 seconds
│ │ │ │ +     -- .000015327 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000014046, 205891132094649}
│ │ │ │ +o2 = Time{.000015327, 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" => 3.2.1
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.10.5
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.1.0-38-amd64
│ │ │ +               "operating system release" => 6.12.48+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 Divisor EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieTypes ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds 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 Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.7
│ │ │                 "readline version" => 8.2
│ │ │                 "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" => 3.2.1
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.10.5
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.1.0-38-amd64
│ │ │ │ +               "operating system release" => 6.12.48+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 3.48009s (cpu); 1.59285s (thread); 0s (gc)
│ │ │ + -- used 3.18775s (cpu); 1.55829s (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 3.48009s (cpu); 1.59285s (thread); 0s (gc)
│ │ │ + -- used 3.18775s (cpu); 1.55829s (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 3.48009s (cpu); 1.59285s (thread); 0s (gc)
│ │ │ │ + -- used 3.18775s (cpu); 1.55829s (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 0.0997549s (cpu); 0.0464881s (thread); 0s (gc)
│ │ │ + -- used 0.0857881s (cpu); 0.0304503s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0187855s (cpu); 0.0200539s (thread); 0s (gc)
│ │ │ + -- used 0.0160105s (cpu); 0.0171603s (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 0.003388s (cpu); 0.000318317s (thread); 0s (gc)
│ │ │ + -- used 0.00080885s (cpu); 0.000303934s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0161736s (cpu); 0.0191588s (thread); 0s (gc)
│ │ │ + -- used 0.0211081s (cpu); 0.0226436s (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 0.00241441s (cpu); 0.00467614s (thread); 0s (gc)
│ │ │ + -- used 0.00797865s (cpu); 0.00488481s (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 0.00100644s (cpu); 0.00207005s (thread); 0s (gc)
│ │ │ + -- used 0.000504913s (cpu); 0.00237945s (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 0.0997549s (cpu); 0.0464881s (thread); 0s (gc)
│ │ │ + -- used 0.0857881s (cpu); 0.0304503s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0187855s (cpu); 0.0200539s (thread); 0s (gc)
│ │ │ + -- used 0.0160105s (cpu); 0.0171603s (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 0.0997549s (cpu); 0.0464881s (thread); 0s (gc)
│ │ │ │ + -- used 0.0857881s (cpu); 0.0304503s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0187855s (cpu); 0.0200539s (thread); 0s (gc)
│ │ │ │ + -- used 0.0160105s (cpu); 0.0171603s (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 0.003388s (cpu); 0.000318317s (thread); 0s (gc)
│ │ │ + -- used 0.00080885s (cpu); 0.000303934s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.0161736s (cpu); 0.0191588s (thread); 0s (gc)
│ │ │ + -- used 0.0211081s (cpu); 0.0226436s (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 0.003388s (cpu); 0.000318317s (thread); 0s (gc)
│ │ │ │ + -- used 0.00080885s (cpu); 0.000303934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0161736s (cpu); 0.0191588s (thread); 0s (gc)
│ │ │ │ + -- used 0.0211081s (cpu); 0.0226436s (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 0.00241441s (cpu); 0.00467614s (thread); 0s (gc)
│ │ │ + -- used 0.00797865s (cpu); 0.00488481s (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 0.00100644s (cpu); 0.00207005s (thread); 0s (gc)
│ │ │ + -- used 0.000504913s (cpu); 0.00237945s (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 0.00241441s (cpu); 0.00467614s (thread); 0s (gc)
│ │ │ │ + -- used 0.00797865s (cpu); 0.00488481s (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 0.00100644s (cpu); 0.00207005s (thread); 0s (gc)
│ │ │ │ + -- used 0.000504913s (cpu); 0.00237945s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:27 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:26 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
│ │ │  ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
│ │ ├── ./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 0.051991s (cpu); 0.0518147s (thread); 0s (gc)
│ │ │ + -- used 0.0685238s (cpu); 0.0684537s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.132906s (cpu); 0.0516232s (thread); 0s (gc)
│ │ │ + -- used 0.134462s (cpu); 0.0619044s (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 0.00036261s (cpu); 0.000204163s (thread); 0s (gc)
│ │ │ + -- used 0.000378291s (cpu); 0.000215795s (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);
│ │ │ - -- .118865s elapsed
│ │ │ + -- .0777218s 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 0.136862s (cpu); 0.057483s (thread); 0s (gc)
│ │ │ + -- used 0.216759s (cpu); 0.135069s (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 1.68221s (cpu); 1.16665s (thread); 0s (gc)
│ │ │ + -- used 1.8057s (cpu); 1.17037s (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 0.0159702s (cpu); 0.0142733s (thread); 0s (gc)
│ │ │ + -- used 0.0160229s (cpu); 0.0147129s (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 4.93483s (cpu); 2.7697s (thread); 0s (gc)
│ │ │ + -- used 5.06374s (cpu); 2.89887s (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 0.000217288s (cpu); 0.000543729s (thread); 0s (gc)
│ │ │ + -- used 0.000777998s (cpu); 0.000569852s (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
│ │ │ - -- .0130746s elapsed
│ │ │ + -- .0134857s 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 0.051991s (cpu); 0.0518147s (thread); 0s (gc)
│ │ │ + -- used 0.0685238s (cpu); 0.0684537s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.132906s (cpu); 0.0516232s (thread); 0s (gc)
│ │ │ + -- used 0.134462s (cpu); 0.0619044s (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 0.00036261s (cpu); 0.000204163s (thread); 0s (gc)
│ │ │ + -- used 0.000378291s (cpu); 0.000215795s (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 0.051991s (cpu); 0.0518147s (thread); 0s (gc)
│ │ │ │ + -- used 0.0685238s (cpu); 0.0684537s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.132906s (cpu); 0.0516232s (thread); 0s (gc)
│ │ │ │ + -- used 0.134462s (cpu); 0.0619044s (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 0.00036261s (cpu); 0.000204163s (thread); 0s (gc)
│ │ │ │ + -- used 0.000378291s (cpu); 0.000215795s (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);
│ │ │ - -- .118865s elapsed
      
│ │ │ + -- .0777218s 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);
│ │ │ │ - -- .118865s elapsed
│ │ │ │ + -- .0777218s 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 0.136862s (cpu); 0.057483s (thread); 0s (gc)
      
│ │ │ + -- used 0.216759s (cpu); 0.135069s (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 0.136862s (cpu); 0.057483s (thread); 0s (gc)
│ │ │ │ + -- used 0.216759s (cpu); 0.135069s (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 1.68221s (cpu); 1.16665s (thread); 0s (gc)
│ │ │ + -- used 1.8057s (cpu); 1.17037s (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 1.68221s (cpu); 1.16665s (thread); 0s (gc)
│ │ │ │ + -- used 1.8057s (cpu); 1.17037s (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 0.0159702s (cpu); 0.0142733s (thread); 0s (gc)
│ │ │ + -- used 0.0160229s (cpu); 0.0147129s (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 4.93483s (cpu); 2.7697s (thread); 0s (gc)
│ │ │ + -- used 5.06374s (cpu); 2.89887s (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 0.0159702s (cpu); 0.0142733s (thread); 0s (gc)
│ │ │ │ + -- used 0.0160229s (cpu); 0.0147129s (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 4.93483s (cpu); 2.7697s (thread); 0s (gc)
│ │ │ │ + -- used 5.06374s (cpu); 2.89887s (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 0.000217288s (cpu); 0.000543729s (thread); 0s (gc)
│ │ │ + -- used 0.000777998s (cpu); 0.000569852s (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 0.000217288s (cpu); 0.000543729s (thread); 0s (gc)
│ │ │ │ + -- used 0.000777998s (cpu); 0.000569852s (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
│ │ │ - -- .0130746s elapsed
│ │ │ + -- .0134857s 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
│ │ │ │ - -- .0130746s elapsed
│ │ │ │ + -- .0134857s 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 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0120229)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00172536)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0329044)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .112853
│ │ │ - -- .136478s elapsed
│ │ │ + --  Time taken : .057011
│ │ │ + -- .0482601s elapsed
│ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000424534s elapsed
│ │ │ + -- .000456878s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0970795s elapsed
│ │ │ + -- .0529937s 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)
│ │ │ - -- .0638501s elapsed
│ │ │ + -- .0765909s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00161997s elapsed
│ │ │ + -- .00228552s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00120717s elapsed
│ │ │ + -- .0014992s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -72,20 +72,20 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0120229)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00172536)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0329044)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .112853
│ │ │ - -- .136478s elapsed
│ │ │ + --  Time taken : .057011
│ │ │ + -- .0482601s elapsed
│ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,21 +11,21 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0120229)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Linear            (time .00172536)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0329044)  #primes = 0 #prunedViaCodim = 0
│ │ │ │    Strategy: Factorization     (time 0)         #primes = 0 #prunedViaCodim = 0
│ │ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and
│ │ │ │  selecting minimal primes...
│ │ │ │ - --  Time taken : .112853
│ │ │ │ - -- .136478s elapsed
│ │ │ │ + --  Time taken : .057011
│ │ │ │ + -- .0482601s elapsed
│ │ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -131,25 +131,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000424534s elapsed
│ │ │ + -- .000456878s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0970795s elapsed
│ │ │ + -- .0529937s 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)
│ │ │ │ - -- .000424534s elapsed
│ │ │ │ + -- .000456878s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0970795s elapsed
│ │ │ │ + -- .0529937s 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)
│ │ │ - -- .0638501s elapsed
│ │ │ + -- .0765909s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00161997s elapsed
│ │ │ + -- .00228552s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00120717s elapsed
│ │ │ + -- .0014992s 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)
│ │ │ │ - -- .0638501s elapsed
│ │ │ │ + -- .0765909s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00161997s elapsed
│ │ │ │ + -- .00228552s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00120717s elapsed
│ │ │ │ + -- .0014992s 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 0.135113s (cpu); 0.0702819s (thread); 0s (gc)
│ │ │ + -- used 0.273466s (cpu); 0.0957844s (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 0.0105631s (cpu); 0.010908s (thread); 0s (gc)
│ │ │ + -- used 0.055153s (cpu); 0.0155352s (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 0.135113s (cpu); 0.0702819s (thread); 0s (gc)
│ │ │ + -- used 0.273466s (cpu); 0.0957844s (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 0.0105631s (cpu); 0.010908s (thread); 0s (gc)
│ │ │ + -- used 0.055153s (cpu); 0.0155352s (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 0.135113s (cpu); 0.0702819s (thread); 0s (gc)
│ │ │ │ + -- used 0.273466s (cpu); 0.0957844s (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 0.0105631s (cpu); 0.010908s (thread); 0s (gc)
│ │ │ │ + -- used 0.055153s (cpu); 0.0155352s (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 0.433561s (cpu); 0.305772s (thread); 0s (gc)
│ │ │ + -- used 0.419997s (cpu); 0.328674s (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 0.677061s (cpu); 0.510136s (thread); 0s (gc)
│ │ │ + -- used 0.694934s (cpu); 0.554217s (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 0.433561s (cpu); 0.305772s (thread); 0s (gc)
│ │ │ + -- used 0.419997s (cpu); 0.328674s (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 0.677061s (cpu); 0.510136s (thread); 0s (gc)
│ │ │ + -- used 0.694934s (cpu); 0.554217s (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 0.433561s (cpu); 0.305772s (thread); 0s (gc)
│ │ │ │ + -- used 0.419997s (cpu); 0.328674s (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 0.677061s (cpu); 0.510136s (thread); 0s (gc)
│ │ │ │ + -- used 0.694934s (cpu); 0.554217s (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})
│ │ │ - -- .00295289s elapsed
│ │ │ - -- .00268838s elapsed
│ │ │ - -- .000358591s elapsed
│ │ │ - -- .0027085s elapsed
│ │ │ - -- .0028867s elapsed
│ │ │ - -- .000229268s elapsed
│ │ │ - -- .00279824s elapsed
│ │ │ - -- .00285023s elapsed
│ │ │ - -- .000230892s elapsed
│ │ │ - -- .00285293s elapsed
│ │ │ - -- .00288533s elapsed
│ │ │ - -- .000249407s elapsed
│ │ │ ---backup directory created: /tmp/M2-88412-0/1
│ │ │ + -- .00353024s elapsed
│ │ │ + -- .00329009s elapsed
│ │ │ + -- .000376634s elapsed
│ │ │ + -- .00328407s elapsed
│ │ │ + -- .00336173s elapsed
│ │ │ + -- .000309655s elapsed
│ │ │ + -- .00328947s elapsed
│ │ │ + -- .00318813s elapsed
│ │ │ + -- .000269364s elapsed
│ │ │ + -- .00334917s elapsed
│ │ │ + -- .00365651s elapsed
│ │ │ + -- .000320772s elapsed
│ │ │ +--backup directory created: /tmp/M2-155212-0/1
│ │ │    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 = {{16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9,
│ │ │ +o9 = {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9,
│ │ │ +     20}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5, 0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15,
│ │ │ +     15, 17}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13, 5, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8,
│ │ │ +     19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14,
│ │ │ +     14, 18, 11, 5, 7}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13,
│ │ │ +     18, 16, 19, 5, 7, 11}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10,
│ │ │ +     9, 1, 16, 4, 0, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5,
│ │ │ +     2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {16, 6, 3, 19, 1, 2, 10, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3, 10, 4, 14, 6, 5,
│ │ │ +     14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {10, 9, 15, 20, 0, 8, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5,
│ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 19, 2, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ +     9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {16, 1, 3, 19, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5,
│ │ │ +     2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {13, 10, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2, 3, 4,
│ │ │ +     15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19, 20}, {10, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3,
│ │ │ +     15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1,
│ │ │ +     {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3,
│ │ │ +     3}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20},
│ │ │ +     5, 7}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     15, 17, 3}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3,
│ │ │ +     12, 17, 3, 15}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14,
│ │ │ +     17, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12,
│ │ │ +     15, 16, 17, 19, 20, 18}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6,
│ │ │ +     14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14,
│ │ │ +     2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ +     12, 13, 14, 18, 16, 19, 15, 17, 3}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10,
│ │ │ +     1, 17, 9, 0, 10, 20, 13, 18, 11, 5, 7}, {6, 10, 7, 13, 2, 20, 8, 18, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3,
│ │ │ +     12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 8, 15, 4, 7, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
│ │ │ +     12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 15, 20, 0, 8, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1,
│ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8,
│ │ │ +     9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12,
│ │ │ +     7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {1, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6,
│ │ │ +     16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19,
│ │ │ +     5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17,
│ │ │ +     10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15, 17},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2,
│ │ │ +     {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12,
│ │ │ +     18}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19},
│ │ │ +     20, 18}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ +     19, 5, 7, 11}, {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3,
│ │ │ +     1, 16, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     20, 13, 18, 11, 5, 7}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20,
│ │ │ +     14, 20, 17, 18, 11, 5, 7}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     17, 14, 11, 3, 5, 19, 20, 18}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     11, 14, 16, 8, 1, 12, 17, 3, 15}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7,
│ │ │ +     13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {3, 1, 6, 5, 16, 2, 0, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10,
│ │ │ +     9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {11, 10, 12, 13, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {16, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {1, 6, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15,
│ │ │ +     19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5,
│ │ │ +     10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {16, 14,
│ │ │ +     17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {0,
│ │ │ +     19, 20}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ +     18, 11, 5, 7}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9, 18, 19,
│ │ │ +     16, 19, 2, 8, 12}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8, 18, 3,
│ │ │ +     8, 0, 12, 17, 3, 15}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}}
│ │ │ +     14, 18, 16, 19, 15, 17, 3}}
│ │ │  
│ │ │  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})
│ │ │ - -- .00295289s elapsed
│ │ │ - -- .00268838s elapsed
│ │ │ - -- .000358591s elapsed
│ │ │ - -- .0027085s elapsed
│ │ │ - -- .0028867s elapsed
│ │ │ - -- .000229268s elapsed
│ │ │ - -- .00279824s elapsed
│ │ │ - -- .00285023s elapsed
│ │ │ - -- .000230892s elapsed
│ │ │ - -- .00285293s elapsed
│ │ │ - -- .00288533s elapsed
│ │ │ - -- .000249407s elapsed
│ │ │ ---backup directory created: /tmp/M2-88412-0/1
│ │ │ + -- .00353024s elapsed
│ │ │ + -- .00329009s elapsed
│ │ │ + -- .000376634s elapsed
│ │ │ + -- .00328407s elapsed
│ │ │ + -- .00336173s elapsed
│ │ │ + -- .000309655s elapsed
│ │ │ + -- .00328947s elapsed
│ │ │ + -- .00318813s elapsed
│ │ │ + -- .000269364s elapsed
│ │ │ + -- .00334917s elapsed
│ │ │ + -- .00365651s elapsed
│ │ │ + -- .000320772s elapsed
│ │ │ +--backup directory created: /tmp/M2-155212-0/1
│ │ │    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})
│ │ │ │ - -- .00295289s elapsed
│ │ │ │ - -- .00268838s elapsed
│ │ │ │ - -- .000358591s elapsed
│ │ │ │ - -- .0027085s elapsed
│ │ │ │ - -- .0028867s elapsed
│ │ │ │ - -- .000229268s elapsed
│ │ │ │ - -- .00279824s elapsed
│ │ │ │ - -- .00285023s elapsed
│ │ │ │ - -- .000230892s elapsed
│ │ │ │ - -- .00285293s elapsed
│ │ │ │ - -- .00288533s elapsed
│ │ │ │ - -- .000249407s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-88412-0/1
│ │ │ │ + -- .00353024s elapsed
│ │ │ │ + -- .00329009s elapsed
│ │ │ │ + -- .000376634s elapsed
│ │ │ │ + -- .00328407s elapsed
│ │ │ │ + -- .00336173s elapsed
│ │ │ │ + -- .000309655s elapsed
│ │ │ │ + -- .00328947s elapsed
│ │ │ │ + -- .00318813s elapsed
│ │ │ │ + -- .000269364s elapsed
│ │ │ │ + -- .00334917s elapsed
│ │ │ │ + -- .00365651s elapsed
│ │ │ │ + -- .000320772s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-155212-0/1
│ │ │ │    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 = {{16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9,
│ │ │ +o9 = {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9,
│ │ │ +     20}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5, 0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15,
│ │ │ +     15, 17}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13, 5, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8,
│ │ │ +     19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14,
│ │ │ +     14, 18, 11, 5, 7}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13,
│ │ │ +     18, 16, 19, 5, 7, 11}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10,
│ │ │ +     9, 1, 16, 4, 0, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5,
│ │ │ +     2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {16, 6, 3, 19, 1, 2, 10, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3, 10, 4, 14, 6, 5,
│ │ │ +     14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {10, 9, 15, 20, 0, 8, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5,
│ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 19, 2, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ +     9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {16, 1, 3, 19, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5,
│ │ │ +     2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {13, 10, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2, 3, 4,
│ │ │ +     15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19, 20}, {10, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3,
│ │ │ +     15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1,
│ │ │ +     {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3,
│ │ │ +     3}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20},
│ │ │ +     5, 7}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     15, 17, 3}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3,
│ │ │ +     12, 17, 3, 15}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14,
│ │ │ +     17, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12,
│ │ │ +     15, 16, 17, 19, 20, 18}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6,
│ │ │ +     14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14,
│ │ │ +     2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ +     12, 13, 14, 18, 16, 19, 15, 17, 3}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10,
│ │ │ +     1, 17, 9, 0, 10, 20, 13, 18, 11, 5, 7}, {6, 10, 7, 13, 2, 20, 8, 18, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3,
│ │ │ +     12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 8, 15, 4, 7, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
│ │ │ +     12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 15, 20, 0, 8, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1,
│ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8,
│ │ │ +     9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12,
│ │ │ +     7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {1, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6,
│ │ │ +     16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19,
│ │ │ +     5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17,
│ │ │ +     10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15, 17},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2,
│ │ │ +     {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12,
│ │ │ +     18}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19},
│ │ │ +     20, 18}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ +     19, 5, 7, 11}, {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3,
│ │ │ +     1, 16, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     20, 13, 18, 11, 5, 7}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20,
│ │ │ +     14, 20, 17, 18, 11, 5, 7}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     17, 14, 11, 3, 5, 19, 20, 18}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     11, 14, 16, 8, 1, 12, 17, 3, 15}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17,
│ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7,
│ │ │ +     13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {3, 1, 6, 5, 16, 2, 0, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10,
│ │ │ +     9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {11, 10, 12, 13, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {16, 6, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {1, 6, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15,
│ │ │ +     19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5,
│ │ │ +     10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {16, 14,
│ │ │ +     17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {0,
│ │ │ +     19, 20}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ +     18, 11, 5, 7}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9, 18, 19,
│ │ │ +     16, 19, 2, 8, 12}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8, 18, 3,
│ │ │ +     8, 0, 12, 17, 3, 15}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 17}}
│ │ │ +     14, 18, 16, 19, 15, 17, 3}}
│ │ │  
│ │ │  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 = {{16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9,
│ │ │ │ +o9 = {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9,
│ │ │ │ +     20}, {7, 10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 11, 1, 16, 5, 0, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15,
│ │ │ │ +     15, 17}, {7, 1, 9, 0, 12, 17, 2, 3, 4, 8, 10, 15, 6, 11, 14, 13, 5, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8,
│ │ │ │ +     19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14,
│ │ │ │ +     14, 18, 11, 5, 7}, {0, 1, 15, 20, 4, 8, 6, 12, 17, 9, 10, 2, 3, 13, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13,
│ │ │ │ +     18, 16, 19, 5, 7, 11}, {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10,
│ │ │ │ +     9, 1, 16, 4, 0, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0,
│ │ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {0, 9, 15, 20, 10, 8, 14, 12, 17, 1, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5,
│ │ │ │ +     2, 3, 13, 6, 18, 16, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3,
│ │ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {16, 6, 3, 19, 1, 2, 10, 8, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3, 10, 4, 14, 6, 5,
│ │ │ │ +     14, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {10, 9, 15, 20, 0, 8, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5,
│ │ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 19, 2, 6, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ │ +     9, 7, 20, 4, 14, 11, 18, 16, 1, 8, 0, 12, 17, 3, 15}, {16, 1, 3, 19, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5,
│ │ │ │ +     2, 4, 8, 15, 6, 10, 12, 17, 0, 14, 20, 13, 18, 11, 5, 7}, {13, 10, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {0, 1, 2, 3, 4,
│ │ │ │ +     15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 18, 19, 20}, {10, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3,
│ │ │ │ +     15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ │ +     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {0, 1,
│ │ │ │ +     {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3,
│ │ │ │ +     3}, {19, 14, 4, 16, 2, 3, 8, 15, 6, 12, 1, 17, 9, 20, 10, 0, 18, 13, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20},
│ │ │ │ +     5, 7}, {0, 1, 6, 20, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13, 14, 18, 16, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ │ +     15, 17, 3}, {13, 1, 19, 2, 6, 5, 9, 7, 20, 4, 10, 11, 18, 16, 14, 8, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3,
│ │ │ │ +     12, 17, 3, 15}, {3, 1, 2, 19, 4, 16, 6, 0, 8, 9, 10, 13, 12, 15, 14, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14,
│ │ │ │ +     17, 18, 11, 5, 7}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12,
│ │ │ │ +     15, 16, 17, 19, 20, 18}, {0, 1, 6, 3, 7, 2, 11, 8, 9, 5, 10, 12, 4, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6,
│ │ │ │ +     14, 15, 16, 17, 18, 19, 20}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14,
│ │ │ │ +     2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ │ +     12, 13, 14, 18, 16, 19, 15, 17, 3}, {16, 14, 4, 19, 2, 3, 8, 15, 6, 12,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10,
│ │ │ │ +     1, 17, 9, 0, 10, 20, 13, 18, 11, 5, 7}, {6, 10, 7, 13, 2, 20, 8, 18, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3,
│ │ │ │ +     12, 14, 19, 5, 9, 1, 16, 4, 0, 3, 15, 17}, {13, 10, 8, 15, 4, 7, 6, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
│ │ │ │ +     12, 9, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {10, 9, 15, 20, 0, 8, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1,
│ │ │ │ +     12, 17, 16, 4, 2, 3, 14, 6, 18, 1, 19, 5, 7, 11}, {13, 10, 8, 15, 6, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8,
│ │ │ │ +     9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {0, 1, 2, 20, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12,
│ │ │ │ +     7, 6, 11, 8, 9, 10, 5, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {1, 6, 3, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6,
│ │ │ │ +     16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19,
│ │ │ │ +     5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18}, {11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17,
│ │ │ │ +     10, 8, 13, 6, 20, 9, 18, 12, 4, 14, 19, 2, 5, 1, 16, 7, 0, 3, 15, 17},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2,
│ │ │ │ +     {13, 10, 8, 15, 6, 7, 9, 11, 12, 4, 14, 5, 2, 16, 1, 17, 0, 3, 19, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12,
│ │ │ │ +     18}, {15, 9, 0, 7, 10, 8, 14, 12, 13, 1, 4, 2, 16, 17, 6, 11, 3, 5, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19},
│ │ │ │ +     20, 18}, {10, 9, 15, 20, 0, 8, 13, 12, 17, 16, 4, 2, 3, 14, 6, 18, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ │ +     19, 5, 7, 11}, {10, 9, 7, 0, 12, 17, 2, 3, 11, 8, 4, 15, 5, 14, 6, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3,
│ │ │ │ +     1, 16, 19, 20, 18}, {16, 6, 3, 19, 10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ │ +     20, 13, 18, 11, 5, 7}, {3, 1, 6, 19, 0, 2, 13, 8, 9, 16, 10, 12, 4, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20,
│ │ │ │ +     14, 20, 17, 18, 11, 5, 7}, {15, 1, 0, 7, 4, 8, 6, 12, 13, 9, 10, 2, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ │ +     17, 14, 11, 3, 5, 19, 20, 18}, {10, 0, 5, 2, 6, 19, 9, 20, 7, 4, 13, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ │ +     11, 14, 16, 8, 1, 12, 17, 3, 15}, {13, 10, 8, 15, 6, 7, 9, 11, 12, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17,
│ │ │ │ +     14, 5, 2, 16, 1, 17, 0, 3, 19, 20, 18}, {1, 6, 3, 19, 16, 2, 0, 8, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7,
│ │ │ │ +     13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {3, 1, 6, 5, 16, 2, 0, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10,
│ │ │ │ +     9, 13, 10, 12, 4, 15, 14, 7, 17, 11, 18, 19, 20}, {11, 10, 12, 13, 6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ │ +     20, 9, 18, 2, 4, 14, 19, 8, 5, 1, 16, 7, 0, 3, 15, 17}, {16, 6, 3, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19,
│ │ │ │ +     10, 2, 14, 8, 15, 1, 9, 12, 17, 0, 4, 20, 13, 18, 11, 5, 7}, {1, 6, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15,
│ │ │ │ +     19, 16, 2, 0, 8, 15, 13, 9, 12, 17, 10, 4, 20, 14, 18, 11, 5, 7}, {13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5,
│ │ │ │ +     10, 5, 15, 4, 12, 6, 2, 7, 9, 14, 8, 11, 16, 1, 17, 0, 3, 19, 20, 18},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ │ +     {6, 10, 12, 13, 11, 20, 5, 18, 2, 7, 14, 19, 8, 9, 1, 16, 4, 0, 3, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {16, 14,
│ │ │ │ +     17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {0,
│ │ │ │ +     19, 20}, {3, 6, 1, 19, 16, 2, 0, 8, 10, 13, 9, 12, 14, 15, 4, 20, 17,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ │ +     18, 11, 5, 7}, {0, 1, 6, 20, 7, 15, 11, 17, 9, 5, 10, 3, 4, 13, 14, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9, 18, 19,
│ │ │ │ +     16, 19, 2, 8, 12}, {13, 10, 19, 2, 6, 5, 9, 7, 20, 4, 14, 11, 18, 16, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8, 18, 3,
│ │ │ │ +     8, 0, 12, 17, 3, 15}, {0, 1, 2, 20, 4, 7, 6, 11, 8, 9, 10, 5, 12, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 17}}
│ │ │ │ +     14, 18, 16, 19, 15, 17, 3}}
│ │ │ │  
│ │ │ │  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/M2-148190-0/0-in.ms -o /tmp/M2-148190-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-269296-0/0-in.ms -o /tmp/M2-269296-0/0-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 = 1156623833
│ │ │ +Initial prime = 1159533211
│ │ │  
│ │ │  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         0.05 | 0.18         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.13 sec
│ │ │ -overall(cpu)            0.39 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.01 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │  symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.07 sec  57.4%
│ │ │ -convert                 0.05 sec  42.5%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +update                  0.00 sec  95.4%
│ │ │ +convert                 0.00 sec   0.3%
│ │ │ +linear algebra          0.00 sec   0.2%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -78,28 +78,28 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1076274691
│ │ │ +New prime = 1199500661
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.06 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%>
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.36 sec (elapsed) /  1.08 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │   
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  Max coeff. bitsize                1
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/_msolve__Real__Solutions.out
│ │ │ @@ -11,103 +11,107 @@
│ │ │               2       2
│ │ │  o2 = ideal (x  - x, y  - 5)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │  
│ │ │ -                                2795809011                       
│ │ │ -o3 = {{{- ------------------------------------------------------,
│ │ │ -          383123885216472214589586756787577295904684780545900544 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -                            452948303                            9603838835 
│ │ │ -     ------------------------------------------------------}, {- ----------,
│ │ │ -     383123885216472214589586756787577295904684780545900544      4294967296 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       4801919417                             58033413                     
│ │ │ -     - ----------}}, {{- -------------------------------------------------,
│ │ │ -       2147483648        2923003274661805836407369665432566039311865085952 
│ │ │ +                               15357363263                     
│ │ │ +o3 = {{{- ----------------------------------------------------,
│ │ │ +          1496577676626844588240573268701473812127674924007424 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                          12038187159                        4801919417 
│ │ │ +                          10540648493                        4801919417 
│ │ │       ----------------------------------------------------}, {----------,
│ │ │ -     5986310706507378352962293074805895248510699696029696    2147483648 
│ │ │ +     1496577676626844588240573268701473812127674924007424    2147483648 
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9603838835      8589934591  8589934593      9603838835    4801919417   
│ │ │ -     ----------}}, {{----------, ----------}, {- ----------, - ----------}},
│ │ │ -     4294967296      8589934592  8589934592      4294967296    2147483648   
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       8589934591  8589934593    4801919417  9603838835
│ │ │ -     {{----------, ----------}, {----------, ----------}}}
│ │ │ -       8589934592  8589934592    2147483648  4294967296
│ │ │ +     9603838835      8589934591  8589934593    4801919417  9603838835       
│ │ │ +     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
│ │ │ +     4294967296      8589934592  8589934592    2147483648  4294967296       
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                                  11523779795                             
│ │ │ +     --------------------------------------------------------------------,
│ │ │ +     13479973333575319897333507543509815336818572211270286240551805124608 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                                  3564748679                                 
│ │ │ +     -------------------------------------------------------------------}, {-
│ │ │ +     6739986666787659948666753771754907668409286105635143120275902562304     
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     9603838835    4801919417      8589934591  8589934593      9603838835   
│ │ │ +     ----------, - ----------}}, {{----------, ----------}, {- ----------, -
│ │ │ +     4294967296    2147483648      8589934592  8589934592      4294967296   
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     4801919417
│ │ │ +     ----------}}}
│ │ │ +     2147483648
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -                           442070429                        19207677669      
│ │ │ -o4 = {{-------------------------------------------------, - -----------}, {1,
│ │ │ -       1461501637330902918203684832716283019655932542976     8589934592      
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       19207677669                        532313171                    
│ │ │ -     - -----------}, {------------------------------------------------,
│ │ │ -        8589934592    730750818665451459101842416358141509827966271488 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     19207677669       19207677669
│ │ │ -     -----------}, {1, -----------}}
│ │ │ -      8589934592        8589934592
│ │ │ +                              2408357385                       19207677669  
│ │ │ +o4 = {{- ----------------------------------------------------, -----------},
│ │ │ +         1496577676626844588240573268701473812127674924007424   8589934592  
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +         19207677669                                   4394282437            
│ │ │ +     {1, -----------}, {- ---------------------------------------------------
│ │ │ +          8589934592      269599466671506397946670150870196306736371444225405
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                          19207677669         19207677669
│ │ │ +     -----------------, - -----------}, {1, - -----------}}
│ │ │ +     72481103610249216     8589934592          8589934592
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o5 = {{[-2.54268e-40,7.63397e-41], [-2.23607,-2.23607]}, {[1,1],
│ │ │ +o5 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [-2.23607,-2.23607]}, {[-1.68492e-41,3.3486e-41], [2.23607,2.23607]},
│ │ │ +     [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[1,1], [2.23607,2.23607]}}
│ │ │ +     {[1,1], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │  
│ │ │ -o6 = {{[-2.56438e-42,6.84394e-42], [-2.23633,-2.23535]}, {[.999512,1.00049],
│ │ │ +o6 = {{[-1.02631e-41,7.04573e-42], [2.23535,2.23633]}, {[.999512,1.00049],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [-2.23633,-2.23535]}, {[-7.41161e-58,1.00626e-57], [2.23535,2.23633]},
│ │ │ +     [2.23535,2.23633]}, {[-8.55042e-58,5.28956e-58], [-2.23633,-2.23535]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[.999512,1.00049], [2.23535,2.23633]}}
│ │ │ +     {[.999512,1.00049], [-2.23633,-2.23535]}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │ -o7 = {{1.88872e-59, -2.23607}, {1, -2.23607}, {-9.4134e-41, 2.23607}, {1,
│ │ │ +o7 = {{-1.60924e-42, 2.23607}, {1, 2.23607}, {-1.62993e-58, -2.23607}, {1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2.23607}}
│ │ │ +     -2.23607}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │  
│ │ │ -o8 = {{-9.01315e-42, 2.23584}, {1, 2.23584}, {-4.39447e-41, -2.23584}, {1,
│ │ │ +o8 = {{-1.60869e-42, 2.23584}, {1, 2.23584}, {-1.63043e-58, -2.23584}, {1,
│ │ │       ------------------------------------------------------------------------
│ │ │       -2.23584}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : I = ideal {(x-1)*x^3, (y^2-5)^2}
│ │ │  
│ │ │               4    3   4      2
│ │ │  o9 = ideal (x  - x , y  - 10y  + 25)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o10 = {{[-1.6606e-40,1.81154e-40], [-2.23607,-2.23607]}, {[1,1],
│ │ │ +o10 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [-2.23607,-2.23607]}, {[-2.33787e-40,2.43282e-40], [2.23607,2.23607]},
│ │ │ +      [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {[1,1], [2.23607,2.23607]}}
│ │ │ +      {[1,1], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o10 : List
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html
│ │ │ @@ -100,102 +100,106 @@
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │  
│ │ │ -                                2795809011                       
│ │ │ -o3 = {{{- ------------------------------------------------------,
│ │ │ -          383123885216472214589586756787577295904684780545900544 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -                            452948303                            9603838835 
│ │ │ -     ------------------------------------------------------}, {- ----------,
│ │ │ -     383123885216472214589586756787577295904684780545900544      4294967296 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       4801919417                             58033413                     
│ │ │ -     - ----------}}, {{- -------------------------------------------------,
│ │ │ -       2147483648        2923003274661805836407369665432566039311865085952 
│ │ │ +                               15357363263                     
│ │ │ +o3 = {{{- ----------------------------------------------------,
│ │ │ +          1496577676626844588240573268701473812127674924007424 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                          12038187159                        4801919417 
│ │ │ +                          10540648493                        4801919417 
│ │ │       ----------------------------------------------------}, {----------,
│ │ │ -     5986310706507378352962293074805895248510699696029696    2147483648 
│ │ │ +     1496577676626844588240573268701473812127674924007424    2147483648 
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9603838835      8589934591  8589934593      9603838835    4801919417   
│ │ │ -     ----------}}, {{----------, ----------}, {- ----------, - ----------}},
│ │ │ -     4294967296      8589934592  8589934592      4294967296    2147483648   
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       8589934591  8589934593    4801919417  9603838835
│ │ │ -     {{----------, ----------}, {----------, ----------}}}
│ │ │ -       8589934592  8589934592    2147483648  4294967296
│ │ │ +     9603838835      8589934591  8589934593    4801919417  9603838835       
│ │ │ +     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
│ │ │ +     4294967296      8589934592  8589934592    2147483648  4294967296       
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                                  11523779795                             
│ │ │ +     --------------------------------------------------------------------,
│ │ │ +     13479973333575319897333507543509815336818572211270286240551805124608 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                                  3564748679                                 
│ │ │ +     -------------------------------------------------------------------}, {-
│ │ │ +     6739986666787659948666753771754907668409286105635143120275902562304     
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     9603838835    4801919417      8589934591  8589934593      9603838835   
│ │ │ +     ----------, - ----------}}, {{----------, ----------}, {- ----------, -
│ │ │ +     4294967296    2147483648      8589934592  8589934592      4294967296   
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     4801919417
│ │ │ +     ----------}}}
│ │ │ +     2147483648
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -                           442070429                        19207677669      
│ │ │ -o4 = {{-------------------------------------------------, - -----------}, {1,
│ │ │ -       1461501637330902918203684832716283019655932542976     8589934592      
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -       19207677669                        532313171                    
│ │ │ -     - -----------}, {------------------------------------------------,
│ │ │ -        8589934592    730750818665451459101842416358141509827966271488 
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     19207677669       19207677669
│ │ │ -     -----------}, {1, -----------}}
│ │ │ -      8589934592        8589934592
│ │ │ +                              2408357385                       19207677669  
│ │ │ +o4 = {{- ----------------------------------------------------, -----------},
│ │ │ +         1496577676626844588240573268701473812127674924007424   8589934592  
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +         19207677669                                   4394282437            
│ │ │ +     {1, -----------}, {- ---------------------------------------------------
│ │ │ +          8589934592      269599466671506397946670150870196306736371444225405
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +                          19207677669         19207677669
│ │ │ +     -----------------, - -----------}, {1, - -----------}}
│ │ │ +     72481103610249216     8589934592          8589934592
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o5 = {{[-2.54268e-40,7.63397e-41], [-2.23607,-2.23607]}, {[1,1],
│ │ │ +o5 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [-2.23607,-2.23607]}, {[-1.68492e-41,3.3486e-41], [2.23607,2.23607]},
│ │ │ +     [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[1,1], [2.23607,2.23607]}}
│ │ │ +     {[1,1], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │  
│ │ │ -o6 = {{[-2.56438e-42,6.84394e-42], [-2.23633,-2.23535]}, {[.999512,1.00049],
│ │ │ +o6 = {{[-1.02631e-41,7.04573e-42], [2.23535,2.23633]}, {[.999512,1.00049],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [-2.23633,-2.23535]}, {[-7.41161e-58,1.00626e-57], [2.23535,2.23633]},
│ │ │ +     [2.23535,2.23633]}, {[-8.55042e-58,5.28956e-58], [-2.23633,-2.23535]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[.999512,1.00049], [2.23535,2.23633]}}
│ │ │ +     {[.999512,1.00049], [-2.23633,-2.23535]}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │ -o7 = {{1.88872e-59, -2.23607}, {1, -2.23607}, {-9.4134e-41, 2.23607}, {1,
│ │ │ +o7 = {{-1.60924e-42, 2.23607}, {1, 2.23607}, {-1.62993e-58, -2.23607}, {1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2.23607}}
│ │ │ +     -2.23607}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │  
│ │ │ -o8 = {{-9.01315e-42, 2.23584}, {1, 2.23584}, {-4.39447e-41, -2.23584}, {1,
│ │ │ +o8 = {{-1.60869e-42, 2.23584}, {1, 2.23584}, {-1.63043e-58, -2.23584}, {1,
│ │ │       ------------------------------------------------------------------------
│ │ │       -2.23584}}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
    
│ │ │ @@ -213,19 +217,19 @@
│ │ │  o9 : Ideal of R
│ │ │              
    
│ │ │                
    i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │  
│ │ │ -o10 = {{[-1.6606e-40,1.81154e-40], [-2.23607,-2.23607]}, {[1,1],
│ │ │ +o10 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [-2.23607,-2.23607]}, {[-2.33787e-40,2.43282e-40], [2.23607,2.23607]},
│ │ │ +      [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {[1,1], [2.23607,2.23607]}}
│ │ │ +      {[1,1], [-2.23607,-2.23607]}}
│ │ │  
│ │ │  o10 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,82 +43,86 @@
│ │ │ │  
│ │ │ │               2       2
│ │ │ │  o2 = ideal (x  - x, y  - 5)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : rationalIntervalSols = msolveRealSolutions I
│ │ │ │  
│ │ │ │ -                                2795809011
│ │ │ │ -o3 = {{{- ------------------------------------------------------,
│ │ │ │ -          383123885216472214589586756787577295904684780545900544
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -                            452948303                            9603838835
│ │ │ │ -     ------------------------------------------------------}, {- ----------,
│ │ │ │ -     383123885216472214589586756787577295904684780545900544      4294967296
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -       4801919417                             58033413
│ │ │ │ -     - ----------}}, {{- -------------------------------------------------,
│ │ │ │ -       2147483648        2923003274661805836407369665432566039311865085952
│ │ │ │ +                               15357363263
│ │ │ │ +o3 = {{{- ----------------------------------------------------,
│ │ │ │ +          1496577676626844588240573268701473812127674924007424
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                          12038187159                        4801919417
│ │ │ │ +                          10540648493                        4801919417
│ │ │ │       ----------------------------------------------------}, {----------,
│ │ │ │ -     5986310706507378352962293074805895248510699696029696    2147483648
│ │ │ │ +     1496577676626844588240573268701473812127674924007424    2147483648
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9603838835      8589934591  8589934593      9603838835    4801919417
│ │ │ │ -     ----------}}, {{----------, ----------}, {- ----------, - ----------}},
│ │ │ │ -     4294967296      8589934592  8589934592      4294967296    2147483648
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -       8589934591  8589934593    4801919417  9603838835
│ │ │ │ -     {{----------, ----------}, {----------, ----------}}}
│ │ │ │ -       8589934592  8589934592    2147483648  4294967296
│ │ │ │ +     9603838835      8589934591  8589934593    4801919417  9603838835
│ │ │ │ +     ----------}}, {{----------, ----------}, {----------, ----------}}, {{-
│ │ │ │ +     4294967296      8589934592  8589934592    2147483648  4294967296
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +                                  11523779795
│ │ │ │ +     --------------------------------------------------------------------,
│ │ │ │ +     13479973333575319897333507543509815336818572211270286240551805124608
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +                                  3564748679
│ │ │ │ +     -------------------------------------------------------------------}, {-
│ │ │ │ +     6739986666787659948666753771754907668409286105635143120275902562304
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     9603838835    4801919417      8589934591  8589934593      9603838835
│ │ │ │ +     ----------, - ----------}}, {{----------, ----------}, {- ----------, -
│ │ │ │ +     4294967296    2147483648      8589934592  8589934592      4294967296
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     4801919417
│ │ │ │ +     ----------}}}
│ │ │ │ +     2147483648
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │ │  
│ │ │ │ -                           442070429                        19207677669
│ │ │ │ -o4 = {{-------------------------------------------------, - -----------}, {1,
│ │ │ │ -       1461501637330902918203684832716283019655932542976     8589934592
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -       19207677669                        532313171
│ │ │ │ -     - -----------}, {------------------------------------------------,
│ │ │ │ -        8589934592    730750818665451459101842416358141509827966271488
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     19207677669       19207677669
│ │ │ │ -     -----------}, {1, -----------}}
│ │ │ │ -      8589934592        8589934592
│ │ │ │ +                              2408357385                       19207677669
│ │ │ │ +o4 = {{- ----------------------------------------------------, -----------},
│ │ │ │ +         1496577676626844588240573268701473812127674924007424   8589934592
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +         19207677669                                   4394282437
│ │ │ │ +     {1, -----------}, {- ---------------------------------------------------
│ │ │ │ +          8589934592      269599466671506397946670150870196306736371444225405
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +                          19207677669         19207677669
│ │ │ │ +     -----------------, - -----------}, {1, - -----------}}
│ │ │ │ +     72481103610249216     8589934592          8589934592
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
│ │ │ │  
│ │ │ │ -o5 = {{[-2.54268e-40,7.63397e-41], [-2.23607,-2.23607]}, {[1,1],
│ │ │ │ +o5 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     [-2.23607,-2.23607]}, {[-1.68492e-41,3.3486e-41], [2.23607,2.23607]},
│ │ │ │ +     [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {[1,1], [2.23607,2.23607]}}
│ │ │ │ +     {[1,1], [-2.23607,-2.23607]}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
│ │ │ │  
│ │ │ │ -o6 = {{[-2.56438e-42,6.84394e-42], [-2.23633,-2.23535]}, {[.999512,1.00049],
│ │ │ │ +o6 = {{[-1.02631e-41,7.04573e-42], [2.23535,2.23633]}, {[.999512,1.00049],
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     [-2.23633,-2.23535]}, {[-7.41161e-58,1.00626e-57], [2.23535,2.23633]},
│ │ │ │ +     [2.23535,2.23633]}, {[-8.55042e-58,5.28956e-58], [-2.23633,-2.23535]},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {[.999512,1.00049], [2.23535,2.23633]}}
│ │ │ │ +     {[.999512,1.00049], [-2.23633,-2.23535]}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │ │  
│ │ │ │ -o7 = {{1.88872e-59, -2.23607}, {1, -2.23607}, {-9.4134e-41, 2.23607}, {1,
│ │ │ │ +o7 = {{-1.60924e-42, 2.23607}, {1, 2.23607}, {-1.62993e-58, -2.23607}, {1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2.23607}}
│ │ │ │ +     -2.23607}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
│ │ │ │  
│ │ │ │ -o8 = {{-9.01315e-42, 2.23584}, {1, 2.23584}, {-4.39447e-41, -2.23584}, {1,
│ │ │ │ +o8 = {{-1.60869e-42, 2.23584}, {1, 2.23584}, {-1.63043e-58, -2.23584}, {1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -2.23584}}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  Note in cases where solutions have multiplicity this is not reflected in the
│ │ │ │  output. While the solver does not return multiplicities, it reliably outputs
│ │ │ │  the verified isolating intervals for multiple solutions.
│ │ │ │ @@ -126,19 +130,19 @@
│ │ │ │  
│ │ │ │               4    3   4      2
│ │ │ │  o9 = ideal (x  - x , y  - 10y  + 25)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : floatApproxSols = msolveRealSolutions(I, RRi)
│ │ │ │  
│ │ │ │ -o10 = {{[-1.6606e-40,1.81154e-40], [-2.23607,-2.23607]}, {[1,1],
│ │ │ │ +o10 = {{[-1.02617e-41,7.04317e-42], [2.23607,2.23607]}, {[1,1],
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      [-2.23607,-2.23607]}, {[-2.33787e-40,2.43282e-40], [2.23607,2.23607]},
│ │ │ │ +      [2.23607,2.23607]}, {[-8.54881e-58,5.28896e-58], [-2.23607,-2.23607]},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {[1,1], [2.23607,2.23607]}}
│ │ │ │ +      {[1,1], [-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/M2-148190-0/0-in.ms -o /tmp/M2-148190-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-269296-0/0-in.ms -o /tmp/M2-269296-0/0-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 = 1156623833
│ │ │ +Initial prime = 1159533211
│ │ │  
│ │ │  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         0.05 | 0.18         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.13 sec
│ │ │ -overall(cpu)            0.39 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.01 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │  symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.07 sec  57.4%
│ │ │ -convert                 0.05 sec  42.5%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +update                  0.00 sec  95.4%
│ │ │ +convert                 0.00 sec   0.3%
│ │ │ +linear algebra          0.00 sec   0.2%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -147,28 +147,28 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1076274691
│ │ │ +New prime = 1199500661
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.06 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%>
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.36 sec (elapsed) /  1.08 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │   
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  Max coeff. bitsize                1
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-148190-0/0-in.ms -o /
│ │ │ │ -tmp/M2-148190-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-269296-0/0-in.ms -o /
│ │ │ │ +tmp/M2-269296-0/0-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 = 1156623833
│ │ │ │ +Initial prime = 1159533211
│ │ │ │  
│ │ │ │  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
│ │ │ │ -0.05 | 0.18
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.13 sec
│ │ │ │ -overall(cpu)            0.39 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.01 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │  symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.07 sec  57.4%
│ │ │ │ -convert                 0.05 sec  42.5%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +update                  0.00 sec  95.4%
│ │ │ │ +convert                 0.00 sec   0.3%
│ │ │ │ +linear algebra          0.00 sec   0.2%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -106,30 +106,30 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1076274691
│ │ │ │ +New prime = 1199500661
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.06 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%>
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.36 sec (elapsed) /  1.08 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -20,16 +20,16 @@
│ │ │  o4 = map (S, R, {t s , t s , t s , t s , t s , t s })
│ │ │                    1 1   1 2   1 3   2 1   2 2   2 3
│ │ │  
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │ - -- .00253166s elapsed
│ │ │ - -- .00917226s elapsed
│ │ │ + -- .00245917s elapsed
│ │ │ + -- .0117132s elapsed
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -114,16 +114,16 @@
│ │ │  o4 : RingMap S <-- R
    │ │ │
    │ │ │ 
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │ - -- .00253166s elapsed
│ │ │ - -- .00917226s elapsed
│ │ │ + -- .00245917s elapsed
│ │ │ + -- .0117132s elapsed
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,16 +48,16 @@
│ │ │ │  o4 = map (S, R, {t s , t s , t s , t s , t s , t s })
│ │ │ │                    1 1   1 2   1 3   2 1   2 2   2 3
│ │ │ │  
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │ - -- .00253166s elapsed
│ │ │ │ - -- .00917226s elapsed
│ │ │ │ + -- .00245917s elapsed
│ │ │ │ + -- .0117132s elapsed
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:26 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:27 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=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ ├── ./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
│ │ │ - -- .0937701s elapsed
│ │ │ + -- .050861s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .196612s elapsed
│ │ │ + -- .153266s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .152482s elapsed
│ │ │ + -- .164015s 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
│ │ │ - -- .21113s elapsed
│ │ │ + -- .154545s 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
│ │ │ - -- .246486s elapsed
│ │ │ + -- .186207s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.45868s elapsed
│ │ │ + -- 1.46241s 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
│ │ │ - -- .0937701s elapsed
│ │ │ + -- .050861s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │
    │ │ │ 
    i4 : elapsedTime monjMult I
│ │ │ - -- .196612s elapsed
│ │ │ + -- .153266s elapsed
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │ 
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .152482s elapsed
│ │ │ + -- .164015s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    │ │ │
    │ │ │ 
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .21113s elapsed
│ │ │ + -- .154545s 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 │ │ │ │ - -- .21113s elapsed │ │ │ │ + -- .154545s 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
│ │ │ - -- .246486s elapsed
│ │ │ + -- .186207s elapsed
│ │ │  
│ │ │  o3 = 192
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime jMult I
│ │ │ - -- 1.45868s elapsed
│ │ │ + -- 1.46241s 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 │ │ │ │ - -- .246486s elapsed │ │ │ │ + -- .186207s elapsed │ │ │ │ │ │ │ │ o3 = 192 │ │ │ │ i4 : elapsedTime jMult I │ │ │ │ - -- 1.45868s elapsed │ │ │ │ + -- 1.46241s 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 3.18819s (cpu); 1.79242s (thread); 0s (gc) │ │ │ + -- used 3.29199s (cpu); 1.91042s (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 3.24858s (cpu); 1.95299s (thread); 0s (gc) │ │ │ + -- used 3.8289s (cpu); 2.07558s (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 7.95558s (cpu); 4.85491s (thread); 0s (gc) │ │ │ + -- used 6.88385s (cpu); 4.99227s (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 5.11651s (cpu); 3.76995s (thread); 0s (gc) │ │ │ + -- used 4.53612s (cpu); 3.88967s (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 0.0996701s (cpu); 0.0991169s (thread); 0s (gc) │ │ │ + -- used 0.161512s (cpu); 0.140129s (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 1.93661s (cpu); 1.32686s (thread); 0s (gc) │ │ │ + -- used 2.02148s (cpu); 1.28678s (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 3.60509s (cpu); 2.34063s (thread); 0s (gc) │ │ │ + -- used 4.42444s (cpu); 2.67204s (thread); 0s (gc) │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ - -- used 0.336985s (cpu); 0.268253s (thread); 0s (gc) │ │ │ + -- used 0.364536s (cpu); 0.296807s (thread); 0s (gc) │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : time degree Phi │ │ │ - -- used 0.272327s (cpu); 0.230404s (thread); 0s (gc) │ │ │ + -- used 0.334696s (cpu); 0.269259s (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 0.442484s (cpu); 0.361777s (thread); 0s (gc) │ │ │ + -- used 0.580944s (cpu); 0.410309s (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 0.000311614s (cpu); 0.000167294s (thread); 0s (gc) │ │ │ + -- used 0.0017867s (cpu); 0.000171259s (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 0.00114434s (cpu); 0.000225412s (thread); 0s (gc) │ │ │ + -- used 0.00229552s (cpu); 0.000259652s (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 1.36555s (cpu); 1.07435s (thread); 0s (gc) │ │ │ + -- used 1.20728s (cpu); 1.01733s (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 1.5099e-05s (cpu); 0.000385823s (thread); 0s (gc) │ │ │ + -- used 0.000126961s (cpu); 0.000425204s (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 0.126166s (cpu); 0.0742722s (thread); 0s (gc) │ │ │ + -- used 0.177544s (cpu); 0.0552381s (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 0.037857s (cpu); 0.0392192s (thread); 0s (gc) │ │ │ + -- used 0.0532142s (cpu); 0.04297s (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 0.22123s (cpu); 0.167286s (thread); 0s (gc) │ │ │ + -- used 0.293397s (cpu); 0.166609s (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 0.206519s (cpu); 0.130282s (thread); 0s (gc) │ │ │ + -- used 0.172518s (cpu); 0.160845s (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 6.14478s (cpu); 2.61305s (thread); 0s (gc) │ │ │ + -- used 10.7568s (cpu); 2.88365s (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 0.368667s (cpu); 0.308957s (thread); 0s (gc) │ │ │ + -- used 0.431024s (cpu); 0.324297s (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 1.62165s (cpu); 1.07916s (thread); 0s (gc) │ │ │ + -- used 1.54946s (cpu); 1.07391s (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 0.154011s (cpu); 0.0900489s (thread); 0s (gc) │ │ │ + -- used 0.197594s (cpu); 0.0824089s (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 0.175753s (cpu); 0.08594s (thread); 0s (gc) │ │ │ + -- used 0.25675s (cpu); 0.103813s (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 0.463963s (cpu); 0.287138s (thread); 0s (gc) │ │ │ + -- used 0.557366s (cpu); 0.34502s (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 0.00073598s (cpu); 8.807e-06s (thread); 0s (gc) │ │ │ + -- used 0.0027703s (cpu); 7.825e-06s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : Psi = first graph Phi; │ │ │ │ │ │ o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3) │ │ │ │ │ │ i6 : time isIsomorphism Psi │ │ │ - -- used 0.31381s (cpu); 0.173043s (thread); 0s (gc) │ │ │ + -- used 0.452926s (cpu); 0.199696s (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 1.60038s (cpu); 0.808017s (thread); 0s (gc) │ │ │ + -- used 1.65286s (cpu); 0.873886s (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 0.44319s (cpu); 0.298111s (thread); 0s (gc) │ │ │ + -- used 0.514003s (cpu); 0.306168s (thread); 0s (gc) │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : time Psi = first graph Phi; │ │ │ - -- used 0.165063s (cpu); 0.102805s (thread); 0s (gc) │ │ │ + -- used 0.0805611s (cpu); 0.0707923s (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 4.29055s (cpu); 3.2286s (thread); 0s (gc) │ │ │ + -- used 3.86793s (cpu); 3.29765s (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 0.0141178s (cpu); 0.0157305s (thread); 0s (gc) │ │ │ + -- used 0.0120845s (cpu); 0.0105237s (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 0.58166s (cpu); 0.464182s (thread); 0s (gc) │ │ │ + -- used 0.55557s (cpu); 0.483732s (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 0.442751s (cpu); 0.356718s (thread); 0s (gc) │ │ │ + -- used 0.589973s (cpu); 0.392058s (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 0.00400213s (cpu); 0.00132019s (thread); 0s (gc) │ │ │ - -- used 0.303402s (cpu); 0.15101s (thread); 0s (gc) │ │ │ - -- used 0.223519s (cpu); 0.149965s (thread); 0s (gc) │ │ │ - -- used 0.211836s (cpu); 0.152658s (thread); 0s (gc) │ │ │ - -- used 0.179095s (cpu); 0.108861s (thread); 0s (gc) │ │ │ + -- used 0.00401521s (cpu); 0.00136884s (thread); 0s (gc) │ │ │ + -- used 0.230725s (cpu); 0.152553s (thread); 0s (gc) │ │ │ + -- used 0.260404s (cpu); 0.188912s (thread); 0s (gc) │ │ │ + -- used 0.267176s (cpu); 0.200434s (thread); 0s (gc) │ │ │ + -- used 0.22506s (cpu); 0.147878s (thread); 0s (gc) │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ - -- used 0.14899s (cpu); 0.0961644s (thread); 0s (gc) │ │ │ + -- used 0.202538s (cpu); 0.109859s (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 0.0160105s (cpu); 0.0167943s (thread); 0s (gc) │ │ │ + -- used 0.0326467s (cpu); 0.0183692s (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 1.84176s (cpu); 0.999012s (thread); 0s (gc) │ │ │ + -- used 1.42723s (cpu); 0.972464s (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 0.7435s (cpu); 0.52297s (thread); 0s (gc) │ │ │ + -- used 1.35477s (cpu); 0.707661s (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 0.315126s (cpu); 0.174179s (thread); 0s (gc) │ │ │ + -- used 0.328476s (cpu); 0.174112s (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 3.18819s (cpu); 1.79242s (thread); 0s (gc)
│ │ │ + -- used 3.29199s (cpu); 1.91042s (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 3.24858s (cpu); 1.95299s (thread); 0s (gc)
│ │ │ + -- used 3.8289s (cpu); 2.07558s (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 7.95558s (cpu); 4.85491s (thread); 0s (gc)
│ │ │ + -- used 6.88385s (cpu); 4.99227s (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 3.18819s (cpu); 1.79242s (thread); 0s (gc) │ │ │ │ + -- used 3.29199s (cpu); 1.91042s (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 3.24858s (cpu); 1.95299s (thread); 0s (gc) │ │ │ │ + -- used 3.8289s (cpu); 2.07558s (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 7.95558s (cpu); 4.85491s (thread); 0s (gc) │ │ │ │ + -- used 6.88385s (cpu); 4.99227s (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 5.11651s (cpu); 3.76995s (thread); 0s (gc)
│ │ │ + -- used 4.53612s (cpu); 3.88967s (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 5.11651s (cpu); 3.76995s (thread); 0s (gc)
│ │ │ │ + -- used 4.53612s (cpu); 3.88967s (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 0.0996701s (cpu); 0.0991169s (thread); 0s (gc)
│ │ │ + -- used 0.161512s (cpu); 0.140129s (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 1.93661s (cpu); 1.32686s (thread); 0s (gc)
│ │ │ + -- used 2.02148s (cpu); 1.28678s (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 0.0996701s (cpu); 0.0991169s (thread); 0s (gc) │ │ │ │ + -- used 0.161512s (cpu); 0.140129s (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 1.93661s (cpu); 1.32686s (thread); 0s (gc) │ │ │ │ + -- used 2.02148s (cpu); 1.28678s (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 3.60509s (cpu); 2.34063s (thread); 0s (gc)
│ │ │ + -- used 4.42444s (cpu); 2.67204s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.336985s (cpu); 0.268253s (thread); 0s (gc)
│ │ │ + -- used 0.364536s (cpu); 0.296807s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time degree Phi
│ │ │ - -- used 0.272327s (cpu); 0.230404s (thread); 0s (gc)
│ │ │ + -- used 0.334696s (cpu); 0.269259s (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 3.60509s (cpu); 2.34063s (thread); 0s (gc) │ │ │ │ + -- used 4.42444s (cpu); 2.67204s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.336985s (cpu); 0.268253s (thread); 0s (gc) │ │ │ │ + -- used 0.364536s (cpu); 0.296807s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.272327s (cpu); 0.230404s (thread); 0s (gc) │ │ │ │ + -- used 0.334696s (cpu); 0.269259s (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 0.442484s (cpu); 0.361777s (thread); 0s (gc)
│ │ │ + -- used 0.580944s (cpu); 0.410309s (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 0.442484s (cpu); 0.361777s (thread); 0s (gc) │ │ │ │ + -- used 0.580944s (cpu); 0.410309s (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 0.000311614s (cpu); 0.000167294s (thread); 0s (gc)
│ │ │ + -- used 0.0017867s (cpu); 0.000171259s (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 0.00114434s (cpu); 0.000225412s (thread); 0s (gc)
│ │ │ + -- used 0.00229552s (cpu); 0.000259652s (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 1.36555s (cpu); 1.07435s (thread); 0s (gc)
│ │ │ + -- used 1.20728s (cpu); 1.01733s (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 1.5099e-05s (cpu); 0.000385823s (thread); 0s (gc)
│ │ │ + -- used 0.000126961s (cpu); 0.000425204s (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 0.000311614s (cpu); 0.000167294s (thread); 0s (gc)
│ │ │ │ + -- used 0.0017867s (cpu); 0.000171259s (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 0.00114434s (cpu); 0.000225412s (thread); 0s (gc)
│ │ │ │ + -- used 0.00229552s (cpu); 0.000259652s (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 1.36555s (cpu); 1.07435s (thread); 0s (gc)
│ │ │ │ + -- used 1.20728s (cpu); 1.01733s (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 1.5099e-05s (cpu); 0.000385823s (thread); 0s (gc)
│ │ │ │ + -- used 0.000126961s (cpu); 0.000425204s (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 0.126166s (cpu); 0.0742722s (thread); 0s (gc)
│ │ │ + -- used 0.177544s (cpu); 0.0552381s (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 0.037857s (cpu); 0.0392192s (thread); 0s (gc)
│ │ │ + -- used 0.0532142s (cpu); 0.04297s (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 0.22123s (cpu); 0.167286s (thread); 0s (gc)
│ │ │ + -- used 0.293397s (cpu); 0.166609s (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 0.126166s (cpu); 0.0742722s (thread); 0s (gc) │ │ │ │ + -- used 0.177544s (cpu); 0.0552381s (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 0.037857s (cpu); 0.0392192s (thread); 0s (gc) │ │ │ │ + -- used 0.0532142s (cpu); 0.04297s (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 0.22123s (cpu); 0.167286s (thread); 0s (gc) │ │ │ │ + -- used 0.293397s (cpu); 0.166609s (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 0.206519s (cpu); 0.130282s (thread); 0s (gc)
│ │ │ + -- used 0.172518s (cpu); 0.160845s (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 0.442751s (cpu); 0.356718s (thread); 0s (gc)
│ │ │ │ + -- used 0.589973s (cpu); 0.392058s (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 6.14478s (cpu); 2.61305s (thread); 0s (gc)
│ │ │ + -- used 10.7568s (cpu); 2.88365s (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 0.206519s (cpu); 0.130282s (thread); 0s (gc)
│ │ │ │ + -- used 0.172518s (cpu); 0.160845s (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 6.14478s (cpu); 2.61305s (thread); 0s (gc)
│ │ │ │ + -- used 10.7568s (cpu); 2.88365s (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 0.368667s (cpu); 0.308957s (thread); 0s (gc)
│ │ │ + -- used 0.431024s (cpu); 0.324297s (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 1.62165s (cpu); 1.07916s (thread); 0s (gc)
│ │ │ + -- used 1.54946s (cpu); 1.07391s (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 0.368667s (cpu); 0.308957s (thread); 0s (gc)
│ │ │ │ + -- used 0.431024s (cpu); 0.324297s (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 1.62165s (cpu); 1.07916s (thread); 0s (gc)
│ │ │ │ + -- used 1.54946s (cpu); 1.07391s (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 0.154011s (cpu); 0.0900489s (thread); 0s (gc)
│ │ │ + -- used 0.197594s (cpu); 0.0824089s (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 0.175753s (cpu); 0.08594s (thread); 0s (gc)
│ │ │ + -- used 0.25675s (cpu); 0.103813s (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 0.463963s (cpu); 0.287138s (thread); 0s (gc)
│ │ │ + -- used 0.557366s (cpu); 0.34502s (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 0.154011s (cpu); 0.0900489s (thread); 0s (gc)
│ │ │ │ + -- used 0.197594s (cpu); 0.0824089s (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 0.175753s (cpu); 0.08594s (thread); 0s (gc)
│ │ │ │ + -- used 0.25675s (cpu); 0.103813s (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 0.463963s (cpu); 0.287138s (thread); 0s (gc)
│ │ │ │ + -- used 0.557366s (cpu); 0.34502s (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 0.00073598s (cpu); 8.807e-06s (thread); 0s (gc)
│ │ │ + -- used 0.0027703s (cpu); 7.825e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
    
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │              
    
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.31381s (cpu); 0.173043s (thread); 0s (gc)
│ │ │ + -- used 0.452926s (cpu); 0.199696s (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 1.60038s (cpu); 0.808017s (thread); 0s (gc)
│ │ │ + -- used 1.65286s (cpu); 0.873886s (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 0.00073598s (cpu); 8.807e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.0027703s (cpu); 7.825e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.31381s (cpu); 0.173043s (thread); 0s (gc)
│ │ │ │ + -- used 0.452926s (cpu); 0.199696s (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 1.60038s (cpu); 0.808017s (thread); 0s (gc)
│ │ │ │ + -- used 1.65286s (cpu); 0.873886s (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 0.44319s (cpu); 0.298111s (thread); 0s (gc)
│ │ │ + -- used 0.514003s (cpu); 0.306168s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.165063s (cpu); 0.102805s (thread); 0s (gc)
│ │ │ + -- used 0.0805611s (cpu); 0.0707923s (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 4.29055s (cpu); 3.2286s (thread); 0s (gc)
│ │ │ + -- used 3.86793s (cpu); 3.29765s (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 0.44319s (cpu); 0.298111s (thread); 0s (gc)
│ │ │ │ + -- used 0.514003s (cpu); 0.306168s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.165063s (cpu); 0.102805s (thread); 0s (gc)
│ │ │ │ + -- used 0.0805611s (cpu); 0.0707923s (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 4.29055s (cpu); 3.2286s (thread); 0s (gc)
│ │ │ │ + -- used 3.86793s (cpu); 3.29765s (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 0.0141178s (cpu); 0.0157305s (thread); 0s (gc)
│ │ │ + -- used 0.0120845s (cpu); 0.0105237s (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 0.58166s (cpu); 0.464182s (thread); 0s (gc)
│ │ │ + -- used 0.55557s (cpu); 0.483732s (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 0.0141178s (cpu); 0.0157305s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120845s (cpu); 0.0105237s (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 0.58166s (cpu); 0.464182s (thread); 0s (gc)
│ │ │ │ + -- used 0.55557s (cpu); 0.483732s (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 0.442751s (cpu); 0.356718s (thread); 0s (gc)
│ │ │ + -- used 0.589973s (cpu); 0.392058s (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 0.00400213s (cpu); 0.00132019s (thread); 0s (gc)
│ │ │ - -- used 0.303402s (cpu); 0.15101s (thread); 0s (gc)
│ │ │ - -- used 0.223519s (cpu); 0.149965s (thread); 0s (gc)
│ │ │ - -- used 0.211836s (cpu); 0.152658s (thread); 0s (gc)
│ │ │ - -- used 0.179095s (cpu); 0.108861s (thread); 0s (gc)
│ │ │ + -- used 0.00401521s (cpu); 0.00136884s (thread); 0s (gc)
│ │ │ + -- used 0.230725s (cpu); 0.152553s (thread); 0s (gc)
│ │ │ + -- used 0.260404s (cpu); 0.188912s (thread); 0s (gc)
│ │ │ + -- used 0.267176s (cpu); 0.200434s (thread); 0s (gc)
│ │ │ + -- used 0.22506s (cpu); 0.147878s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.14899s (cpu); 0.0961644s (thread); 0s (gc)
    
│ │ │ + -- used 0.202538s (cpu); 0.109859s (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 0.00400213s (cpu); 0.00132019s (thread); 0s (gc) │ │ │ │ - -- used 0.303402s (cpu); 0.15101s (thread); 0s (gc) │ │ │ │ - -- used 0.223519s (cpu); 0.149965s (thread); 0s (gc) │ │ │ │ - -- used 0.211836s (cpu); 0.152658s (thread); 0s (gc) │ │ │ │ - -- used 0.179095s (cpu); 0.108861s (thread); 0s (gc) │ │ │ │ + -- used 0.00401521s (cpu); 0.00136884s (thread); 0s (gc) │ │ │ │ + -- used 0.230725s (cpu); 0.152553s (thread); 0s (gc) │ │ │ │ + -- used 0.260404s (cpu); 0.188912s (thread); 0s (gc) │ │ │ │ + -- used 0.267176s (cpu); 0.200434s (thread); 0s (gc) │ │ │ │ + -- used 0.22506s (cpu); 0.147878s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.14899s (cpu); 0.0961644s (thread); 0s (gc) │ │ │ │ + -- used 0.202538s (cpu); 0.109859s (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 0.0160105s (cpu); 0.0167943s (thread); 0s (gc)
│ │ │ + -- used 0.0326467s (cpu); 0.0183692s (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 1.84176s (cpu); 0.999012s (thread); 0s (gc)
│ │ │ + -- used 1.42723s (cpu); 0.972464s (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 0.0160105s (cpu); 0.0167943s (thread); 0s (gc) │ │ │ │ + -- used 0.0326467s (cpu); 0.0183692s (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 1.84176s (cpu); 0.999012s (thread); 0s (gc) │ │ │ │ + -- used 1.42723s (cpu); 0.972464s (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 0.7435s (cpu); 0.52297s (thread); 0s (gc)
│ │ │ + -- used 1.35477s (cpu); 0.707661s (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 0.7435s (cpu); 0.52297s (thread); 0s (gc)
│ │ │ │ + -- used 1.35477s (cpu); 0.707661s (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 0.315126s (cpu); 0.174179s (thread); 0s (gc)
│ │ │ + -- used 0.328476s (cpu); 0.174112s (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 0.315126s (cpu); 0.174179s (thread); 0s (gc)
│ │ │ │ + -- used 0.328476s (cpu); 0.174112s (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 = (71, 85, 86, 90, 92, 98, 97, 95, 92, 98, 97, 95, 96, 97, 99, 98, 96, 99,
│ │ │ +o9 = (65, 80, 88, 92, 92, 93, 94, 95, 98, 97, 98, 96, 99, 98, 99, 95, 98, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     99, 99, 99, 97, 97, 99, 99, 95, 100, 98, 98)
│ │ │ +     95, 98, 97, 96, 99, 97, 97, 96, 97, 98, 96)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (20, 8, 7, 2, 2, 5, 1, 4, 3, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1,
│ │ │ +o10 = (21, 11, 8, 8, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 1, 1, 0, 0)
│ │ │ +      0, 0, 0, 0, 1, 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 = {DOK, DIk, DUW, DdG, Dm_}
│ │ │ +o2 = {DY[, DOk, DKG, Dvs, DA[}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, 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 0.000433423s (cpu); 0.000568886s (thread); 0s (gc)
│ │ │ + -- used 0.000663586s (cpu); 0.000523396s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0452254s (cpu); 0.0435819s (thread); 0s (gc)
│ │ │ + -- used 0.0588814s (cpu); 0.0563825s (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 = (71, 85, 86, 90, 92, 98, 97, 95, 92, 98, 97, 95, 96, 97, 99, 98, 96, 99,
│ │ │ +o9 = (65, 80, 88, 92, 92, 93, 94, 95, 98, 97, 98, 96, 99, 98, 99, 95, 98, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     99, 99, 99, 97, 97, 99, 99, 95, 100, 98, 98)
│ │ │ +     95, 98, 97, 96, 99, 97, 97, 96, 97, 98, 96)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (20, 8, 7, 2, 2, 5, 1, 4, 3, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1,
│ │ │ +o10 = (21, 11, 8, 8, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 1, 1, 0, 0)
│ │ │ +      0, 0, 0, 0, 1, 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 = (71, 85, 86, 90, 92, 98, 97, 95, 92, 98, 97, 95, 96, 97, 99, 98, 96, 99, │ │ │ │ +o9 = (65, 80, 88, 92, 92, 93, 94, 95, 98, 97, 98, 96, 99, 98, 99, 95, 98, 98, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 99, 99, 99, 97, 97, 99, 99, 95, 100, 98, 98) │ │ │ │ + 95, 98, 97, 96, 99, 97, 97, 96, 97, 98, 96) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (20, 8, 7, 2, 2, 5, 1, 4, 3, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, │ │ │ │ +o10 = (21, 11, 8, 8, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 1, 0, 1, 1, 0, 0) │ │ │ │ + 0, 0, 0, 0, 1, 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 = {DOK, DIk, DUW, DdG, Dm_}
│ │ │ +o2 = {DY[, DOk, DKG, Dvs, DA[}
│ │ │  
│ │ │  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 = {DOK, DIk, DUW, DdG, Dm_}
│ │ │ │ +o2 = {DY[, DOk, DKG, Dvs, DA[}
│ │ │ │  
│ │ │ │  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" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, 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" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, 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 0.000433423s (cpu); 0.000568886s (thread); 0s (gc)
    
│ │ │ + -- used 0.000663586s (cpu); 0.000523396s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0452254s (cpu); 0.0435819s (thread); 0s (gc)
    │ │ │ + -- used 0.0588814s (cpu); 0.0563825s (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 0.000433423s (cpu); 0.000568886s (thread); 0s (gc) │ │ │ │ + -- used 0.000663586s (cpu); 0.000523396s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.0452254s (cpu); 0.0435819s (thread); 0s (gc) │ │ │ │ + -- used 0.0588814s (cpu); 0.0563825s (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 = (74, 84, 89, 90, 93, 98, 96, 97, 94, 98, 99, 97, 98, 93, 95, 97, 97, 98, │ │ │ +o9 = (71, 80, 86, 93, 97, 93, 92, 97, 98, 97, 99, 93, 99, 99, 96, 98, 96, 97, │ │ │ ------------------------------------------------------------------------ │ │ │ - 95, 96, 96, 99, 99, 97, 99, 98, 98, 98, 99) │ │ │ + 98, 99, 98, 98, 97, 99, 99, 98, 100, 98, 99) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (19, 10, 4, 5, 5, 4, 1, 0, 3, 2, 2, 1, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 0, │ │ │ +o10 = (20, 9, 8, 3, 4, 2, 2, 2, 2, 1, 2, 3, 2, 0, 0, 1, 2, 1, 0, 1, 2, 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 = {DES, DTO, DQg, DRK, DEw} │ │ │ +o2 = {DU?, DLo, DV?, DHO, DMw} │ │ │ │ │ │ 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 = {Dbg, DdW, DkK} │ │ │ +o1 = {DMg, DqK, DkK} │ │ │ │ │ │ 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 0.000532408s (cpu); 0.000439804s (thread); 0s (gc) │ │ │ + -- used 0.00070049s (cpu); 0.000610902s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.0536394s (cpu); 0.0520112s (thread); 0s (gc) │ │ │ + -- used 0.0639375s (cpu); 0.061548s (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 = (74, 84, 89, 90, 93, 98, 96, 97, 94, 98, 99, 97, 98, 93, 95, 97, 97, 98,
│ │ │ +o9 = (71, 80, 86, 93, 97, 93, 92, 97, 98, 97, 99, 93, 99, 99, 96, 98, 96, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     95, 96, 96, 99, 99, 97, 99, 98, 98, 98, 99)
│ │ │ +     98, 99, 98, 98, 97, 99, 99, 98, 100, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (19, 10, 4, 5, 5, 4, 1, 0, 3, 2, 2, 1, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 0,
│ │ │ +o10 = (20, 9, 8, 3, 4, 2, 2, 2, 2, 1, 2, 3, 2, 0, 0, 1, 2, 1, 0, 1, 2, 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 = (74, 84, 89, 90, 93, 98, 96, 97, 94, 98, 99, 97, 98, 93, 95, 97, 97, 98, │ │ │ │ +o9 = (71, 80, 86, 93, 97, 93, 92, 97, 98, 97, 99, 93, 99, 99, 96, 98, 96, 97, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 95, 96, 96, 99, 99, 97, 99, 98, 98, 98, 99) │ │ │ │ + 98, 99, 98, 98, 97, 99, 99, 98, 100, 98, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (19, 10, 4, 5, 5, 4, 1, 0, 3, 2, 2, 1, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 0, │ │ │ │ +o10 = (20, 9, 8, 3, 4, 2, 2, 2, 2, 1, 2, 3, 2, 0, 0, 1, 2, 1, 0, 1, 2, 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 = {DES, DTO, DQg, DRK, DEw}
│ │ │ +o2 = {DU?, DLo, DV?, DHO, DMw}
│ │ │  
│ │ │  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 = {DES, DTO, DQg, DRK, DEw}
│ │ │ │ +o2 = {DU?, DLo, DV?, DHO, DMw}
│ │ │ │  
│ │ │ │  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 = {Dbg, DdW, DkK}
│ │ │ +o1 = {DMg, DqK, DkK}
│ │ │  
│ │ │  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 = {Dbg, DdW, DkK} │ │ │ │ +o1 = {DMg, DqK, DkK} │ │ │ │ │ │ │ │ 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 0.000532408s (cpu); 0.000439804s (thread); 0s (gc)
    │ │ │ + -- used 0.00070049s (cpu); 0.000610902s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0536394s (cpu); 0.0520112s (thread); 0s (gc)
    │ │ │ + -- used 0.0639375s (cpu); 0.061548s (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 0.000532408s (cpu); 0.000439804s (thread); 0s (gc) │ │ │ │ + -- used 0.00070049s (cpu); 0.000610902s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.0536394s (cpu); 0.0520112s (thread); 0s (gc) │ │ │ │ + -- used 0.0639375s (cpu); 0.061548s (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") │ │ │ - -- .130718s elapsed │ │ │ + -- .11392s 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")
│ │ │ - -- .130718s elapsed
│ │ │ + -- .11392s 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") │ │ │ │ - -- .130718s elapsed │ │ │ │ + -- .11392s 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 0.280491s (cpu); 0.279713s (thread); 0s (gc) │ │ │ + -- used 0.247898s (cpu); 0.249121s (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 0.000834565s (cpu); 0.00160969s (thread); 0s (gc) │ │ │ - -- used 2.5468e-05s (cpu); 0.000108473s (thread); 0s (gc) │ │ │ - -- used 1.2474e-05s (cpu); 9.8024e-05s (thread); 0s (gc) │ │ │ - -- used 1.1561e-05s (cpu); 8.8126e-05s (thread); 0s (gc) │ │ │ - -- used 1.1131e-05s (cpu); 8.8305e-05s (thread); 0s (gc) │ │ │ - -- used 1.1462e-05s (cpu); 8.2585e-05s (thread); 0s (gc) │ │ │ + -- used 0.00393371s (cpu); 0.0016751s (thread); 0s (gc) │ │ │ + -- used 2.6653e-05s (cpu); 0.000107973s (thread); 0s (gc) │ │ │ + -- used 1.082e-05s (cpu); 8.4393e-05s (thread); 0s (gc) │ │ │ + -- used 9.06e-06s (cpu); 7.0336e-05s (thread); 0s (gc) │ │ │ + -- used 9.082e-06s (cpu); 6.4006e-05s (thread); 0s (gc) │ │ │ + -- used 8.735e-06s (cpu); 6.7854e-05s (thread); 0s (gc) │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ o19 : List │ │ │ │ │ │ i20 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -1,29 +1,29 @@ │ │ │ -- -*- M2-comint -*- hash: 16408385764843695632 │ │ │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ - -- setting random seed to 1756126388 │ │ │ + -- setting random seed to 1759553998 │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {1, 2, 3} │ │ │ +o3 = {0, 5, 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, 2, 5, 7, 9} │ │ │ +o7 = {1, 2, 5, 6, 8} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ @@ -6,61 +6,61 @@ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ - -- .0678242s elapsed │ │ │ + -- .0309636s 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)) │ │ │ - -- .00188643s elapsed │ │ │ + -- .00151641s 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 │ │ │ - -- .0573703s elapsed │ │ │ + -- .0442928s 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)) │ │ │ - -- .00137864s elapsed │ │ │ + -- .0012905s 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 │ │ │ - -- 40.1745s elapsed │ │ │ + -- 28.7147s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .0278313s elapsed │ │ │ + -- .0314044s elapsed │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ @@ -24,19 +24,19 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : X = normalToricVariety F; │ │ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ - -- used 0.000200005s (cpu); 2.5037e-05s (thread); 0s (gc) │ │ │ + -- used 0.00185524s (cpu); 3.6168e-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 0.21419s (cpu); 0.137653s (thread); 0s (gc) │ │ │ + -- used 0.169063s (cpu); 0.0888592s (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 0.0257115s (cpu); 0.02417s (thread); 0s (gc) │ │ │ + -- used 0.023985s (cpu); 0.0248124s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 9.8134e-05s (cpu); 0.00216945s (thread); 0s (gc) │ │ │ + -- used 0.000483177s (cpu); 0.00266589s (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 0.280491s (cpu); 0.279713s (thread); 0s (gc)
    │ │ │ + -- used 0.247898s (cpu); 0.249121s (thread); 0s (gc) │ │ │
    │ │ │ 
    i15 : A2 = intersectionRing Y;
    │ │ │
    │ │ │ 
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.000834565s (cpu); 0.00160969s (thread); 0s (gc)
│ │ │ - -- used 2.5468e-05s (cpu); 0.000108473s (thread); 0s (gc)
│ │ │ - -- used 1.2474e-05s (cpu); 9.8024e-05s (thread); 0s (gc)
│ │ │ - -- used 1.1561e-05s (cpu); 8.8126e-05s (thread); 0s (gc)
│ │ │ - -- used 1.1131e-05s (cpu); 8.8305e-05s (thread); 0s (gc)
│ │ │ - -- used 1.1462e-05s (cpu); 8.2585e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00393371s (cpu); 0.0016751s (thread); 0s (gc)
│ │ │ + -- used 2.6653e-05s (cpu); 0.000107973s (thread); 0s (gc)
│ │ │ + -- used 1.082e-05s (cpu); 8.4393e-05s (thread); 0s (gc)
│ │ │ + -- used 9.06e-06s (cpu); 7.0336e-05s (thread); 0s (gc)
│ │ │ + -- used 9.082e-06s (cpu); 6.4006e-05s (thread); 0s (gc)
│ │ │ + -- used 8.735e-06s (cpu); 6.7854e-05s (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 0.280491s (cpu); 0.279713s (thread); 0s (gc) │ │ │ │ + -- used 0.247898s (cpu); 0.249121s (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 0.000834565s (cpu); 0.00160969s (thread); 0s (gc) │ │ │ │ - -- used 2.5468e-05s (cpu); 0.000108473s (thread); 0s (gc) │ │ │ │ - -- used 1.2474e-05s (cpu); 9.8024e-05s (thread); 0s (gc) │ │ │ │ - -- used 1.1561e-05s (cpu); 8.8126e-05s (thread); 0s (gc) │ │ │ │ - -- used 1.1131e-05s (cpu); 8.8305e-05s (thread); 0s (gc) │ │ │ │ - -- used 1.1462e-05s (cpu); 8.2585e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00393371s (cpu); 0.0016751s (thread); 0s (gc) │ │ │ │ + -- used 2.6653e-05s (cpu); 0.000107973s (thread); 0s (gc) │ │ │ │ + -- used 1.082e-05s (cpu); 8.4393e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.06e-06s (cpu); 7.0336e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.082e-06s (cpu); 6.4006e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.735e-06s (cpu); 6.7854e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -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 1756126388
    │ │ │ + -- setting random seed to 1759553998 │ │ │
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {1, 2, 3}
│ │ │ +o3 = {0, 5, 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, 2, 5, 7, 9}
│ │ │ +o7 = {1, 2, 5, 6, 8}
│ │ │  
│ │ │  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 1756126388 │ │ │ │ + -- setting random seed to 1759553998 │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {1, 2, 3} │ │ │ │ +o3 = {0, 5, 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, 2, 5, 7, 9} │ │ │ │ +o7 = {1, 2, 5, 6, 8} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ The next ten examples illustrate various ways that two lists can fail to define │ │ │ │ a normal toric variety. By making the current debugging level greater than one, │ │ │ │ one gets some addition information about the nature of the failure. │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │
    │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0678242s elapsed
│ │ │ + -- .0309636s 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))
│ │ │ - -- .00188643s elapsed
    │ │ │ + -- .00151641s 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
│ │ │ - -- .0573703s elapsed
│ │ │ + -- .0442928s 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))
│ │ │ - -- .00137864s elapsed
    │ │ │ + -- .0012905s elapsed │ │ │
    │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 40.1745s elapsed
│ │ │ + -- 28.7147s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0278313s elapsed
    │ │ │ + -- .0314044s 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 │ │ │ │ - -- .0678242s elapsed │ │ │ │ + -- .0309636s 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)) │ │ │ │ - -- .00188643s elapsed │ │ │ │ + -- .00151641s 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 │ │ │ │ - -- .0573703s elapsed │ │ │ │ + -- .0442928s 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)) │ │ │ │ - -- .00137864s elapsed │ │ │ │ + -- .0012905s 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 │ │ │ │ - -- 40.1745s elapsed │ │ │ │ + -- 28.7147s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .0278313s elapsed │ │ │ │ + -- .0314044s 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 0.000200005s (cpu); 2.5037e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00185524s (cpu); 3.6168e-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 0.21419s (cpu); 0.137653s (thread); 0s (gc) │ │ │ │ + -- used 0.169063s (cpu); 0.0888592s (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 0.0257115s (cpu); 0.02417s (thread); 0s (gc) │ │ │ │ + -- used 0.023985s (cpu); 0.0248124s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 9.8134e-05s (cpu); 0.00216945s (thread); 0s (gc) │ │ │ │ + -- used 0.000483177s (cpu); 0.00266589s (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 │ │ │ @@ -23,19 +23,19 @@ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ Sampling image points ... │ │ │ - -- used .00399984 seconds │ │ │ + -- used .00805238 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0040008 seconds │ │ │ + -- used .00797898 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00399909 seconds │ │ │ + -- used .00399026 seconds │ │ │ Computing numerical kernel ... │ │ │ -- used 0 seconds │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ @@ -13,21 +13,21 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ Sampling image points ... │ │ │ - -- used .0048782 seconds │ │ │ + -- used .00399877 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00274079 seconds │ │ │ + -- used .0040235 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00112381 seconds │ │ │ + -- used 0 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000418326 seconds │ │ │ + -- used 0 seconds │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ 53 0 3 53 0 3 │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ @@ -13,19 +13,19 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ Sampling image points ... │ │ │ - -- used .0122087 seconds │ │ │ + -- used .0159119 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0119958 seconds │ │ │ + -- used .0120009 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00753363 seconds │ │ │ + -- used .00800026 seconds │ │ │ Computing numerical kernel ... │ │ │ -- used 0 seconds │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ @@ -34,19 +34,19 @@ │ │ │ │ │ │ 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 .00400037 seconds │ │ │ + -- used .0039442 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00799956 seconds │ │ │ + -- used .00800513 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00400037 seconds │ │ │ + -- used 0 seconds │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ @@ -22,12 +22,12 @@ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ - -- used 0.0630907s (cpu); 0.0656429s (thread); 0s (gc) │ │ │ + -- used 0.0735472s (cpu); 0.0739705s (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) │ │ │ - -- .618807s elapsed │ │ │ + -- .469095s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -102,19 +102,19 @@ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -40,21 +40,21 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0048782 seconds │ │ │ │ + -- used .00399877 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00274079 seconds │ │ │ │ + -- used .0040235 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00112381 seconds │ │ │ │ + -- used 0 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000418326 seconds │ │ │ │ + -- used 0 seconds │ │ │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ │ 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 │ │ │ @@ -109,19 +109,19 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -59,19 +59,19 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0122087 seconds │ │ │ │ + -- used .0159119 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0119958 seconds │ │ │ │ + -- used .0120009 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00753363 seconds │ │ │ │ + -- used .00800026 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ -- used 0 seconds │ │ │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ @@ -79,19 +79,19 @@ │ │ │ │ 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 .00400037 seconds │ │ │ │ + -- used .0039442 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00799956 seconds │ │ │ │ + -- used .00800513 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00400037 seconds │ │ │ │ + -- used 0 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -142,15 +142,15 @@ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 0.000200005s (cpu); 2.5037e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00185524s (cpu); 3.6168e-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 0.21419s (cpu); 0.137653s (thread); 0s (gc)
    │ │ │ + -- used 0.169063s (cpu); 0.0888592s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0257115s (cpu); 0.02417s (thread); 0s (gc)
    │ │ │ + -- used 0.023985s (cpu); 0.0248124s (thread); 0s (gc) │ │ │
    │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 9.8134e-05s (cpu); 0.00216945s (thread); 0s (gc)
    │ │ │ + -- used 0.000483177s (cpu); 0.00266589s (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 .00399984 seconds
│ │ │ +     -- used .00805238 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0040008 seconds
│ │ │ +     -- used .00797898 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00399909 seconds
│ │ │ +     -- used .00399026 seconds
│ │ │  Computing numerical kernel ...
│ │ │       -- used 0 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,19 +35,19 @@ │ │ │ │ == 0 │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00399984 seconds │ │ │ │ + -- used .00805238 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0040008 seconds │ │ │ │ + -- used .00797898 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00399909 seconds │ │ │ │ + -- used .00399026 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ -- used 0 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -104,21 +104,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │
    │ │ │ 
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0048782 seconds
│ │ │ +     -- used .00399877 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00274079 seconds
│ │ │ +     -- used .0040235 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00112381 seconds
│ │ │ +     -- used 0 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000418326 seconds
│ │ │ +     -- used 0 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
    │ │ │
    │ │ │ 
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0122087 seconds
│ │ │ +     -- used .0159119 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0119958 seconds
│ │ │ +     -- used .0120009 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00753363 seconds
│ │ │ +     -- used .00800026 seconds
│ │ │  Computing numerical kernel ...
│ │ │       -- used 0 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    │ │ │ @@ -149,19 +149,19 @@ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │
    │ │ │ 
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00400037 seconds
│ │ │ +     -- used .0039442 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00799956 seconds
│ │ │ +     -- used .00800513 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00400037 seconds
│ │ │ +     -- used 0 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.0630907s (cpu); 0.0656429s (thread); 0s (gc)
│ │ │ + -- used 0.0735472s (cpu); 0.0739705s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -45,15 +45,15 @@ │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ │ │ │ │ 1 70 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ │ - -- used 0.0630907s (cpu); 0.0656429s (thread); 0s (gc) │ │ │ │ + -- used 0.0735472s (cpu); 0.0739705s (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)
│ │ │ - -- .618807s elapsed
│ │ │ + -- .469095s 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) │ │ │ │ - -- .618807s elapsed │ │ │ │ + -- .469095s elapsed │ │ │ │ │ │ │ │ o7 = p │ │ │ │ │ │ │ │ o7 : Point │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ │ │ │ │ │ │ o8 = | .722359 .289465 -.295808 .591752 -.454678 | │ │ ├── ./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 │ │ │ - -- .119318s elapsed │ │ │ + -- .114906s elapsed │ │ │ flatteningRelations │ │ │ - -- .108313s elapsed │ │ │ + -- .0868281s elapsed │ │ │ next gb │ │ │ - -- .00064129s elapsed │ │ │ + -- .0007216s elapsed │ │ │ true │ │ │ unfolding │ │ │ - -- .145883s elapsed │ │ │ + -- .0927667s elapsed │ │ │ flatteningRelations │ │ │ - -- .140356s elapsed │ │ │ + -- .0894087s elapsed │ │ │ next gb │ │ │ - -- .000577611s elapsed │ │ │ + -- .000538565s elapsed │ │ │ true │ │ │ - -- 1.39574s elapsed │ │ │ + -- 1.08999s elapsed │ │ │ │ │ │ o6 = {} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0)) │ │ │ - -- 1.28528s elapsed │ │ │ + -- 1.04377s elapsed │ │ │ │ │ │ o7 = {} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : LL7b=={} │ │ │ │ │ │ @@ -75,22 +75,22 @@ │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true) │ │ │ (13, 1) │ │ │ {5, 8, 11, 12} │ │ │ unfolding │ │ │ - -- .222902s elapsed │ │ │ + -- .215354s elapsed │ │ │ flatteningRelations │ │ │ - -- .153112s elapsed │ │ │ + -- .112626s elapsed │ │ │ next gb │ │ │ - -- .000957172s elapsed │ │ │ + -- .000921893s elapsed │ │ │ true │ │ │ - -- .676471s elapsed │ │ │ -(5, 8, all semigroups are smoothable) -- .707383s elapsed │ │ │ + -- .539775s elapsed │ │ │ +(5, 8, all semigroups are smoothable) -- .571815s 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) │ │ │ - -- .0488101s elapsed │ │ │ + -- .054725s elapsed │ │ │ 6 │ │ │ false │ │ │ 5 │ │ │ false │ │ │ 4 │ │ │ decompose │ │ │ - -- .29765s elapsed │ │ │ + -- .264469s 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)) │ │ │ - -- 3.29347s elapsed │ │ │ + -- 2.73939s 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) │ │ │ - -- .906899s elapsed │ │ │ + -- .790528s elapsed │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ - -- 4.15531s elapsed │ │ │ + -- 3.09997s elapsed │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Weierstrass__Semigroup.out │ │ │ @@ -7,12 +7,12 @@ │ │ │ o1 : List │ │ │ │ │ │ i2 : genus L │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : elapsedTime isWeierstrassSemigroup(L,0.15) │ │ │ - -- 3.79755s elapsed │ │ │ + -- 3.02472s elapsed │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 6860996532851631556 │ │ │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ -(6, 7, all semigroups are smoothable) -- 1.34571s elapsed │ │ │ +(6, 7, all semigroups are smoothable) -- 1.09898s 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 │ │ │ - -- .4726s elapsed │ │ │ + -- .322146s elapsed │ │ │ flatteningRelations │ │ │ - -- .154197s elapsed │ │ │ + -- .166252s elapsed │ │ │ next gb │ │ │ - -- .00185489s elapsed │ │ │ + -- .00677188s elapsed │ │ │ true │ │ │ - -- 1.00343s elapsed │ │ │ + -- .800189s elapsed │ │ │ {6, 7, 9, 17} │ │ │ unfolding │ │ │ - -- .422286s elapsed │ │ │ + -- .306425s elapsed │ │ │ flatteningRelations │ │ │ - -- .257661s elapsed │ │ │ + -- .178135s elapsed │ │ │ next gb │ │ │ - -- .00262426s elapsed │ │ │ + -- .00234516s elapsed │ │ │ decompose │ │ │ - -- .173587s elapsed │ │ │ + -- .128054s elapsed │ │ │ number of components: 2 │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ {0, -1} │ │ │ - -- 2.98504s elapsed │ │ │ + -- 2.28569s elapsed │ │ │ {6, 8, 9, 10} │ │ │ unfolding │ │ │ - -- .131991s elapsed │ │ │ + -- .10996s elapsed │ │ │ flatteningRelations │ │ │ - -- .112878s elapsed │ │ │ + -- .10063s elapsed │ │ │ next gb │ │ │ - -- .000491821s elapsed │ │ │ + -- .000426749s elapsed │ │ │ true │ │ │ - -- .641694s elapsed │ │ │ + -- .550181s elapsed │ │ │ {6, 8, 10, 11, 13} │ │ │ unfolding │ │ │ - -- .560936s elapsed │ │ │ + -- .428372s elapsed │ │ │ flatteningRelations │ │ │ - -- .237974s elapsed │ │ │ + -- .170088s elapsed │ │ │ next gb │ │ │ - -- .00884147s elapsed │ │ │ + -- .00385664s elapsed │ │ │ decompose │ │ │ - -- .997776s elapsed │ │ │ + -- .739772s elapsed │ │ │ number of components: 1 │ │ │ support c, codim c: {(5, 1)} │ │ │ {-1} │ │ │ - -- 3.00768s elapsed │ │ │ - -- 7.63798s elapsed │ │ │ + -- 2.24599s elapsed │ │ │ + -- 5.88218s elapsed │ │ │ 0 │ │ │ │ │ │ - -- .000003676s elapsed │ │ │ - -- 7.74659s elapsed │ │ │ + -- .000003346s elapsed │ │ │ + -- 5.94651s 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
│ │ │ - -- .119318s elapsed
│ │ │ + -- .114906s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .108313s elapsed
│ │ │ + -- .0868281s elapsed
│ │ │  next gb
│ │ │ - -- .00064129s elapsed
│ │ │ + -- .0007216s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .145883s elapsed
│ │ │ + -- .0927667s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .140356s elapsed
│ │ │ + -- .0894087s elapsed
│ │ │  next gb
│ │ │ - -- .000577611s elapsed
│ │ │ + -- .000538565s elapsed
│ │ │  true
│ │ │ - -- 1.39574s elapsed
│ │ │ + -- 1.08999s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.28528s elapsed
│ │ │ + -- 1.04377s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -184,22 +184,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .222902s elapsed
│ │ │ + -- .215354s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .153112s elapsed
│ │ │ + -- .112626s elapsed
│ │ │  next gb
│ │ │ - -- .000957172s elapsed
│ │ │ + -- .000921893s elapsed
│ │ │  true
│ │ │ - -- .676471s elapsed
│ │ │ -(5, 8,  all semigroups are smoothable) -- .707383s elapsed
│ │ │ + -- .539775s elapsed
│ │ │ +(5, 8,  all semigroups are smoothable) -- .571815s elapsed
│ │ │  
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ @@ -223,22 +223,22 @@ │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0488101s elapsed
│ │ │ + -- .054725s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .29765s elapsed
│ │ │ + -- .264469s 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 │ │ │ │ - -- .119318s elapsed │ │ │ │ + -- .114906s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .108313s elapsed │ │ │ │ + -- .0868281s elapsed │ │ │ │ next gb │ │ │ │ - -- .00064129s elapsed │ │ │ │ + -- .0007216s elapsed │ │ │ │ true │ │ │ │ unfolding │ │ │ │ - -- .145883s elapsed │ │ │ │ + -- .0927667s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .140356s elapsed │ │ │ │ + -- .0894087s elapsed │ │ │ │ next gb │ │ │ │ - -- .000577611s elapsed │ │ │ │ + -- .000538565s elapsed │ │ │ │ true │ │ │ │ - -- 1.39574s elapsed │ │ │ │ + -- 1.08999s elapsed │ │ │ │ │ │ │ │ o6 = {} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0)) │ │ │ │ - -- 1.28528s elapsed │ │ │ │ + -- 1.04377s elapsed │ │ │ │ │ │ │ │ o7 = {} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : LL7b=={} │ │ │ │ │ │ │ │ o8 = true │ │ │ │ @@ -92,22 +92,22 @@ │ │ │ │ o10 = (5, 8) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true) │ │ │ │ (13, 1) │ │ │ │ {5, 8, 11, 12} │ │ │ │ unfolding │ │ │ │ - -- .222902s elapsed │ │ │ │ + -- .215354s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .153112s elapsed │ │ │ │ + -- .112626s elapsed │ │ │ │ next gb │ │ │ │ - -- .000957172s elapsed │ │ │ │ + -- .000921893s elapsed │ │ │ │ true │ │ │ │ - -- .676471s elapsed │ │ │ │ -(5, 8, all semigroups are smoothable) -- .707383s elapsed │ │ │ │ + -- .539775s elapsed │ │ │ │ +(5, 8, all semigroups are smoothable) -- .571815s 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 │ │ │ │ @@ -120,22 +120,22 @@ │ │ │ │ o12 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : genus L │ │ │ │ │ │ │ │ o13 = 8 │ │ │ │ i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true) │ │ │ │ - -- .0488101s elapsed │ │ │ │ + -- .054725s elapsed │ │ │ │ 6 │ │ │ │ false │ │ │ │ 5 │ │ │ │ false │ │ │ │ 4 │ │ │ │ decompose │ │ │ │ - -- .29765s elapsed │ │ │ │ + -- .264469s 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))
│ │ │ - -- 3.29347s elapsed
│ │ │ + -- 2.73939s 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)) │ │ │ │ - -- 3.29347s elapsed │ │ │ │ + -- 2.73939s 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)
│ │ │ - -- .906899s elapsed
│ │ │ + -- .790528s elapsed
│ │ │  
│ │ │  o3 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.15531s elapsed
│ │ │ + -- 3.09997s 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) │ │ │ │ - -- .906899s elapsed │ │ │ │ + -- .790528s elapsed │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ │ - -- 4.15531s elapsed │ │ │ │ + -- 3.09997s 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)
│ │ │ - -- 3.79755s elapsed
│ │ │ + -- 3.02472s 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) │ │ │ │ - -- 3.79755s elapsed │ │ │ │ + -- 3.02472s 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 │ │ │ @@ -78,15 +78,15 @@ │ │ │
    │ │ │

    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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -101,61 +101,61 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -21,76 +21,76 @@ │ │ │ │ LLdifficult │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ We test which semigroups of multiplicity m and genus g are smoothable. If no │ │ │ │ smoothing was found then L is a candidate for a non Weierstrass semigroup. In │ │ │ │ this search certain semigroups L in LLdifficult, where the computation is │ │ │ │ particular heavy are excluded. │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ │ -(6, 7, all semigroups are smoothable) -- 1.34571s elapsed │ │ │ │ +(6, 7, all semigroups are smoothable) -- 1.09898s 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 │ │ │ │ - -- .4726s elapsed │ │ │ │ + -- .322146s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .154197s elapsed │ │ │ │ + -- .166252s elapsed │ │ │ │ next gb │ │ │ │ - -- .00185489s elapsed │ │ │ │ + -- .00677188s elapsed │ │ │ │ true │ │ │ │ - -- 1.00343s elapsed │ │ │ │ + -- .800189s elapsed │ │ │ │ {6, 7, 9, 17} │ │ │ │ unfolding │ │ │ │ - -- .422286s elapsed │ │ │ │ + -- .306425s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .257661s elapsed │ │ │ │ + -- .178135s elapsed │ │ │ │ next gb │ │ │ │ - -- .00262426s elapsed │ │ │ │ + -- .00234516s elapsed │ │ │ │ decompose │ │ │ │ - -- .173587s elapsed │ │ │ │ + -- .128054s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ │ {0, -1} │ │ │ │ - -- 2.98504s elapsed │ │ │ │ + -- 2.28569s elapsed │ │ │ │ {6, 8, 9, 10} │ │ │ │ unfolding │ │ │ │ - -- .131991s elapsed │ │ │ │ + -- .10996s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .112878s elapsed │ │ │ │ + -- .10063s elapsed │ │ │ │ next gb │ │ │ │ - -- .000491821s elapsed │ │ │ │ + -- .000426749s elapsed │ │ │ │ true │ │ │ │ - -- .641694s elapsed │ │ │ │ + -- .550181s elapsed │ │ │ │ {6, 8, 10, 11, 13} │ │ │ │ unfolding │ │ │ │ - -- .560936s elapsed │ │ │ │ + -- .428372s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .237974s elapsed │ │ │ │ + -- .170088s elapsed │ │ │ │ next gb │ │ │ │ - -- .00884147s elapsed │ │ │ │ + -- .00385664s elapsed │ │ │ │ decompose │ │ │ │ - -- .997776s elapsed │ │ │ │ + -- .739772s elapsed │ │ │ │ number of components: 1 │ │ │ │ support c, codim c: {(5, 1)} │ │ │ │ {-1} │ │ │ │ - -- 3.00768s elapsed │ │ │ │ - -- 7.63798s elapsed │ │ │ │ + -- 2.24599s elapsed │ │ │ │ + -- 5.88218s elapsed │ │ │ │ 0 │ │ │ │ │ │ │ │ - -- .000003676s elapsed │ │ │ │ - -- 7.74659s elapsed │ │ │ │ + -- .000003346s elapsed │ │ │ │ + -- 5.94651s elapsed │ │ │ │ {} │ │ │ │ │ │ │ │ o3 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (17,5), indicates that there 17 semigroups of │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ │ │ i5 : C = oiRes({b}, 2) │ │ │ │ │ │ o5 = 0: (e0, {3}, {-2}) │ │ │ 1: (e1, {5, 5}, {-4, -3}) │ │ │ - 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -2, -3, -4, -4, -3, -5, -4, -5}) │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -3, -4, -5, -4, -3, -5, -4, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : phi = C.dd_1 │ │ │ │ │ │ o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ ├── ./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 0.150409s (cpu); 0.117628s (thread); 0s (gc) │ │ │ + -- used 0.149491s (cpu); 0.0980672s (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 0.146353s (cpu); 0.0861848s (thread); 0s (gc) │ │ │ + -- used 0.15106s (cpu); 0.101754s (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 0.346392s (cpu); 0.252546s (thread); 0s (gc) │ │ │ + -- used 0.604507s (cpu); 0.296792s (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 0.136261s (cpu); 0.0905376s (thread); 0s (gc) │ │ │ + -- used 0.143736s (cpu); 0.0998037s (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 0.146369s (cpu); 0.106697s (thread); 0s (gc) │ │ │ + -- used 0.180343s (cpu); 0.129662s (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/_image_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,19 +10,19 @@ │ │ │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ │ │ i7 : image phi │ │ │ │ │ │ -o7 = {-x e0 + x e0 + x e0 - │ │ │ - 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, 4},1 2,3 5,{1, 2, 5},1 │ │ │ +o7 = {x x e0 - x x e0 - x x e0 │ │ │ + 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 2,3 1,2 5,{1, 4, │ │ │ ------------------------------------------------------------------------ │ │ │ - x e0 , x x e0 - x x e0 - │ │ │ - 2,3 5,{1, 2, 4},1 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 │ │ │ + + x x e0 , -x e0 + x e0 │ │ │ + 5},1 2,4 1,2 5,{1, 3, 5},1 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, │ │ │ ------------------------------------------------------------------------ │ │ │ - x x e0 + x x e0 } │ │ │ - 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ + + x e0 - x e0 } │ │ │ + 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./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 0.349276s (cpu); 0.250698s (thread); 0s (gc) │ │ │ + -- used 0.368178s (cpu); 0.267471s (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 0.0200002s (cpu); 0.0226859s (thread); 0s (gc) │ │ │ + -- used 0.0240144s (cpu); 0.02568s (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 0.0239813s (cpu); 0.0250935s (thread); 0s (gc) │ │ │ + -- used 0.027954s (cpu); 0.0293941s (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 │ │ │ │ │ │ @@ -42,16 +42,16 @@ │ │ │ 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : minimizeOIGB C -- an element gets removed │ │ │ │ │ │ │ │ │ -o14 = {x e + x e , x x e + x x e , │ │ │ - 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ +o14 = {x x e + x x e , x e + x e , │ │ │ + 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 1,1 1,{1},1 2,1 1,{1},2 │ │ │ ----------------------------------------------------------------------- │ │ │ 2 │ │ │ x x x e - x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ o14 : List │ │ ├── ./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 0.180402s (cpu); 0.118358s (thread); 0s (gc) │ │ │ + -- used 0.152451s (cpu); 0.0977736s (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 0.0279994s (cpu); 0.0261102s (thread); 0s (gc) │ │ │ + -- used 0.0319093s (cpu); 0.0303633s (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 0.325754s (cpu); 0.234878s (thread); 0s (gc) │ │ │ + -- used 0.376309s (cpu); 0.26982s (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 0.170315s (cpu); 0.109167s (thread); 0s (gc) │ │ │ + -- used 0.176313s (cpu); 0.128463s (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/example-output/_terms_lp__Vector__In__Width_rp.out │ │ │ @@ -13,13 +13,13 @@ │ │ │ │ │ │ 4 │ │ │ o4 : (QQ[x , x , x , x ]) in width 2 │ │ │ 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i5 : terms f │ │ │ │ │ │ -o5 = {-x e , x e , -x e , x e } │ │ │ - 1,2 2,{2},1 2,1 2,{1},2 2,2 2,{2},2 1,1 2,{1},1 │ │ │ +o5 = {x e , -x e , x e , -x e } │ │ │ + 1,1 2,{1},1 1,2 2,{2},1 2,1 2,{1},2 2,2 2,{2},2 │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 0.150409s (cpu); 0.117628s (thread); 0s (gc) │ │ │ │ + -- used 0.149491s (cpu); 0.0980672s (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 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 0.346392s (cpu); 0.252546s (thread); 0s (gc) │ │ │ │ + -- used 0.604507s (cpu); 0.296792s (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 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │ -(6, 7,  all semigroups are smoothable) -- 1.34571s elapsed
│ │ │ +(6, 7,  all semigroups are smoothable) -- 1.09898s elapsed
│ │ │  
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │ 
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .4726s elapsed
│ │ │ + -- .322146s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .154197s elapsed
│ │ │ + -- .166252s elapsed
│ │ │  next gb
│ │ │ - -- .00185489s elapsed
│ │ │ + -- .00677188s elapsed
│ │ │  true
│ │ │ - -- 1.00343s elapsed
│ │ │ + -- .800189s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .422286s elapsed
│ │ │ + -- .306425s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .257661s elapsed
│ │ │ + -- .178135s elapsed
│ │ │  next gb
│ │ │ - -- .00262426s elapsed
│ │ │ + -- .00234516s elapsed
│ │ │  decompose
│ │ │ - -- .173587s elapsed
│ │ │ + -- .128054s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.98504s elapsed
│ │ │ + -- 2.28569s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .131991s elapsed
│ │ │ + -- .10996s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .112878s elapsed
│ │ │ + -- .10063s elapsed
│ │ │  next gb
│ │ │ - -- .000491821s elapsed
│ │ │ + -- .000426749s elapsed
│ │ │  true
│ │ │ - -- .641694s elapsed
│ │ │ + -- .550181s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .560936s elapsed
│ │ │ + -- .428372s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .237974s elapsed
│ │ │ + -- .170088s elapsed
│ │ │  next gb
│ │ │ - -- .00884147s elapsed
│ │ │ + -- .00385664s elapsed
│ │ │  decompose
│ │ │ - -- .997776s elapsed
│ │ │ + -- .739772s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.00768s elapsed
│ │ │ - -- 7.63798s elapsed
│ │ │ + -- 2.24599s elapsed
│ │ │ + -- 5.88218s elapsed
│ │ │  0
│ │ │  
│ │ │ - -- .000003676s elapsed
│ │ │ - -- 7.74659s elapsed
│ │ │ + -- .000003346s elapsed
│ │ │ + -- 5.94651s elapsed
│ │ │  {}
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -2, -3, -4, -4, -3, -5, -4, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -3, -4, -5, -4, -3, -5, -4, -5})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │
    │ │ │ 
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -2, -3, -4, -4, -3, -5, -4, -5})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -3, -4, -5, -4, -3, -5, -4, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./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 0.150409s (cpu); 0.117628s (thread); 0s (gc)
│ │ │ + -- used 0.149491s (cpu); 0.0980672s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │
    │ │ │ 
    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 0.146353s (cpu); 0.0861848s (thread); 0s (gc)
    │ │ │ + -- used 0.15106s (cpu); 0.101754s (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 0.146353s (cpu); 0.0861848s (thread); 0s (gc)
│ │ │ │ + -- used 0.15106s (cpu); 0.101754s (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 0.346392s (cpu); 0.252546s (thread); 0s (gc)
│ │ │ + -- used 0.604507s (cpu); 0.296792s (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
    │ │ │     		│ │ │ 
    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 0.136261s (cpu); 0.0905376s (thread); 0s (gc)
    │ │ │ + -- used 0.143736s (cpu); 0.0998037s (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 0.136261s (cpu); 0.0905376s (thread); 0s (gc)
│ │ │ │ + -- used 0.143736s (cpu); 0.0998037s (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 0.146369s (cpu); 0.106697s (thread); 0s (gc)
    │ │ │ + -- used 0.180343s (cpu); 0.129662s (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 0.146369s (cpu); 0.106697s (thread); 0s (gc)
│ │ │ │ + -- used 0.180343s (cpu); 0.129662s (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/_image_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │
    │ │ │ 
    i7 : image phi
│ │ │  
│ │ │ -o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ -        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1  
│ │ │ +o7 = {x   x   e0              - x   x   e0              - x   x   e0        
│ │ │ +       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1  
│ │ │ +          + x   x   e0             , -x   e0              + x   e0        
│ │ │ +     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              + x   x   e0             }
│ │ │ -      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │ +          + x   e0              - x   e0             }
│ │ │ +     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,22 +18,22 @@ │ │ │ │ i2 : F = makeFreeOIModule(e, {1,2}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 3); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ i7 : image phi │ │ │ │ │ │ │ │ -o7 = {-x e0 + x e0 + x e0 - │ │ │ │ - 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, 4},1 2,3 5,{1, 2, 5},1 │ │ │ │ +o7 = {x x e0 - x x e0 - x x e0 │ │ │ │ + 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 2,3 1,2 5,{1, 4, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x e0 , x x e0 - x x e0 - │ │ │ │ - 2,3 5,{1, 2, 4},1 2,3 1,1 5,{2, 4, 5},1 2,4 1,1 5,{2, 3, 5},1 │ │ │ │ + + x x e0 , -x e0 + x e0 │ │ │ │ + 5},1 2,4 1,2 5,{1, 3, 5},1 2,2 5,{1, 3, 5},1 2,2 5,{1, 3, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x e0 + x x e0 } │ │ │ │ - 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ │ + + x e0 - x e0 } │ │ │ │ + 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},1 │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_m_a_g_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- get the basis element images of a free OI- │ │ │ │ module map │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./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 0.349276s (cpu); 0.250698s (thread); 0s (gc)
│ │ │ + -- used 0.368178s (cpu); 0.267471s (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 0.349276s (cpu); 0.250698s (thread); 0s (gc) │ │ │ │ + -- used 0.368178s (cpu); 0.267471s (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 0.0200002s (cpu); 0.0226859s (thread); 0s (gc)
│ │ │ + -- used 0.0240144s (cpu); 0.02568s (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 0.0200002s (cpu); 0.0226859s (thread); 0s (gc)
│ │ │ │ + -- used 0.0240144s (cpu); 0.02568s (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 0.0239813s (cpu); 0.0250935s (thread); 0s (gc)
│ │ │ + -- used 0.027954s (cpu); 0.0293941s (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
│ │ │  
│ │ │ @@ -149,16 +149,16 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │                                                                             
│ │ │ -o14 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ -        1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │ +o14 = {x   x   e        + x   x   e          , x   e        + x   e       ,
│ │ │ +        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   1,1 1,{1},1    2,1 1,{1},2 
│ │ │        -----------------------------------------------------------------------
│ │ │                                       2
│ │ │        x   x   x   e           - x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │ ├── 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 0.0239813s (cpu); 0.0250935s (thread); 0s (gc)
│ │ │ │ + -- used 0.027954s (cpu); 0.0293941s (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
│ │ │ │  
│ │ │ │ @@ -50,16 +50,16 @@
│ │ │ │             - x   x   e          }
│ │ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : minimizeOIGB C -- an element gets removed
│ │ │ │  
│ │ │ │  
│ │ │ │ -o14 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │ -        1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │ +o14 = {x   x   e        + x   x   e          , x   e        + x   e       ,
│ │ │ │ +        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                                       2
│ │ │ │        x   x   x   e           - x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1, 3},3
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** WWaayyss ttoo uussee mmiinniimmiizzeeOOIIGGBB:: **********
│ │ ├── ./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 0.180402s (cpu); 0.118358s (thread); 0s (gc)
    
│ │ │ + -- used 0.152451s (cpu); 0.0977736s (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 0.180402s (cpu); 0.118358s (thread); 0s (gc)
│ │ │ │ + -- used 0.152451s (cpu); 0.0977736s (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 0.0279994s (cpu); 0.0261102s (thread); 0s (gc)
│ │ │ + -- used 0.0319093s (cpu); 0.0303633s (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 0.0279994s (cpu); 0.0261102s (thread); 0s (gc)
│ │ │ │ + -- used 0.0319093s (cpu); 0.0303633s (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 0.325754s (cpu); 0.234878s (thread); 0s (gc)
│ │ │ + -- used 0.376309s (cpu); 0.26982s (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 0.325754s (cpu); 0.234878s (thread); 0s (gc)
│ │ │ │ + -- used 0.376309s (cpu); 0.26982s (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 0.170315s (cpu); 0.109167s (thread); 0s (gc)
│ │ │ + -- used 0.176313s (cpu); 0.128463s (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 0.170315s (cpu); 0.109167s (thread); 0s (gc)
│ │ │ │ + -- used 0.176313s (cpu); 0.128463s (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/_terms_lp__Vector__In__Width_rp.html
│ │ │ @@ -99,16 +99,16 @@
│ │ │            2,2   2,1   1,2   1,1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : terms f
│ │ │  
│ │ │ -o5 = {-x   e       , x   e       , -x   e       , x   e       }
│ │ │ -        1,2 2,{2},1   2,1 2,{1},2    2,2 2,{2},2   1,1 2,{1},1
│ │ │ +o5 = {x   e       , -x   e       , x   e       , -x   e       }
│ │ │ +       1,1 2,{1},1    1,2 2,{2},1   2,1 2,{1},2    2,2 2,{2},2
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -24,16 +24,16 @@ │ │ │ │ 1,2 2,{2},1 1,1 2,{1},1 2,2 2,{2},2 2,1 2,{1},2 │ │ │ │ │ │ │ │ 4 │ │ │ │ o4 : (QQ[x , x , x , x ]) in width 2 │ │ │ │ 2,2 2,1 1,2 1,1 │ │ │ │ i5 : terms f │ │ │ │ │ │ │ │ -o5 = {-x e , x e , -x e , x e } │ │ │ │ - 1,2 2,{2},1 2,1 2,{1},2 2,2 2,{2},2 1,1 2,{1},1 │ │ │ │ +o5 = {x e , -x e , x e , -x e } │ │ │ │ + 1,1 2,{1},1 1,2 2,{2},1 2,1 2,{1},2 2,2 2,{2},2 │ │ │ │ │ │ │ │ o5 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The list of terms need not be in order. To find the lead term of an element, │ │ │ │ see _l_e_a_d_T_e_r_m_(_V_e_c_t_o_r_I_n_W_i_d_t_h_). │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _t_e_r_m_s_(_V_e_c_t_o_r_I_n_W_i_d_t_h_) -- get the terms of an element of a free OI-module │ │ ├── ./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) │ │ │ - -- 2.27486s elapsed │ │ │ + -- 2.47772s 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_*) │ │ │ - -- 1.71676s elapsed │ │ │ + -- 1.27629s 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) │ │ │ - -- 2.38774s elapsed │ │ │ + -- 2.54783s 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 1.05216s (cpu); 0.844843s (thread); 0s (gc) │ │ │ - -- used 0.386363s (cpu); 0.319325s (thread); 0s (gc) │ │ │ + -- used 1.05864s (cpu); 0.990682s (thread); 0s (gc) │ │ │ + -- used 0.964514s (cpu); 0.820608s (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)
│ │ │ - -- 2.27486s elapsed
│ │ │ + -- 2.47772s 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_*)
│ │ │ - -- 1.71676s elapsed
│ │ │ + -- 1.27629s 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) │ │ │ │ - -- 2.27486s elapsed │ │ │ │ + -- 2.47772s 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_*) │ │ │ │ - -- 1.71676s elapsed │ │ │ │ + -- 1.27629s 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)
│ │ │ - -- 2.38774s elapsed
│ │ │ + -- 2.54783s 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) │ │ │ │ - -- 2.38774s elapsed │ │ │ │ + -- 2.54783s 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 1.05216s (cpu); 0.844843s (thread); 0s (gc) │ │ │ - -- used 0.386363s (cpu); 0.319325s (thread); 0s (gc) │ │ │ + -- used 1.05864s (cpu); 0.990682s (thread); 0s (gc) │ │ │ + -- used 0.964514s (cpu); 0.820608s (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 1.05216s (cpu); 0.844843s (thread); 0s (gc) │ │ │ │ - -- used 0.386363s (cpu); 0.319325s (thread); 0s (gc) │ │ │ │ + -- used 1.05864s (cpu); 0.990682s (thread); 0s (gc) │ │ │ │ + -- used 0.964514s (cpu); 0.820608s (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/OldPolyhedra/example-output/_hirzebruch.out │ │ │ @@ -7,13 +7,13 @@ │ │ │ number of rays => 4 │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ o1 : Fan │ │ │ │ │ │ i2 : apply(maxCones F,rays) │ │ │ │ │ │ -o2 = {| 0 -1 |, | 1 0 |, | 1 0 |, | 0 -1 |} │ │ │ - | 1 3 | | 0 1 | | 0 -1 | | -1 3 | │ │ │ +o2 = {| 1 0 |, | 0 -1 |, | 0 -1 |, | 1 0 |} │ │ │ + | 0 -1 | | -1 3 | | 1 3 | | 0 1 | │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/example-output/_skeleton.out │ │ │ @@ -27,17 +27,17 @@ │ │ │ number of rays => 4 │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ o3 : Fan │ │ │ │ │ │ i4 : apply(maxCones F1,rays) │ │ │ │ │ │ -o4 = {| -1 0 |, | 1 -1 |, | 0 -1 |, | 1 0 |, | 1 0 |, | 0 0 |} │ │ │ - | -1 0 | | 0 -1 | | 1 -1 | | 0 1 | | 0 0 | | 1 0 | │ │ │ - | -1 1 | | 0 -1 | | 0 -1 | | 0 0 | | 0 1 | | 0 1 | │ │ │ +o4 = {| 1 -1 |, | 0 -1 |, | 1 0 |, | 0 0 |, | 1 0 |, | -1 0 |} │ │ │ + | 0 -1 | | 1 -1 | | 0 0 | | 1 0 | | 0 1 | | -1 0 | │ │ │ + | 0 -1 | | 0 -1 | | 0 1 | | 0 1 | | 0 0 | | -1 1 | │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : PC = polyhedralComplex hypercube 3 │ │ │ │ │ │ o5 = {ambient dimension => 3 } │ │ │ number of generating polyhedra => 1 │ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/_hirzebruch.html │ │ │ @@ -81,16 +81,16 @@ │ │ │ o1 : Fan │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : apply(maxCones F,rays)
│ │ │  
│ │ │ -o2 = {| 0 -1 |, | 1 0 |, | 1 0  |, | 0  -1 |}
│ │ │ -      | 1 3  |  | 0 1 |  | 0 -1 |  | -1 3  |
│ │ │ +o2 = {| 1 0  |, | 0  -1 |, | 0 -1 |, | 1 0 |}
│ │ │ +      | 0 -1 |  | -1 3  |  | 1 3  |  | 0 1 |
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,16 +19,16 @@ │ │ │ │ number of generating cones => 4 │ │ │ │ number of rays => 4 │ │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ │ │ o1 : Fan │ │ │ │ i2 : apply(maxCones F,rays) │ │ │ │ │ │ │ │ -o2 = {| 0 -1 |, | 1 0 |, | 1 0 |, | 0 -1 |} │ │ │ │ - | 1 3 | | 0 1 | | 0 -1 | | -1 3 | │ │ │ │ +o2 = {| 1 0 |, | 0 -1 |, | 0 -1 |, | 1 0 |} │ │ │ │ + | 0 -1 | | -1 3 | | 1 3 | | 0 1 | │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** WWaayyss ttoo uussee hhiirrzzeebbrruucchh:: ********** │ │ │ │ * hirzebruch(ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _h_i_r_z_e_b_r_u_c_h is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/_skeleton.html │ │ │ @@ -111,17 +111,17 @@ │ │ │ o3 : Fan │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : apply(maxCones F1,rays)
│ │ │  
│ │ │ -o4 = {| -1 0 |, | 1 -1 |, | 0 -1 |, | 1 0 |, | 1 0 |, | 0 0 |}
│ │ │ -      | -1 0 |  | 0 -1 |  | 1 -1 |  | 0 1 |  | 0 0 |  | 1 0 |
│ │ │ -      | -1 1 |  | 0 -1 |  | 0 -1 |  | 0 0 |  | 0 1 |  | 0 1 |
│ │ │ +o4 = {| 1 -1 |, | 0 -1 |, | 1 0 |, | 0 0 |, | 1 0 |, | -1 0 |}
│ │ │ +      | 0 -1 |  | 1 -1 |  | 0 0 |  | 1 0 |  | 0 1 |  | -1 0 |
│ │ │ +      | 0 -1 |  | 0 -1 |  | 0 1 |  | 0 1 |  | 0 0 |  | -1 1 |
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │

    │ │ │ For a PolyhedralComplex PC and an integer k between 0 and the dimension of PC, skeleton computes the k-skeleton of the PolyhedralComplex PC, i.e. the PolyhedralComplex PC1 generated by all polyhedra of dimension k in PC. │ │ │ ├── html2text {} │ │ │ │ @@ -43,17 +43,17 @@ │ │ │ │ number of generating cones => 6 │ │ │ │ number of rays => 4 │ │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ │ │ o3 : Fan │ │ │ │ i4 : apply(maxCones F1,rays) │ │ │ │ │ │ │ │ -o4 = {| -1 0 |, | 1 -1 |, | 0 -1 |, | 1 0 |, | 1 0 |, | 0 0 |} │ │ │ │ - | -1 0 | | 0 -1 | | 1 -1 | | 0 1 | | 0 0 | | 1 0 | │ │ │ │ - | -1 1 | | 0 -1 | | 0 -1 | | 0 0 | | 0 1 | | 0 1 | │ │ │ │ +o4 = {| 1 -1 |, | 0 -1 |, | 1 0 |, | 0 0 |, | 1 0 |, | -1 0 |} │ │ │ │ + | 0 -1 | | 1 -1 | | 0 0 | | 1 0 | | 0 1 | | -1 0 | │ │ │ │ + | 0 -1 | | 0 -1 | | 0 1 | | 0 1 | | 0 0 | | -1 1 | │ │ │ │ │ │ │ │ o4 : List │ │ │ │ For a _P_o_l_y_h_e_d_r_a_l_C_o_m_p_l_e_x PC and an integer k between 0 and the dimension of PC, │ │ │ │ skeleton computes the k-skeleton of the _P_o_l_y_h_e_d_r_a_l_C_o_m_p_l_e_x PC, i.e. the │ │ │ │ _P_o_l_y_h_e_d_r_a_l_C_o_m_p_l_e_x PC1 generated by all polyhedra of dimension k in PC. │ │ │ │ i5 : PC = polyhedralComplex hypercube 3 │ │ ├── ./usr/share/doc/Macaulay2/OldToricVectorBundles/example-output/_projective__Space__Fan.out │ │ │ @@ -7,13 +7,13 @@ │ │ │ number of rays => 3 │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ o1 : Fan │ │ │ │ │ │ i2 : apply(maxCones F, rays) │ │ │ │ │ │ -o2 = {| 1 -1 |, | 1 0 |, | -1 0 |} │ │ │ - | 0 -1 | | 0 1 | | -1 1 | │ │ │ +o2 = {| 1 0 |, | -1 0 |, | 1 -1 |} │ │ │ + | 0 1 | | -1 1 | | 0 -1 | │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/OldToricVectorBundles/html/_projective__Space__Fan.html │ │ │ @@ -81,16 +81,16 @@ │ │ │ o1 : Fan │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : apply(maxCones F, rays)
│ │ │  
│ │ │ -o2 = {| 1 -1 |, | 1 0 |, | -1 0 |}
│ │ │ -      | 0 -1 |  | 0 1 |  | -1 1 |
│ │ │ +o2 = {| 1 0 |, | -1 0 |, | 1 -1 |}
│ │ │ +      | 0 1 |  | -1 1 |  | 0 -1 |
│ │ │  
│ │ │  o2 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,16 +18,16 @@ │ │ │ │ number of generating cones => 3 │ │ │ │ number of rays => 3 │ │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ │ │ o1 : Fan │ │ │ │ i2 : apply(maxCones F, rays) │ │ │ │ │ │ │ │ -o2 = {| 1 -1 |, | 1 0 |, | -1 0 |} │ │ │ │ - | 0 -1 | | 0 1 | | -1 1 | │ │ │ │ +o2 = {| 1 0 |, | -1 0 |, | 1 -1 |} │ │ │ │ + | 0 1 | | -1 1 | | 0 -1 | │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _F_a_n -- the class of all fans │ │ │ │ * _h_i_r_z_e_b_r_u_c_h_F_a_n -- the fan of the n-th Hirzebruch surface │ │ │ │ * _p_p_1_P_r_o_d_u_c_t_F_a_n -- the fan of n products of PP^1 │ │ │ │ ********** WWaayyss ttoo uussee pprroojjeeccttiivveeSSppaacceeFFaann:: ********** │ │ ├── ./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} │ │ │ ); │ │ │ - -- .415678s elapsed │ │ │ - -- .496835s elapsed │ │ │ - -- .715173s elapsed │ │ │ - -- .405844s elapsed │ │ │ - -- .513434s elapsed │ │ │ - -- .459835s elapsed │ │ │ - -- .849636s elapsed │ │ │ - -- .702988s elapsed │ │ │ - -- .771172s elapsed │ │ │ - -- .621818s elapsed │ │ │ - -- .361019s elapsed │ │ │ + -- .326808s elapsed │ │ │ + -- .326454s elapsed │ │ │ + -- .490403s elapsed │ │ │ + -- .264515s elapsed │ │ │ + -- .284275s elapsed │ │ │ + -- .348206s elapsed │ │ │ + -- .640868s elapsed │ │ │ + -- .503931s elapsed │ │ │ + -- .506835s elapsed │ │ │ + -- .337078s elapsed │ │ │ + -- .226935s 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} │ │ │ ); │ │ │ - -- .69364s elapsed │ │ │ - -- .797714s elapsed │ │ │ - -- 1.48339s elapsed │ │ │ - -- 1.94383s elapsed │ │ │ - -- 1.11895s elapsed │ │ │ - -- 1.38109s elapsed │ │ │ - -- 1.53039s elapsed │ │ │ - -- 1.65589s elapsed │ │ │ - -- 1.35415s elapsed │ │ │ - -- 1.16803s elapsed │ │ │ - -- .520728s elapsed │ │ │ - -- .699905s elapsed │ │ │ - -- .667573s elapsed │ │ │ - -- .91576s elapsed │ │ │ - -- 1.05251s elapsed │ │ │ - -- 1.61894s elapsed │ │ │ - -- 1.73195s elapsed │ │ │ - -- 1.68105s elapsed │ │ │ - -- 1.82739s elapsed │ │ │ - -- 1.52885s elapsed │ │ │ - -- 1.02047s elapsed │ │ │ - -- 1.11083s elapsed │ │ │ - -- 2.09686s elapsed │ │ │ - -- 1.96734s elapsed │ │ │ - -- .953695s elapsed │ │ │ - -- 1.28279s elapsed │ │ │ - -- 2.00452s elapsed │ │ │ - -- 1.25212s elapsed │ │ │ - -- .912155s elapsed │ │ │ - -- 1.44258s elapsed │ │ │ - -- 1.70126s elapsed │ │ │ - -- 1.47787s elapsed │ │ │ - -- .991996s elapsed │ │ │ - -- 1.75162s elapsed │ │ │ - -- 1.31039s elapsed │ │ │ - -- 1.90954s elapsed │ │ │ - -- 1.6628s elapsed │ │ │ - -- 2.0586s elapsed │ │ │ - -- 2.2701s elapsed │ │ │ - -- 1.39832s elapsed │ │ │ - -- 1.1299s elapsed │ │ │ - -- 1.84896s elapsed │ │ │ - -- 2.37253s elapsed │ │ │ - -- 2.47875s elapsed │ │ │ - -- 1.51195s elapsed │ │ │ - -- 1.73262s elapsed │ │ │ - -- 2.33948s elapsed │ │ │ - -- 2.14844s elapsed │ │ │ - -- 1.72814s elapsed │ │ │ - -- 1.83218s elapsed │ │ │ - -- 1.71131s elapsed │ │ │ - -- 1.43648s elapsed │ │ │ - -- 1.48611s elapsed │ │ │ - -- 1.73027s elapsed │ │ │ - -- 1.08176s elapsed │ │ │ - -- 1.86789s elapsed │ │ │ - -- 2.16645s elapsed │ │ │ - -- 2.29611s elapsed │ │ │ - -- 1.2988s elapsed │ │ │ - -- .89606s elapsed │ │ │ - -- .666112s elapsed │ │ │ + -- .4081s elapsed │ │ │ + -- .481081s elapsed │ │ │ + -- .946793s elapsed │ │ │ + -- 1.26722s elapsed │ │ │ + -- .714664s elapsed │ │ │ + -- .904213s elapsed │ │ │ + -- 1.01174s elapsed │ │ │ + -- 1.08082s elapsed │ │ │ + -- .712759s elapsed │ │ │ + -- .72132s elapsed │ │ │ + -- .350734s elapsed │ │ │ + -- .444452s elapsed │ │ │ + -- .498302s elapsed │ │ │ + -- .63701s elapsed │ │ │ + -- .844186s elapsed │ │ │ + -- 1.15191s elapsed │ │ │ + -- .993004s elapsed │ │ │ + -- .953207s elapsed │ │ │ + -- 1.23231s elapsed │ │ │ + -- 1.00686s elapsed │ │ │ + -- .821857s elapsed │ │ │ + -- .912833s elapsed │ │ │ + -- 1.27644s elapsed │ │ │ + -- 1.25323s elapsed │ │ │ + -- .547972s elapsed │ │ │ + -- .700954s elapsed │ │ │ + -- 1.32865s elapsed │ │ │ + -- .706727s elapsed │ │ │ + -- .593716s elapsed │ │ │ + -- .84837s elapsed │ │ │ + -- .999105s elapsed │ │ │ + -- .81637s elapsed │ │ │ + -- .544839s elapsed │ │ │ + -- .993164s elapsed │ │ │ + -- .785499s elapsed │ │ │ + -- 1.05677s elapsed │ │ │ + -- .915761s elapsed │ │ │ + -- 1.10507s elapsed │ │ │ + -- 1.18055s elapsed │ │ │ + -- .723905s elapsed │ │ │ + -- .611094s elapsed │ │ │ + -- 1.05558s elapsed │ │ │ + -- 1.31913s elapsed │ │ │ + -- 1.55224s elapsed │ │ │ + -- 1.052s elapsed │ │ │ + -- .987815s elapsed │ │ │ + -- 1.34239s elapsed │ │ │ + -- 1.20152s elapsed │ │ │ + -- .975699s elapsed │ │ │ + -- 1.01835s elapsed │ │ │ + -- 1.07078s elapsed │ │ │ + -- .760947s elapsed │ │ │ + -- .782478s elapsed │ │ │ + -- .954493s elapsed │ │ │ + -- .629618s elapsed │ │ │ + -- 1.06576s elapsed │ │ │ + -- 1.17816s elapsed │ │ │ + -- 1.27111s elapsed │ │ │ + -- .735223s elapsed │ │ │ + -- .460188s elapsed │ │ │ + -- .355052s 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 │ │ │ @@ -41,16 +41,16 @@ │ │ │ .954}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime stablesolsPent = showExoticSolutions Pent │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ - -- .902s elapsed │ │ │ - -- 1s elapsed │ │ │ + -- .744s elapsed │ │ │ + -- .804s 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| │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ @@ -45,25 +45,25 @@ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ o5 = 3 │ │ │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ - -- .799s elapsed │ │ │ - -- .703s elapsed │ │ │ - -- .841s elapsed │ │ │ - -- .954s elapsed │ │ │ - -- 1.14s elapsed │ │ │ - -- 1.3s elapsed │ │ │ - -- 1.64s elapsed │ │ │ - -- 1.25s elapsed │ │ │ - -- 1.42s elapsed │ │ │ - -- 1.58s elapsed │ │ │ - -- 1.34s elapsed │ │ │ + -- .579s elapsed │ │ │ + -- .538s elapsed │ │ │ + -- .643s elapsed │ │ │ + -- .833s elapsed │ │ │ + -- .992s elapsed │ │ │ + -- 1.28s elapsed │ │ │ + -- 1.37s elapsed │ │ │ + -- 1.11s elapsed │ │ │ + -- 1.1s elapsed │ │ │ + -- 1.23s elapsed │ │ │ + -- 1.1s elapsed │ │ │ 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} │ │ │ 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} │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ 1 => {3, 4} │ │ │ 2 => {0, 4} │ │ │ 3 => {0, 1} │ │ │ 4 => {2, 1} │ │ ├── ./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) │ │ │ - -- .296874s elapsed │ │ │ + -- .193921s 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) │ │ │ - -- .95871s elapsed │ │ │ + -- .722488s 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 │ │ │ - -- 1.25533s elapsed │ │ │ + -- 1.02052s 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}
│ │ │            );
│ │ │ - -- .415678s elapsed
│ │ │ - -- .496835s elapsed
│ │ │ - -- .715173s elapsed
│ │ │ - -- .405844s elapsed
│ │ │ - -- .513434s elapsed
│ │ │ - -- .459835s elapsed
│ │ │ - -- .849636s elapsed
│ │ │ - -- .702988s elapsed
│ │ │ - -- .771172s elapsed
│ │ │ - -- .621818s elapsed
│ │ │ - -- .361019s elapsed
    │ │ │ + -- .326808s elapsed │ │ │ + -- .326454s elapsed │ │ │ + -- .490403s elapsed │ │ │ + -- .264515s elapsed │ │ │ + -- .284275s elapsed │ │ │ + -- .348206s elapsed │ │ │ + -- .640868s elapsed │ │ │ + -- .503931s elapsed │ │ │ + -- .506835s elapsed │ │ │ + -- .337078s elapsed │ │ │ + -- .226935s 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}
│ │ │            );
│ │ │ - -- .69364s elapsed
│ │ │ - -- .797714s elapsed
│ │ │ - -- 1.48339s elapsed
│ │ │ - -- 1.94383s elapsed
│ │ │ - -- 1.11895s elapsed
│ │ │ - -- 1.38109s elapsed
│ │ │ - -- 1.53039s elapsed
│ │ │ - -- 1.65589s elapsed
│ │ │ - -- 1.35415s elapsed
│ │ │ - -- 1.16803s elapsed
│ │ │ - -- .520728s elapsed
│ │ │ - -- .699905s elapsed
│ │ │ - -- .667573s elapsed
│ │ │ - -- .91576s elapsed
│ │ │ - -- 1.05251s elapsed
│ │ │ - -- 1.61894s elapsed
│ │ │ - -- 1.73195s elapsed
│ │ │ - -- 1.68105s elapsed
│ │ │ - -- 1.82739s elapsed
│ │ │ - -- 1.52885s elapsed
│ │ │ - -- 1.02047s elapsed
│ │ │ - -- 1.11083s elapsed
│ │ │ - -- 2.09686s elapsed
│ │ │ - -- 1.96734s elapsed
│ │ │ - -- .953695s elapsed
│ │ │ - -- 1.28279s elapsed
│ │ │ - -- 2.00452s elapsed
│ │ │ - -- 1.25212s elapsed
│ │ │ - -- .912155s elapsed
│ │ │ - -- 1.44258s elapsed
│ │ │ - -- 1.70126s elapsed
│ │ │ - -- 1.47787s elapsed
│ │ │ - -- .991996s elapsed
│ │ │ - -- 1.75162s elapsed
│ │ │ - -- 1.31039s elapsed
│ │ │ - -- 1.90954s elapsed
│ │ │ - -- 1.6628s elapsed
│ │ │ - -- 2.0586s elapsed
│ │ │ - -- 2.2701s elapsed
│ │ │ - -- 1.39832s elapsed
│ │ │ - -- 1.1299s elapsed
│ │ │ - -- 1.84896s elapsed
│ │ │ - -- 2.37253s elapsed
│ │ │ - -- 2.47875s elapsed
│ │ │ - -- 1.51195s elapsed
│ │ │ - -- 1.73262s elapsed
│ │ │ - -- 2.33948s elapsed
│ │ │ - -- 2.14844s elapsed
│ │ │ - -- 1.72814s elapsed
│ │ │ - -- 1.83218s elapsed
│ │ │ - -- 1.71131s elapsed
│ │ │ - -- 1.43648s elapsed
│ │ │ - -- 1.48611s elapsed
│ │ │ - -- 1.73027s elapsed
│ │ │ - -- 1.08176s elapsed
│ │ │ - -- 1.86789s elapsed
│ │ │ - -- 2.16645s elapsed
│ │ │ - -- 2.29611s elapsed
│ │ │ - -- 1.2988s elapsed
│ │ │ - -- .89606s elapsed
│ │ │ - -- .666112s elapsed
    
│ │ │ + -- .4081s elapsed
│ │ │ + -- .481081s elapsed
│ │ │ + -- .946793s elapsed
│ │ │ + -- 1.26722s elapsed
│ │ │ + -- .714664s elapsed
│ │ │ + -- .904213s elapsed
│ │ │ + -- 1.01174s elapsed
│ │ │ + -- 1.08082s elapsed
│ │ │ + -- .712759s elapsed
│ │ │ + -- .72132s elapsed
│ │ │ + -- .350734s elapsed
│ │ │ + -- .444452s elapsed
│ │ │ + -- .498302s elapsed
│ │ │ + -- .63701s elapsed
│ │ │ + -- .844186s elapsed
│ │ │ + -- 1.15191s elapsed
│ │ │ + -- .993004s elapsed
│ │ │ + -- .953207s elapsed
│ │ │ + -- 1.23231s elapsed
│ │ │ + -- 1.00686s elapsed
│ │ │ + -- .821857s elapsed
│ │ │ + -- .912833s elapsed
│ │ │ + -- 1.27644s elapsed
│ │ │ + -- 1.25323s elapsed
│ │ │ + -- .547972s elapsed
│ │ │ + -- .700954s elapsed
│ │ │ + -- 1.32865s elapsed
│ │ │ + -- .706727s elapsed
│ │ │ + -- .593716s elapsed
│ │ │ + -- .84837s elapsed
│ │ │ + -- .999105s elapsed
│ │ │ + -- .81637s elapsed
│ │ │ + -- .544839s elapsed
│ │ │ + -- .993164s elapsed
│ │ │ + -- .785499s elapsed
│ │ │ + -- 1.05677s elapsed
│ │ │ + -- .915761s elapsed
│ │ │ + -- 1.10507s elapsed
│ │ │ + -- 1.18055s elapsed
│ │ │ + -- .723905s elapsed
│ │ │ + -- .611094s elapsed
│ │ │ + -- 1.05558s elapsed
│ │ │ + -- 1.31913s elapsed
│ │ │ + -- 1.55224s elapsed
│ │ │ + -- 1.052s elapsed
│ │ │ + -- .987815s elapsed
│ │ │ + -- 1.34239s elapsed
│ │ │ + -- 1.20152s elapsed
│ │ │ + -- .975699s elapsed
│ │ │ + -- 1.01835s elapsed
│ │ │ + -- 1.07078s elapsed
│ │ │ + -- .760947s elapsed
│ │ │ + -- .782478s elapsed
│ │ │ + -- .954493s elapsed
│ │ │ + -- .629618s elapsed
│ │ │ + -- 1.06576s elapsed
│ │ │ + -- 1.17816s elapsed
│ │ │ + -- 1.27111s elapsed
│ │ │ + -- .735223s elapsed
│ │ │ + -- .460188s elapsed
│ │ │ + -- .355052s 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}
│ │ │ │            );
│ │ │ │ - -- .415678s elapsed
│ │ │ │ - -- .496835s elapsed
│ │ │ │ - -- .715173s elapsed
│ │ │ │ - -- .405844s elapsed
│ │ │ │ - -- .513434s elapsed
│ │ │ │ - -- .459835s elapsed
│ │ │ │ - -- .849636s elapsed
│ │ │ │ - -- .702988s elapsed
│ │ │ │ - -- .771172s elapsed
│ │ │ │ - -- .621818s elapsed
│ │ │ │ - -- .361019s elapsed
│ │ │ │ + -- .326808s elapsed
│ │ │ │ + -- .326454s elapsed
│ │ │ │ + -- .490403s elapsed
│ │ │ │ + -- .264515s elapsed
│ │ │ │ + -- .284275s elapsed
│ │ │ │ + -- .348206s elapsed
│ │ │ │ + -- .640868s elapsed
│ │ │ │ + -- .503931s elapsed
│ │ │ │ + -- .506835s elapsed
│ │ │ │ + -- .337078s elapsed
│ │ │ │ + -- .226935s 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}
│ │ │ │            );
│ │ │ │ - -- .69364s elapsed
│ │ │ │ - -- .797714s elapsed
│ │ │ │ - -- 1.48339s elapsed
│ │ │ │ - -- 1.94383s elapsed
│ │ │ │ - -- 1.11895s elapsed
│ │ │ │ - -- 1.38109s elapsed
│ │ │ │ - -- 1.53039s elapsed
│ │ │ │ - -- 1.65589s elapsed
│ │ │ │ - -- 1.35415s elapsed
│ │ │ │ - -- 1.16803s elapsed
│ │ │ │ - -- .520728s elapsed
│ │ │ │ - -- .699905s elapsed
│ │ │ │ - -- .667573s elapsed
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│ │ │ │ - -- 1.05251s elapsed
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│ │ │ │ - -- 1.52885s elapsed
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│ │ │ │ - -- 2.09686s elapsed
│ │ │ │ - -- 1.96734s elapsed
│ │ │ │ - -- .953695s elapsed
│ │ │ │ - -- 1.28279s elapsed
│ │ │ │ - -- 2.00452s elapsed
│ │ │ │ - -- 1.25212s elapsed
│ │ │ │ - -- .912155s elapsed
│ │ │ │ - -- 1.44258s elapsed
│ │ │ │ - -- 1.70126s elapsed
│ │ │ │ - -- 1.47787s elapsed
│ │ │ │ - -- .991996s elapsed
│ │ │ │ - -- 1.75162s elapsed
│ │ │ │ - -- 1.31039s elapsed
│ │ │ │ - -- 1.90954s elapsed
│ │ │ │ - -- 1.6628s elapsed
│ │ │ │ - -- 2.0586s elapsed
│ │ │ │ - -- 2.2701s elapsed
│ │ │ │ - -- 1.39832s elapsed
│ │ │ │ - -- 1.1299s elapsed
│ │ │ │ - -- 1.84896s elapsed
│ │ │ │ - -- 2.37253s elapsed
│ │ │ │ - -- 2.47875s elapsed
│ │ │ │ - -- 1.51195s elapsed
│ │ │ │ - -- 1.73262s elapsed
│ │ │ │ - -- 2.33948s elapsed
│ │ │ │ - -- 2.14844s elapsed
│ │ │ │ - -- 1.72814s elapsed
│ │ │ │ - -- 1.83218s elapsed
│ │ │ │ - -- 1.71131s elapsed
│ │ │ │ - -- 1.43648s elapsed
│ │ │ │ - -- 1.48611s elapsed
│ │ │ │ - -- 1.73027s elapsed
│ │ │ │ - -- 1.08176s elapsed
│ │ │ │ - -- 1.86789s elapsed
│ │ │ │ - -- 2.16645s elapsed
│ │ │ │ - -- 2.29611s elapsed
│ │ │ │ - -- 1.2988s elapsed
│ │ │ │ - -- .89606s elapsed
│ │ │ │ - -- .666112s elapsed
│ │ │ │ + -- .4081s elapsed
│ │ │ │ + -- .481081s elapsed
│ │ │ │ + -- .946793s elapsed
│ │ │ │ + -- 1.26722s elapsed
│ │ │ │ + -- .714664s elapsed
│ │ │ │ + -- .904213s elapsed
│ │ │ │ + -- 1.01174s elapsed
│ │ │ │ + -- 1.08082s elapsed
│ │ │ │ + -- .712759s elapsed
│ │ │ │ + -- .72132s elapsed
│ │ │ │ + -- .350734s elapsed
│ │ │ │ + -- .444452s elapsed
│ │ │ │ + -- .498302s elapsed
│ │ │ │ + -- .63701s elapsed
│ │ │ │ + -- .844186s elapsed
│ │ │ │ + -- 1.15191s elapsed
│ │ │ │ + -- .993004s elapsed
│ │ │ │ + -- .953207s elapsed
│ │ │ │ + -- 1.23231s elapsed
│ │ │ │ + -- 1.00686s elapsed
│ │ │ │ + -- .821857s elapsed
│ │ │ │ + -- .912833s elapsed
│ │ │ │ + -- 1.27644s elapsed
│ │ │ │ + -- 1.25323s elapsed
│ │ │ │ + -- .547972s elapsed
│ │ │ │ + -- .700954s elapsed
│ │ │ │ + -- 1.32865s elapsed
│ │ │ │ + -- .706727s elapsed
│ │ │ │ + -- .593716s elapsed
│ │ │ │ + -- .84837s elapsed
│ │ │ │ + -- .999105s elapsed
│ │ │ │ + -- .81637s elapsed
│ │ │ │ + -- .544839s elapsed
│ │ │ │ + -- .993164s elapsed
│ │ │ │ + -- .785499s elapsed
│ │ │ │ + -- 1.05677s elapsed
│ │ │ │ + -- .915761s elapsed
│ │ │ │ + -- 1.10507s elapsed
│ │ │ │ + -- 1.18055s elapsed
│ │ │ │ + -- .723905s elapsed
│ │ │ │ + -- .611094s elapsed
│ │ │ │ + -- 1.05558s elapsed
│ │ │ │ + -- 1.31913s elapsed
│ │ │ │ + -- 1.55224s elapsed
│ │ │ │ + -- 1.052s elapsed
│ │ │ │ + -- .987815s elapsed
│ │ │ │ + -- 1.34239s elapsed
│ │ │ │ + -- 1.20152s elapsed
│ │ │ │ + -- .975699s elapsed
│ │ │ │ + -- 1.01835s elapsed
│ │ │ │ + -- 1.07078s elapsed
│ │ │ │ + -- .760947s elapsed
│ │ │ │ + -- .782478s elapsed
│ │ │ │ + -- .954493s elapsed
│ │ │ │ + -- .629618s elapsed
│ │ │ │ + -- 1.06576s elapsed
│ │ │ │ + -- 1.17816s elapsed
│ │ │ │ + -- 1.27111s elapsed
│ │ │ │ + -- .735223s elapsed
│ │ │ │ + -- .460188s elapsed
│ │ │ │ + -- .355052s 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
│ │ │ @@ -112,16 +112,16 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .902s elapsed
│ │ │ - -- 1s elapsed
│ │ │ + -- .744s elapsed
│ │ │ + -- .804s 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|
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,16 +45,16 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .954}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .902s elapsed
│ │ │ │ - -- 1s elapsed
│ │ │ │ + -- .744s elapsed
│ │ │ │ + -- .804s 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|
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html
│ │ │ @@ -116,25 +116,25 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .799s elapsed
│ │ │ - -- .703s elapsed
│ │ │ - -- .841s elapsed
│ │ │ - -- .954s elapsed
│ │ │ - -- 1.14s elapsed
│ │ │ - -- 1.3s elapsed
│ │ │ - -- 1.64s elapsed
│ │ │ - -- 1.25s elapsed
│ │ │ - -- 1.42s elapsed
│ │ │ - -- 1.58s elapsed
│ │ │ - -- 1.34s elapsed
│ │ │ + -- .579s elapsed
│ │ │ + -- .538s elapsed
│ │ │ + -- .643s elapsed
│ │ │ + -- .833s elapsed
│ │ │ + -- .992s elapsed
│ │ │ + -- 1.28s elapsed
│ │ │ + -- 1.37s elapsed
│ │ │ + -- 1.11s elapsed
│ │ │ + -- 1.1s elapsed
│ │ │ + -- 1.23s elapsed
│ │ │ + -- 1.1s elapsed
│ │ │  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}
│ │ │  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}
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -47,25 +47,25 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : printingPrecision = 3
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .799s elapsed
│ │ │ │ - -- .703s elapsed
│ │ │ │ - -- .841s elapsed
│ │ │ │ - -- .954s elapsed
│ │ │ │ - -- 1.14s elapsed
│ │ │ │ - -- 1.3s elapsed
│ │ │ │ - -- 1.64s elapsed
│ │ │ │ - -- 1.25s elapsed
│ │ │ │ - -- 1.42s elapsed
│ │ │ │ - -- 1.58s elapsed
│ │ │ │ - -- 1.34s elapsed
│ │ │ │ + -- .579s elapsed
│ │ │ │ + -- .538s elapsed
│ │ │ │ + -- .643s elapsed
│ │ │ │ + -- .833s elapsed
│ │ │ │ + -- .992s elapsed
│ │ │ │ + -- 1.28s elapsed
│ │ │ │ + -- 1.37s elapsed
│ │ │ │ + -- 1.11s elapsed
│ │ │ │ + -- 1.1s elapsed
│ │ │ │ + -- 1.23s elapsed
│ │ │ │ + -- 1.1s elapsed
│ │ │ │  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}
│ │ │ │  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}
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ ├── ./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)
│ │ │ - -- .296874s elapsed
│ │ │ + -- .193921s 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)
│ │ │ │ - -- .296874s elapsed
│ │ │ │ + -- .193921s 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)
│ │ │ - -- .95871s elapsed
│ │ │ + -- .722488s 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
│ │ │ - -- 1.25533s elapsed
│ │ │ + -- 1.02052s 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)
│ │ │ │ - -- .95871s elapsed
│ │ │ │ + -- .722488s 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
│ │ │ │ - -- 1.25533s elapsed
│ │ │ │ + -- 1.02052s 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/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);
│ │ │ - -- 1.37136s elapsed
│ │ │ + -- .990732s 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 ();
│ │ │ - -- .477556s elapsed
│ │ │ + -- .344486s 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
│ │ │ - -- 21.9954s elapsed
│ │ │ + -- 13.4586s 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);
│ │ │ - -- .591281s elapsed
│ │ │ + -- .554717s 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);
│ │ │ - -- 2.17336s elapsed
│ │ │ + -- 1.80113s 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);
│ │ │ - -- 1.37136s elapsed
    
│ │ │ + -- .990732s 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 ();
│ │ │ - -- .477556s elapsed
    
│ │ │ + -- .344486s 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
│ │ │ - -- 21.9954s elapsed
    
│ │ │ + -- 13.4586s 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);
│ │ │ │ - -- 1.37136s elapsed
│ │ │ │ + -- .990732s 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 ();
│ │ │ │ - -- .477556s elapsed
│ │ │ │ + -- .344486s 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
│ │ │ │ - -- 21.9954s elapsed
│ │ │ │ + -- 13.4586s 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);
│ │ │ - -- .591281s elapsed
    │ │ │ + -- .554717s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.17336s elapsed
    │ │ │ + -- 1.80113s 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);
│ │ │ │ - -- .591281s elapsed
│ │ │ │ + -- .554717s 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);
│ │ │ │ - -- 2.17336s elapsed
│ │ │ │ + -- 1.80113s 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);
│ │ │ - -- 2.39895s elapsed
│ │ │ + -- 1.60077s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.07782s elapsed
│ │ │ + -- 3.84634s 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);
│ │ │ - -- 2.39895s elapsed
    │ │ │ + -- 1.60077s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.07782s elapsed
    │ │ │ + -- 3.84634s 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); │ │ │ │ - -- 2.39895s elapsed │ │ │ │ + -- 1.60077s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 4.07782s elapsed │ │ │ │ + -- 3.84634s 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/PolyominoIdeals/example-output/___Field.out │ │ │ @@ -7,16 +7,16 @@ │ │ │ o1 : GaloisField │ │ │ │ │ │ i2 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}}; │ │ │ │ │ │ i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex) │ │ │ │ │ │ o3 = ideal (- x x + x x , - x x + x x , - x x + │ │ │ - 3,1 2,3 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 │ │ │ + 2,1 1,2 2,2 1,1 3,1 1,2 3,2 1,1 3,2 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , - x x + x x , - x x + x x ) │ │ │ - 3,2 1,1 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 │ │ │ + 3,3 2,2 3,1 2,2 3,2 2,1 3,1 2,3 3,3 2,1 │ │ │ │ │ │ o3 : Ideal of F[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/_polyo__Ideal.out │ │ │ @@ -5,44 +5,44 @@ │ │ │ o1 = {{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{2, 2}, {3, 3}}} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ + 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x ) │ │ │ - 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ + 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i3 : Q={{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{3, 1}, {4, 2}}, {{3, 2}, {4, 3}}, {{3, 3}, {4, 4}}, {{2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 2}, {2, 3}}}; │ │ │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 │ │ │ + 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 │ │ │ + 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 2,2 1,3 2,4 1,1 │ │ │ + 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ - 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 │ │ │ + 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 │ │ │ + 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 │ │ │ + 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x ) │ │ │ - 3,1 1,2 │ │ │ + 4,2 3,3 │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ 4,4 4,3 4,2 4,1 3,4 3,3 3,2 3,1 2,4 2,3 2,2 2,1 1,4 1,3 1,2 1,1 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/___Field.html │ │ │ @@ -67,18 +67,18 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex)
│ │ │  
│ │ │  o3 = ideal (- x   x    + x   x   , - x   x    + x   x   , - x   x    +
│ │ │ -               3,1 2,3    3,3 2,1     2,1 1,2    2,2 1,1     3,1 1,2  
│ │ │ +               2,1 1,2    2,2 1,1     3,1 1,2    3,2 1,1     3,2 2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , - x   x    + x   x   , - x   x    + x   x   )
│ │ │ -      3,2 1,1     3,2 2,3    3,3 2,2     3,1 2,2    3,2 2,1
│ │ │ +      3,3 2,2     3,1 2,2    3,2 2,1     3,1 2,3    3,3 2,1
│ │ │  
│ │ │  o3 : Ideal of F[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                   3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -12,18 +12,18 @@ │ │ │ │ o1 = F │ │ │ │ │ │ │ │ o1 : GaloisField │ │ │ │ i2 : Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}}; │ │ │ │ i3 : I = polyoIdeal(Q,Field=> F,RingChoice=>1,TermOrder=> GRevLex) │ │ │ │ │ │ │ │ o3 = ideal (- x x + x x , - x x + x x , - x x + │ │ │ │ - 3,1 2,3 3,3 2,1 2,1 1,2 2,2 1,1 3,1 1,2 │ │ │ │ + 2,1 1,2 2,2 1,1 3,1 1,2 3,2 1,1 3,2 2,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , - x x + x x , - x x + x x ) │ │ │ │ - 3,2 1,1 3,2 2,3 3,3 2,2 3,1 2,2 3,2 2,1 │ │ │ │ + 3,3 2,2 3,1 2,2 3,2 2,1 3,1 2,3 3,3 2,1 │ │ │ │ │ │ │ │ o3 : Ideal of F[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _p_o_l_y_o_I_d_e_a_l -- Ideal of inner 2-minors of a collection of cells │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd FFiieelldd:: ********** │ │ │ │ * polyoIdeal(...,Field=>...) │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/_polyo__Ideal.html │ │ │ @@ -84,18 +84,18 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : I = polyoIdeal Q
│ │ │  
│ │ │  o2 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             3,3 2,1    3,1 2,3   3,2 2,1    3,1 2,2   2,2 1,1    2,1 1,2 
│ │ │ +             2,2 1,1    2,1 1,2   3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   )
│ │ │ -      3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2
│ │ │ +      3,3 2,1    3,1 2,3   3,2 2,1    3,1 2,2
│ │ │  
│ │ │  o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │

    │ │ │ @@ -105,33 +105,33 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : I = polyoIdeal Q
│ │ │  
│ │ │  o4 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             4,4 2,3    4,3 2,4   2,2 1,1    2,1 1,2   4,2 3,1    4,1 3,2 
│ │ │ +             3,4 1,3    3,3 1,4   3,2 2,1    3,1 2,2   4,2 1,1    4,1 1,2 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      2,3 1,1    2,1 1,3   4,3 3,1    4,1 3,3   4,4 1,3    4,3 1,4   3,4 2,3
│ │ │ +      2,4 1,2    2,2 1,4   4,4 3,2    4,2 3,4   2,4 1,3    2,3 1,4   4,4 3,3
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        3,3 2,4   4,2 2,1    4,1 2,2   2,3 1,2    2,2 1,3   2,4 1,1  
│ │ │ +        4,3 3,4   3,2 1,1    3,1 1,2   4,4 2,3    4,3 2,4   2,2 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -      2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2    4,2 3,3   3,4 1,3    3,3 1,4 
│ │ │ +      2,1 1,2   4,2 3,1    4,1 3,2   2,3 1,1    2,1 1,3   4,3 3,1    4,1 3,3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      3,2 2,1    3,1 2,2   4,2 1,1    4,1 1,2   2,4 1,2    2,2 1,4   4,4 3,2
│ │ │ +      4,4 1,3    4,3 1,4   3,4 2,3    3,3 2,4   4,2 2,1    4,1 2,2   2,3 1,2
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        4,2 3,4   2,4 1,3    2,3 1,4   4,4 3,3    4,3 3,4   3,2 1,1  
│ │ │ +        2,2 1,3   2,4 1,1    2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   )
│ │ │ -      3,1 1,2
│ │ │ +      4,2 3,3
│ │ │  
│ │ │  o4 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    4,4   4,3   4,2   4,1   3,4   3,3   3,2   3,1   2,4   2,3   2,2   2,1   1,4   1,3   1,2   1,1
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -31,47 +31,47 @@ │ │ │ │ │ │ │ │ o1 = {{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{2, 2}, {3, 3}}} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ + 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x ) │ │ │ │ - 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ + 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ │ │ │ │ │ │ i3 : Q={{{1, 1}, {2, 2}}, {{2, 1}, {3, 2}}, {{3, 1}, {4, 2}}, {{3, 2}, {4, 3}}, │ │ │ │ {{3, 3}, {4, 4}}, {{2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 2}, {2, 3}}}; │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 │ │ │ │ + 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 │ │ │ │ + 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 2,2 1,3 2,4 1,1 │ │ │ │ + 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ │ - 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 │ │ │ │ + 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 │ │ │ │ + 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 │ │ │ │ + 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ - 3,1 1,2 │ │ │ │ + 4,2 3,3 │ │ │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x │ │ │ │ , x , x , x , x , x ] │ │ │ │ 4,4 4,3 4,2 4,1 3,4 3,3 3,2 3,1 2,4 2,3 │ │ │ │ 2,2 2,1 1,4 1,3 1,2 1,1 │ │ │ │ ********** WWaayyss ttoo uussee ppoollyyooIIddeeaall:: ********** │ │ │ │ * polyoIdeal(List) │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ @@ -31,27 +31,27 @@ │ │ │ o5 = CacheTable{name => P} │ │ │ │ │ │ i6 : C == P │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : time isDistributive C │ │ │ - -- used 0.00304925s (cpu); 7.173e-06s (thread); 0s (gc) │ │ │ + -- used 0.000349606s (cpu); 4.955e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.43622s (cpu); 4.29279s (thread); 0s (gc) │ │ │ + -- used 6.13273s (cpu); 4.05863s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 0.00027196s (cpu); 3.797e-06s (thread); 0s (gc) │ │ │ + -- used 0.000237436s (cpu); 4.718e-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 0.394099s (cpu); 0.221406s (thread); 0s (gc) │ │ │ + -- used 0.364094s (cpu); 0.220462s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 0.000244068s (cpu); 1.2644e-05s (thread); 0s (gc) │ │ │ + -- used 0.000219417s (cpu); 1.1913e-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 0.00304925s (cpu); 7.173e-06s (thread); 0s (gc)
│ │ │ + -- used 0.000349606s (cpu); 4.955e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.43622s (cpu); 4.29279s (thread); 0s (gc)
│ │ │ + -- used 6.13273s (cpu); 4.05863s (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 0.00027196s (cpu); 3.797e-06s (thread); 0s (gc)
│ │ │ + -- used 0.000237436s (cpu); 4.718e-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 0.00304925s (cpu); 7.173e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.000349606s (cpu); 4.955e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.43622s (cpu); 4.29279s (thread); 0s (gc)
│ │ │ │ + -- used 6.13273s (cpu); 4.05863s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 0.00027196s (cpu); 3.797e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.000237436s (cpu); 4.718e-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 0.394099s (cpu); 0.221406s (thread); 0s (gc)
│ │ │ + -- used 0.364094s (cpu); 0.220462s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 0.000244068s (cpu); 1.2644e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000219417s (cpu); 1.1913e-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 0.394099s (cpu); 0.221406s (thread); 0s (gc)
│ │ │ │ + -- used 0.364094s (cpu); 0.220462s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 0.000244068s (cpu); 1.2644e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000219417s (cpu); 1.1913e-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/_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)
│ │ │ - -- .25306s elapsed
│ │ │ + -- .149366s 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)
│ │ │ - -- .0475166s elapsed
│ │ │ + -- .0233545s 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)
│ │ │ - -- .0689703s elapsed
│ │ │ + -- .0376028s 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
│ │ │ - -- .0995369s elapsed
│ │ │ + -- .0558109s 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)
│ │ │ - -- .00828047s elapsed
│ │ │ + -- .00900301s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -107,41 +107,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .25306s elapsed
│ │ │ + -- .149366s 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)
│ │ │ - -- .0475166s elapsed
│ │ │ + -- .0233545s 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)
│ │ │ - -- .0689703s elapsed
│ │ │ + -- .0376028s 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) │ │ │ │ - -- .25306s elapsed │ │ │ │ + -- .149366s 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) │ │ │ │ - -- .0475166s elapsed │ │ │ │ + -- .0233545s 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) │ │ │ │ - -- .0689703s elapsed │ │ │ │ + -- .0376028s 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
│ │ │ - -- .0995369s elapsed
│ │ │ + -- .0558109s 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)
│ │ │ - -- .00828047s elapsed
│ │ │ + -- .00900301s 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 │ │ │ │ - -- .0995369s elapsed │ │ │ │ + -- .0558109s 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) │ │ │ │ - -- .00828047s elapsed │ │ │ │ + -- .00900301s 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 │ │ │ @@ -4,12 +4,12 @@ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ -o2 = │ │ │ +o2 = │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ -o2 = │ │ │ +o2 = │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ │ │ i3 : next i │ │ │ │ │ │ o3 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ o12 = m2sqrt │ │ │ │ │ │ o12 : FunctionClosure │ │ │ │ │ │ i13 : pysqrt = toPython m2sqrt │ │ │ │ │ │ -o13 = │ │ │ +o13 = │ │ │ │ │ │ o13 : PythonObject of class builtin_function_or_method │ │ │ │ │ │ i14 : pysqrt 2 │ │ │ calling Macaulay2 code from Python! │ │ │ │ │ │ o14 = 1.4142135623730951 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : i = iterator x
│ │ │  
│ │ │ -o2 = <range_iterator object at 0x7f2f6969b540>
│ │ │ +o2 = <range_iterator object at 0x7f78a322b3c0>
│ │ │  
│ │ │  o2 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,15 +17,15 @@ │ │ │ │ i1 : x = pythonValue "range(3)" │ │ │ │ │ │ │ │ o1 = range(0, 3) │ │ │ │ │ │ │ │ o1 : PythonObject of class range │ │ │ │ i2 : i = iterator x │ │ │ │ │ │ │ │ -o2 = │ │ │ │ +o2 = │ │ │ │ │ │ │ │ o2 : PythonObject of class range_iterator │ │ │ │ ********** 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 │ │ │ @@ -77,15 +77,15 @@ │ │ │ o1 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : i = iterator x
│ │ │  
│ │ │ -o2 = <range_iterator object at 0x7fb1c3e6f510>
│ │ │ +o2 = <range_iterator object at 0x7fb2128833c0>
│ │ │  
│ │ │  o2 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : x = pythonValue "range(3)"
│ │ │ │  
│ │ │ │  o1 = range(0, 3)
│ │ │ │  
│ │ │ │  o1 : PythonObject of class range
│ │ │ │  i2 : i = iterator x
│ │ │ │  
│ │ │ │ -o2 = 
│ │ │ │ +o2 = 
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range_iterator
│ │ │ │  i3 : next i
│ │ │ │  
│ │ │ │  o3 = 0
│ │ │ │  
│ │ │ │  o3 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o12 : FunctionClosure
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7fed378e2700>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f2cec6ba700>
│ │ │  
│ │ │  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/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
│ │ │ - -- 3.00441s elapsed
│ │ │ + -- 2.34531s 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;
│ │ │ - -- 2.14119s elapsed
│ │ │ + -- 1.84016s 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 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │  
│ │ │                 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  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ +         2                          2   2              2                  
│ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*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  
│ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*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                2 
│ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2    3
│ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ +             2        2        2      3
│ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -358,81 +358,84 @@
│ │ │            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 + 5d, b - 33d, a - 21d)                                                                            |
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                      |
│ │ │ +      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )                                                     |
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                             2                      2                           2   2                     2 |
│ │ │ +      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                 2          
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ +                                                                             
│ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -         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                  
│ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        3     2                2         2        2      3
│ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ +         2                           2   2                     2
│ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ +o49 = b + 45c + 49d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 2, 3}
│ │ │  
│ │ │  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 = 2
│ │ │  
│ │ │  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 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │  
│ │ │                 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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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                 
│ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*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            
│ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*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
│ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +             2        2     3
│ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -440,80 +443,114 @@
│ │ │            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 - 45d, b + 16d, a + 38d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |                                      2              2 |
│ │ │ +      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ +      +-------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                         2   2                      2                                                                                                             |
│ │ │ +o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ +      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ +      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ +      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ +      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ +      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ +      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ +      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ +      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ +      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  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 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │        -----------------------------------------------------------------------
│ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │  
│ │ │                 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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │  
│ │ │                 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               
│ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                            2   2              2                 
│ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ +         2                             2   2              2                  
│ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 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  
│ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  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        2     3   3                2   
│ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │        -----------------------------------------------------------------------
│ │ │             2        2        2      3
│ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │  
│ │ │  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               
│ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +         2                             2   2              2                  
│ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*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         
│ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*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
│ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +             2        2     3
│ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -531,36 +568,38 @@
│ │ │            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 - 40d, b - 10d, a + 32d)                                                                      |
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                 |
│ │ │ +      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )                                                |
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2                    2                          2   2                    2 |
│ │ │ +      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b  - b*d + 15c*d + 40d )|
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  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)|
│ │ │ -      +---------------------------------+
│ │ │ +      +------------------------------------------------------+
│ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |                                     2              2 |
│ │ │ +      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ +      +------------------------------------------------------+
│ │ │  
│ │ │  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
│ │ │ - -- 3.00441s elapsed
│ │ │ + -- 2.34531s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The Schreyer resolution and minimal Betti numbers
    
│ │ │ @@ -556,15 +556,15 @@
│ │ │  
│ │ │  o39 : Ideal of T
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.14119s elapsed
    │ │ │ + -- 1.84016s 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 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │  
│ │ │                 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  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ +         2                          2   2              2                  
│ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*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  
│ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*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                2 
│ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2    3
│ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ +             2        2        2      3
│ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │  
│ │ │  o45 : Ideal of S
    │ │ │
    │ │ │ 
    i46 : betti res Fa
│ │ │ @@ -638,104 +638,107 @@
│ │ │  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 + 5d, b - 33d, a - 21d) | │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 45c + 49d, a - 22c - 26d, c + 49c*d + 42d ) | │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a + 26b + 37c + 36d, c - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b - 3b*d - 24c*d - 36d )| │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                 2          
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ +                                                                             
│ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -         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                  
│ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        3     2                2         2        2      3
│ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ +         2                           2   2                     2
│ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │  
│ │ │  o48 : List
    │ │ │
    │ │ │ 
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ +o49 = b + 45c + 49d
│ │ │  
│ │ │  o49 : S
    │ │ │
    │ │ │ 
    i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 2, 3}
│ │ │  
│ │ │  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 = 2 │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    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 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │  
│ │ │                 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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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                 
│ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*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            
│ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*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
│ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +             2        2     3
│ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │  
│ │ │  o54 : Ideal of S
    │ │ │
    │ │ │ 
    i55 : betti res Fb
│ │ │ @@ -749,102 +752,136 @@
│ │ │  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 - 45d, b + 16d, a + 38d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 43d, b + 10d, a + 8d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 34d, b + 15d, a + 28d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 11d, b + 39d, a + 23d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b - 32c + 42d, a - 19c - 16d, c - 28c*d - 40d )| │ │ │ + +-------------------------------------------------------+ │ │ │
    │ │ │ 
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
    │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ +o57 = |ideal (a - 7b + 32c + d, c + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b + 28b*d - 32c*d + 16d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 7b + 32c + d, c + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b - 20b*d + 29c*d + 38d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 7b + 32c + d, c - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b - 8b*d - 12c*d - 46d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 3 2 2 3 | │ │ │ + |ideal (b + 23c - 11d, a - 9c + 25d, c - 13c d - 14c*d + 23d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a + 48b - 40c - 20d, b*c - 32c + 43b*d - 21c*d - 12d , b - 14c + 14b*d + 18c*d + 36d , c + 28c d - 20b*d + 42c*d - 50d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a + b + 50c + 26d, b*c - 32c + 34b*d - 36c*d + 14d , b - 14c + 34b*d - 16c*d - 33d , c + 28c d + 39b*d - 28c*d + 4d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a + 28a*d - 13b*d - 25c*d - 35d )| │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a - 46b + 39c - 29d, b*c - 32c + 11b*d - 7c*d - 43d , b - 14c + 29b*d + 43c*d - 41d , c + 28c d + 46b*d - 50c*d - 5d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a - 8a*d - 24b*d + c*d - 22d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a - 44a*d + 16b*d + 11c*d + 12d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    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 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │        -----------------------------------------------------------------------
│ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │  
│ │ │                 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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │  
│ │ │                 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               
│ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                            2   2              2                 
│ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ +         2                             2   2              2                  
│ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 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  
│ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  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        2     3   3                2   
│ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │        -----------------------------------------------------------------------
│ │ │             2        2        2      3
│ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │  
│ │ │  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               
│ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +         2                             2   2              2                  
│ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*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         
│ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*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
│ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +             2        2     3
│ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │  
│ │ │  o61 : Ideal of S
    │ │ │
    │ │ │ 
    i62 : betti res I0
│ │ │ @@ -873,39 +910,41 @@
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    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 - 40d, b - 10d, a + 32d) | │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 10c + 25d, a + 27c - 50d, c - 34c*d - 17d ) | │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 13b - 2c + 29d, c - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b - b*d + 15c*d + 40d )| │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    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)|
│ │ │ -      +---------------------------------+
    │ │ │ + +------------------------------------------------------+ │ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 29d, b - 8d, a + 50d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 16d, b + 39d, a - 32d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 5d, b - 14d, a + 7d) | │ │ │ + +------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b - 40c + 5d, a - 47c + 24d, c - 27c*d + 15d )| │ │ │ + +------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    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
│ │ │ │ - -- 3.00441s elapsed
│ │ │ │ + -- 2.34531s 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;
│ │ │ │ - -- 2.14119s elapsed
│ │ │ │ + -- 1.84016s 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 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │ │  
│ │ │ │                 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  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2   2              2
│ │ │ │ -      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ │ +         2                          2   2              2
│ │ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*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
│ │ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*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                2
│ │ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2        2        2    3
│ │ │ │ -      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ │ +             2        2        2      3
│ │ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │ │  
│ │ │ │  o45 : Ideal of S
│ │ │ │  i46 : betti res Fa
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o46 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -468,172 +468,256 @@
│ │ │ │            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 + 5d, b - 33d, a - 21d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ -      |                             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 )|
│ │ │ │ +------------------------------------+
│ │ │ │ +      |                                      2              2
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )
│ │ │ │ +|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ +------------------------------------+
│ │ │ │ +      |                             2                      2
│ │ │ │ +2   2                     2 |
│ │ │ │ +      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d +
│ │ │ │ +16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------------------+
│ │ │ │  i48 : CFa = minimalPrimes Fa
│ │ │ │  
│ │ │ │ -                                                                 2
│ │ │ │ -o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ │ +
│ │ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         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
│ │ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        3     2                2         2        2      3
│ │ │ │ -      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ │ +         2                           2   2                     2
│ │ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a + 49b - 10c + 39d
│ │ │ │ +o49 = b + 45c + 49d
│ │ │ │  
│ │ │ │  o49 : S
│ │ │ │  i50 : CFa/degree
│ │ │ │  
│ │ │ │ -o50 = {1, 5}
│ │ │ │ +o50 = {1, 2, 3}
│ │ │ │  
│ │ │ │  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 = 2
│ │ │ │  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 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ │  
│ │ │ │                 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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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
│ │ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*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
│ │ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*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
│ │ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2       2      3
│ │ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ │ +             2        2     3
│ │ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │ │  
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : betti res Fb
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o55 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o55 : BettiTally
│ │ │ │  i56 : netList decompose Fb --
│ │ │ │  
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |                                      2              2 |
│ │ │ │ +      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -      |                                      2              2
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                          2                      2
│ │ │ │ +2   2                      2
│ │ │ │  |
│ │ │ │ -o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )
│ │ │ │ +o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d +
│ │ │ │ +18d , b  + 28b*d - 32c*d + 16d )
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -      |        2                             2
│ │ │ │ -2                                   2   2                            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
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d
│ │ │ │ ++ 39d , b  - 20b*d + 29c*d + 38d )
│ │ │ │ +|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ -
│ │ │ │ -o57 = ++
│ │ │ │ -      ++
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                          2                      2
│ │ │ │ +2   2                     2
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d
│ │ │ │ +- 32d , b  - 8b*d - 12c*d - 46d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     3      2         2      3
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     2                      2   2      2
│ │ │ │ +2   3      2         2        2      3
│ │ │ │ +|
│ │ │ │ +      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c
│ │ │ │ ++ 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                   2                      2   2      2
│ │ │ │ +2   3      2         2        2     3
│ │ │ │ +|
│ │ │ │ +      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  +
│ │ │ │ +34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                            2                                 2
│ │ │ │ +2   2                              2                                  2   2
│ │ │ │ +2 |
│ │ │ │ +      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d
│ │ │ │ +, a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b +
│ │ │ │ +3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     2                     2   2      2
│ │ │ │ +2   3      2         2        2     3
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c
│ │ │ │ ++ 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                              2
│ │ │ │ +2                                   2   2                              2
│ │ │ │ +2   2                           2  |
│ │ │ │ +      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d
│ │ │ │ +, a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b -
│ │ │ │ +6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                              2
│ │ │ │ +2                                  2   2                      2
│ │ │ │ +2   2                              2      |
│ │ │ │ +      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d
│ │ │ │ +, a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d -
│ │ │ │ +30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │  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 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │ │  
│ │ │ │                 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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ │  
│ │ │ │                 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
│ │ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                            2   2              2
│ │ │ │ -      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ │ +         2                             2   2              2
│ │ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 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
│ │ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                  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        2     3   3                2
│ │ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │             2        2        2      3
│ │ │ │ -      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │ │  
│ │ │ │  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
│ │ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2   2              2
│ │ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ │ +         2                             2   2              2
│ │ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*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
│ │ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*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
│ │ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2      3
│ │ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ │ +             2        2     3
│ │ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │ │  
│ │ │ │  o61 : Ideal of S
│ │ │ │  i62 : betti res I0
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o62 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -649,42 +733,45 @@
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -o64 = |ideal (c - 42d, b + 26d, a - 30d)
│ │ │ │ +-------------------------------+
│ │ │ │ +o64 = |ideal (c - 40d, b - 10d, a + 32d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -      |ideal (c - 47d, b + 7d, a - 44d)
│ │ │ │ +-------------------------------+
│ │ │ │ +      |                                      2              2
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ -      |                                     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 )|
│ │ │ │ +-------------------------------+
│ │ │ │ +      |                            2                    2
│ │ │ │ +2   2                    2 |
│ │ │ │ +      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d
│ │ │ │ ++ 8d , b  - b*d + 15c*d + 40d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -------------------------+
│ │ │ │ +-------------------------------+
│ │ │ │  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)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |                                     2              2 |
│ │ │ │ +      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │  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 8.21418s (cpu); 5.64783s (thread); 0s (gc)
│ │ │ + -- used 7.45129s (cpu); 5.80722s (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 8.21418s (cpu); 5.64783s (thread); 0s (gc)
│ │ │ + -- used 7.45129s (cpu); 5.80722s (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 8.21418s (cpu); 5.64783s (thread); 0s (gc) │ │ │ │ + -- used 7.45129s (cpu); 5.80722s (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}, 0, .00399974) │ │ │ +o3 = ({5, 2.91596e52, 9}, .00401621, 0) │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00799972, .0359965) │ │ │ +o4 = ({50, 2.30853e454, 98}, .00799782, .0408928) │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ -o5 = {{.00488092, .0120099}, {.00762201, .00429259}, {.00399974, .008}, │ │ │ +o5 = {{.0080279, .0159703}, {.00398161, .00398873}, {.0079648, .0079887}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00400446, .00799509}, {.00800175, .0119993}, {.00495679, .0130886}, │ │ │ + {.0040012, .0080541}, {.00800274, .0159951}, {.00798886, .0120376}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00399784, .00800007}, {.00400827, .00798114}, {.00400158, .00799842}, │ │ │ + {.00797394, .00802261}, {.00402017, .00798805}, {.00397113, .00796722}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00396918, .00799992}} │ │ │ + {.00804335, .0119768}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ -o6 = .004944253799999965 │ │ │ +o6 = .006397568999999948 │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ -o7 = .008936502300000005 │ │ │ +o7 = .00999892030000007 │ │ │ │ │ │ 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}, 0, .00399974)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00401621, 0)
│ │ │  
│ │ │  o3 : Sequence
    │ │ │
    │ │ │ 
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00799972, .0359965)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00799782, .0408928)
│ │ │  
│ │ │  o4 : Sequence
    │ │ │
    │ │ │ 
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00488092, .0120099}, {.00762201, .00429259}, {.00399974, .008},
│ │ │ +o5 = {{.0080279, .0159703}, {.00398161, .00398873}, {.0079648, .0079887},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00400446, .00799509}, {.00800175, .0119993}, {.00495679, .0130886},
│ │ │ +     {.0040012, .0080541}, {.00800274, .0159951}, {.00798886, .0120376},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00399784, .00800007}, {.00400827, .00798114}, {.00400158, .00799842},
│ │ │ +     {.00797394, .00802261}, {.00402017, .00798805}, {.00397113, .00796722},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00396918, .00799992}}
│ │ │ +     {.00804335, .0119768}}
│ │ │  
│ │ │  o5 : List
    │ │ │
    │ │ │ 
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .004944253799999965
│ │ │ +o6 = .006397568999999948
│ │ │  
│ │ │  o6 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008936502300000005
│ │ │ +o7 = .00999892030000007
│ │ │  
│ │ │  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}, 0, .00399974) │ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00401621, 0) │ │ │ │ │ │ │ │ o3 : Sequence │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00799972, .0359965) │ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00799782, .0408928) │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ │ │ -o5 = {{.00488092, .0120099}, {.00762201, .00429259}, {.00399974, .008}, │ │ │ │ +o5 = {{.0080279, .0159703}, {.00398161, .00398873}, {.0079648, .0079887}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00400446, .00799509}, {.00800175, .0119993}, {.00495679, .0130886}, │ │ │ │ + {.0040012, .0080541}, {.00800274, .0159951}, {.00798886, .0120376}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00399784, .00800007}, {.00400827, .00798114}, {.00400158, .00799842}, │ │ │ │ + {.00797394, .00802261}, {.00402017, .00798805}, {.00397113, .00796722}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00396918, .00799992}} │ │ │ │ + {.00804335, .0119768}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ │ │ -o6 = .004944253799999965 │ │ │ │ +o6 = .006397568999999948 │ │ │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ │ │ -o7 = .008936502300000005 │ │ │ │ +o7 = .00999892030000007 │ │ │ │ │ │ │ │ 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 1.48302s (cpu); 1.14005s (thread); 0s (gc) │ │ │ + -- used 1.32969s (cpu); 1.1419s (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 1.48302s (cpu); 1.14005s (thread); 0s (gc) │ │ │ │ + -- used 1.32969s (cpu); 1.1419s (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 1.7509s (cpu); 1.40175s (thread); 0s (gc) │ │ │ + -- used 1.65816s (cpu); 1.39705s (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 1.48302s (cpu); 1.14005s (thread); 0s (gc)
│ │ │ + -- used 1.32969s (cpu); 1.1419s (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 1.7509s (cpu); 1.40175s (thread); 0s (gc)
│ │ │ + -- used 1.65816s (cpu); 1.39705s (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 1.7509s (cpu); 1.40175s (thread); 0s (gc)
│ │ │ │ + -- used 1.65816s (cpu); 1.39705s (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
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1756126631
│ │ │ + -- setting random seed to 1759554190
│ │ │  
│ │ │ -o1 = 1756126631
│ │ │ +o1 = 1759554190
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 1.80571s (cpu); 1.42113s (thread); 0s (gc)
│ │ │ + -- used 1.99308s (cpu); 1.60442s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 42 }
│ │ │ -           5 => 194
│ │ │ -           6 => 180
│ │ │ -           7 => 69
│ │ │ -           8 => 13
│ │ │ -           9 => 2
│ │ │ +o4 = Tally{4 => 50 }
│ │ │ +           5 => 199
│ │ │ +           6 => 178
│ │ │ +           7 => 60
│ │ │ +           8 => 12
│ │ │ +           9 => 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 1756126677
│ │ │ + -- setting random seed to 1759554225
│ │ │  
│ │ │ -o1 = 1756126677
│ │ │ +o1 = 1759554225
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -        2
│ │ │ -o4 = a*b
│ │ │ +      3
│ │ │ +o4 = a
│ │ │  
│ │ │  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 1756126740
│ │ │ + -- setting random seed to 1759554273
│ │ │  
│ │ │ -o1 = 1756126740
│ │ │ +o1 = 1759554273
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -22,18 +22,18 @@
│ │ │  o4 = {3, 5, 7}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*e)
│ │ │ +o5 = ideal(b*d*e)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (b*c, a*c, c*e, a*d, a*b)
│ │ │ +o6 = ideal (b*d, a*c, a*e, a*b, b*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 1756126722
│ │ │ + -- setting random seed to 1759554260
│ │ │  
│ │ │ -o1 = 1756126722
│ │ │ +o1 = 1759554260
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,17 +15,17 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d,
│ │ │ +o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     b*c, a*c}}
│ │ │ +     b*d, b*c, a}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -39,15 +39,15 @@
│ │ │  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 4.42401s (cpu); 3.05863s (thread); 0s (gc)
│ │ │ + -- used 4.31444s (cpu); 3.07733s (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 1756126677
│ │ │ + -- setting random seed to 1759554225
│ │ │  
│ │ │ -o1 = 1756126677
    
│ │ │ +o1 = 1759554225
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -98,16 +98,16 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -        2
│ │ │ -o4 = a*b
│ │ │ +      3
│ │ │ +o4 = a
│ │ │  
│ │ │  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 1756126677
│ │ │ │ + -- setting random seed to 1759554225
│ │ │ │  
│ │ │ │ -o1 = 1756126677
│ │ │ │ +o1 = 1759554225
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -        2
│ │ │ │ -o4 = a*b
│ │ │ │ +      3
│ │ │ │ +o4 = a
│ │ │ │  
│ │ │ │  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 1756126740
│ │ │ + -- setting random seed to 1759554273
│ │ │  
│ │ │ -o1 = 1756126740
    
│ │ │ +o1 = 1759554273
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -108,24 +108,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*e)
│ │ │ +o5 = ideal(b*d*e)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (b*c, a*c, c*e, a*d, a*b)
│ │ │ +o6 = ideal (b*d, a*c, a*e, a*b, b*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 1756126740
│ │ │ │ + -- setting random seed to 1759554273
│ │ │ │  
│ │ │ │ -o1 = 1756126740
│ │ │ │ +o1 = 1759554273
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = {3, 5, 7}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │ -o5 = ideal(a*c*e)
│ │ │ │ +o5 = ideal(b*d*e)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (b*c, a*c, c*e, a*d, a*b)
│ │ │ │ +o6 = ideal (b*d, a*c, a*e, a*b, b*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 1756126722
│ │ │ + -- setting random seed to 1759554260
│ │ │  
│ │ │ -o1 = 1756126722
    
│ │ │ +o1 = 1759554260
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -106,17 +106,17 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d,
│ │ │ +o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     b*c, a*c}}
│ │ │ +     b*d, b*c, a}}
│ │ │  
│ │ │  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 +147,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    │ │ │
    │ │ │ 
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.42401s (cpu); 3.05863s (thread); 0s (gc)
    │ │ │ + -- used 4.31444s (cpu); 3.07733s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,32 +35,32 @@ │ │ │ │ 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 1756126722 │ │ │ │ + -- setting random seed to 1759554260 │ │ │ │ │ │ │ │ -o1 = 1756126722 │ │ │ │ +o1 = 1759554260 │ │ │ │ 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*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d, │ │ │ │ +o4 = {monomialIdeal (a*b, a*c, a*d, b*c*d), {a*b, a*c, a*d, b*c*d}, {c*d, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - b*c, a*c}} │ │ │ │ + b*d, b*c, a}} │ │ │ │ │ │ │ │ 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 +74,15 @@ │ │ │ │ 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 4.42401s (cpu); 3.05863s (thread); 0s (gc) │ │ │ │ + -- used 4.31444s (cpu); 3.07733s (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.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -72,22 +72,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1756126631
│ │ │ + -- setting random seed to 1759554190
│ │ │  
│ │ │ -o1 = 1756126631
    │ │ │ +o1 = 1759554190 │ │ │
    │ │ │ 
    i2 : kk=ZZ/101;
    │ │ │
    │ │ │ 
    i3 : S=kk[vars(0..5)];
    │ │ │
    │ │ │ 
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 1.80571s (cpu); 1.42113s (thread); 0s (gc)
│ │ │ + -- used 1.99308s (cpu); 1.60442s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 42 }
│ │ │ -           5 => 194
│ │ │ -           6 => 180
│ │ │ -           7 => 69
│ │ │ -           8 => 13
│ │ │ -           9 => 2
│ │ │ +o4 = Tally{4 => 50 }
│ │ │ +           5 => 199
│ │ │ +           6 => 178
│ │ │ +           7 => 60
│ │ │ +           8 => 12
│ │ │ +           9 => 1
│ │ │  
│ │ │  o4 : Tally
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -9,28 +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 1756126631 │ │ │ │ + -- setting random seed to 1759554190 │ │ │ │ │ │ │ │ -o1 = 1756126631 │ │ │ │ +o1 = 1759554190 │ │ │ │ i2 : kk=ZZ/101; │ │ │ │ i3 : S=kk[vars(0..5)]; │ │ │ │ i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S) │ │ │ │ - -- used 1.80571s (cpu); 1.42113s (thread); 0s (gc) │ │ │ │ + -- used 1.99308s (cpu); 1.60442s (thread); 0s (gc) │ │ │ │ │ │ │ │ -o4 = Tally{4 => 42 } │ │ │ │ - 5 => 194 │ │ │ │ - 6 => 180 │ │ │ │ - 7 => 69 │ │ │ │ - 8 => 13 │ │ │ │ - 9 => 2 │ │ │ │ +o4 = Tally{4 => 50 } │ │ │ │ + 5 => 199 │ │ │ │ + 6 => 178 │ │ │ │ + 7 => 60 │ │ │ │ + 8 => 12 │ │ │ │ + 9 => 1 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ How does this compare with the case of binomial ideals? or pure binomial │ │ │ │ ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" │ │ │ │ above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger │ │ │ │ numbers of examples. Click the link "Finding Extreme Examples" below to see │ │ │ │ some other, more elaborate ways to search. │ │ ├── ./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) │ │ │ - -- 1.42533s elapsed │ │ │ + -- 1.2402s elapsed │ │ │ │ │ │ o4 = 4 │ │ │ │ │ │ i5 : elapsedTime dim I │ │ │ - -- 3.34931s elapsed │ │ │ + -- 3.03409s 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)) ); │ │ │ - -- 1.84763s elapsed │ │ │ + -- 1.55318s 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) │ │ │ - -- 3.11071s elapsed │ │ │ + -- 2.70333s 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) │ │ │ - -- 2.54466s elapsed │ │ │ + -- 1.96227s 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)
│ │ │ - -- 1.42533s elapsed
│ │ │ + -- 1.2402s elapsed
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : elapsedTime dim I
│ │ │ - -- 3.34931s elapsed
│ │ │ + -- 3.03409s 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) │ │ │ │ - -- 1.42533s elapsed │ │ │ │ + -- 1.2402s elapsed │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : elapsedTime dim I │ │ │ │ - -- 3.34931s elapsed │ │ │ │ + -- 3.03409s 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)) );
│ │ │ - -- 1.84763s elapsed
    │ │ │ + -- 1.55318s 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)) ); │ │ │ │ - -- 1.84763s elapsed │ │ │ │ + -- 1.55318s 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)
│ │ │ - -- 3.11071s elapsed
│ │ │ + -- 2.70333s 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)
│ │ │ - -- 2.54466s elapsed
│ │ │ + -- 1.96227s 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) │ │ │ │ - -- 3.11071s elapsed │ │ │ │ + -- 2.70333s 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) │ │ │ │ - -- 2.54466s elapsed │ │ │ │ + -- 1.96227s 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 0.567139s (cpu); 0.393907s (thread); 0s (gc) │ │ │ + -- used 0.893144s (cpu); 0.496163s (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 0.567139s (cpu); 0.393907s (thread); 0s (gc)
│ │ │ + -- used 0.893144s (cpu); 0.496163s (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 0.567139s (cpu); 0.393907s (thread); 0s (gc) │ │ │ │ + -- used 0.893144s (cpu); 0.496163s (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 0.0798272s (cpu); 0.011392s (thread); 0s (gc) │ │ │ + -- used 0.0643697s (cpu); 0.0172656s (thread); 0s (gc) │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ │ │ o16 : Ideal of QQ[x..z] │ │ │ │ │ │ @@ -141,24 +141,24 @@ │ │ │ o30 : Ideal of R │ │ │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ - -- used 0.895568s (cpu); 0.75913s (thread); 0s (gc) │ │ │ + -- used 0.996855s (cpu); 0.853831s (thread); 0s (gc) │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ │ │ i33 : #oo │ │ │ │ │ │ o33 = 31 │ │ │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ - -- used 0.286803s (cpu); 0.21927s (thread); 0s (gc) │ │ │ + -- used 0.308956s (cpu); 0.233852s (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 0.0798272s (cpu); 0.011392s (thread); 0s (gc)
│ │ │ + -- used 0.0643697s (cpu); 0.0172656s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Over number fields

    │ │ │ @@ -347,15 +347,15 @@ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │ - -- used 0.895568s (cpu); 0.75913s (thread); 0s (gc)
│ │ │ + -- used 0.996855s (cpu); 0.853831s (thread); 0s (gc)
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : #oo
│ │ │  
│ │ │ @@ -373,15 +373,15 @@
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.286803s (cpu); 0.21927s (thread); 0s (gc)
│ │ │ + -- used 0.308956s (cpu); 0.233852s (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 0.0798272s (cpu); 0.011392s (thread); 0s (gc) │ │ │ │ + -- used 0.0643697s (cpu); 0.0172656s (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); │ │ │ │ @@ -196,25 +196,25 @@ │ │ │ │ (2z-qw)(4(x2+y2-z2)+(1+3(5-q2))w2)2"; │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ │ - -- used 0.895568s (cpu); 0.75913s (thread); 0s (gc) │ │ │ │ + -- used 0.996855s (cpu); 0.853831s (thread); 0s (gc) │ │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ i33 : #oo │ │ │ │ │ │ │ │ o33 = 31 │ │ │ │ Still it runs a lot faster when reduced to a positive characteristic. │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ │ - -- used 0.286803s (cpu); 0.21927s (thread); 0s (gc) │ │ │ │ + -- used 0.308956s (cpu); 0.233852s (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 0.877055s (cpu); 0.719897s (thread); 0s (gc) │ │ │ + -- used 0.908387s (cpu); 0.769736s (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 1.09738s (cpu); 0.751584s (thread); 0s (gc) │ │ │ + -- used 1.02199s (cpu); 0.811793s (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 1.5168s (cpu); 1.16318s (thread); 0s (gc) │ │ │ + -- used 1.56589s (cpu); 1.32675s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 1.66687s (cpu); 1.26007s (thread); 0s (gc) │ │ │ + -- used 1.60268s (cpu); 1.3497s (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 0.0282264s (cpu); 0.0279747s (thread); 0s (gc) │ │ │ + -- used 0.0837518s (cpu); 0.0324166s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.121578s (cpu); 0.123297s (thread); 0s (gc) │ │ │ + -- used 0.176489s (cpu); 0.166293s (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 0.0973976s (cpu); 0.0362689s (thread); 0s (gc) │ │ │ + -- used 0.257193s (cpu); 0.0525402s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00398681s (cpu); 0.00748113s (thread); 0s (gc) │ │ │ + -- used 0.0777128s (cpu); 0.0182729s (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 0.877055s (cpu); 0.719897s (thread); 0s (gc) │ │ │ │ + -- used 0.908387s (cpu); 0.769736s (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 0.877055s (cpu); 0.719897s (thread); 0s (gc)
│ │ │ + -- used 0.908387s (cpu); 0.769736s (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 1.09738s (cpu); 0.751584s (thread); 0s (gc)
│ │ │ + -- used 1.02199s (cpu); 0.811793s (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 1.5168s (cpu); 1.16318s (thread); 0s (gc)
│ │ │ + -- used 1.56589s (cpu); 1.32675s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.66687s (cpu); 1.26007s (thread); 0s (gc)
│ │ │ + -- used 1.60268s (cpu); 1.3497s (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 1.09738s (cpu); 0.751584s (thread); 0s (gc) │ │ │ │ + -- used 1.02199s (cpu); 0.811793s (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 1.5168s (cpu); 1.16318s (thread); 0s (gc) │ │ │ │ + -- used 1.56589s (cpu); 1.32675s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 1.66687s (cpu); 1.26007s (thread); 0s (gc) │ │ │ │ + -- used 1.60268s (cpu); 1.3497s (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 0.0282264s (cpu); 0.0279747s (thread); 0s (gc)
│ │ │ + -- used 0.0837518s (cpu); 0.0324166s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.121578s (cpu); 0.123297s (thread); 0s (gc)
│ │ │ + -- used 0.176489s (cpu); 0.166293s (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 0.0973976s (cpu); 0.0362689s (thread); 0s (gc)
│ │ │ + -- used 0.257193s (cpu); 0.0525402s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00398681s (cpu); 0.00748113s (thread); 0s (gc)
│ │ │ + -- used 0.0777128s (cpu); 0.0182729s (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 0.0282264s (cpu); 0.0279747s (thread); 0s (gc) │ │ │ │ + -- used 0.0837518s (cpu); 0.0324166s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.121578s (cpu); 0.123297s (thread); 0s (gc) │ │ │ │ + -- used 0.176489s (cpu); 0.166293s (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 0.0973976s (cpu); 0.0362689s (thread); 0s (gc) │ │ │ │ + -- used 0.257193s (cpu); 0.0525402s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.00398681s (cpu); 0.00748113s (thread); 0s (gc) │ │ │ │ + -- used 0.0777128s (cpu); 0.0182729s (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 = .2545877649999999 │ │ │ +o8 = .244346293 │ │ │ │ │ │ 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 = .07076026763888892 │ │ │ +o11 = .0792295146935484 │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ │ │ i12 : time regularity I2 │ │ │ - -- used 0.000813275s (cpu); 0.00214398s (thread); 0s (gc) │ │ │ + -- used 0.00390207s (cpu); 0.00253289s (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 = .2545877649999999
│ │ │ +o8 = .244346293
│ │ │  
│ │ │  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 = .07076026763888892
│ │ │ +o11 = .0792295146935484
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.000813275s (cpu); 0.00214398s (thread); 0s (gc)
│ │ │ + -- used 0.00390207s (cpu); 0.00253289s (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 = .2545877649999999 │ │ │ │ +o8 = .244346293 │ │ │ │ │ │ │ │ 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 = .07076026763888892 │ │ │ │ +o11 = .0792295146935484 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.000813275s (cpu); 0.00214398s (thread); 0s (gc) │ │ │ │ + -- used 0.00390207s (cpu); 0.00253289s (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 0.0439944s (cpu); 0.0463877s (thread); 0s (gc) │ │ │ + -- used 0.0599842s (cpu); 0.0574975s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.127607s (cpu); 0.0769988s (thread); 0s (gc) │ │ │ + -- used 0.122599s (cpu); 0.0758033s (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 0.156403s (cpu); 0.100372s (thread); 0s (gc) │ │ │ + -- used 0.156654s (cpu); 0.104704s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.157983s (cpu); 0.0925469s (thread); 0s (gc) │ │ │ + -- used 0.160015s (cpu); 0.106295s (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 0.0399817s (cpu); 0.0401994s (thread); 0s (gc) │ │ │ + -- used 0.0478629s (cpu); 0.0498197s (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 0.109535s (cpu); 0.0517928s (thread); 0s (gc) │ │ │ + -- used 0.110749s (cpu); 0.0546887s (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 0.145885s (cpu); 0.093329s (thread); 0s (gc) │ │ │ + -- used 0.154881s (cpu); 0.100139s (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 0.136507s (cpu); 0.0861143s (thread); 0s (gc) │ │ │ + -- used 0.139949s (cpu); 0.0836923s (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 5.00704s (cpu); 4.62662s (thread); 0s (gc) │ │ │ + -- used 5.04364s (cpu); 4.80582s (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 1.27577s (cpu); 1.09674s (thread); 0s (gc) │ │ │ + -- used 1.09673s (cpu); 1.0502s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.225281s (cpu); 0.16014s (thread); 0s (gc) │ │ │ + -- used 0.232004s (cpu); 0.184684s (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 0.31187s (cpu); 0.194538s (thread); 0s (gc) │ │ │ + -- used 0.229954s (cpu); 0.181059s (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 0.00800121s (cpu); 0.00865969s (thread); 0s (gc) │ │ │ + -- used 0.0120158s (cpu); 0.010025s (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 0.00741673s (cpu); 0.00950854s (thread); 0s (gc) │ │ │ + -- used 0.0108181s (cpu); 0.0110375s (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 0.422407s (cpu); 0.420635s (thread); 0s (gc) │ │ │ + -- used 0.415819s (cpu); 0.415159s (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 0.108129s (cpu); 0.106181s (thread); 0s (gc) │ │ │ + -- used 0.136265s (cpu); 0.135585s (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 0.219998s (cpu); 0.157691s (thread); 0s (gc) │ │ │ + -- used 0.2179s (cpu); 0.162679s (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 0.061731s (cpu); 0.0602097s (thread); 0s (gc) │ │ │ + -- used 0.0738155s (cpu); 0.0729647s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.627729s (cpu); 0.578975s (thread); 0s (gc) │ │ │ + -- used 0.740854s (cpu); 0.689631s (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 0.0437722s (cpu); 0.0407153s (thread); 0s (gc) │ │ │ + -- used 0.0518947s (cpu); 0.0499503s (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 0.119026s (cpu); 0.0608633s (thread); 0s (gc) │ │ │ + -- used 0.110451s (cpu); 0.0596955s (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 0.0200484s (cpu); 0.0193455s (thread); 0s (gc) │ │ │ + -- used 0.0199995s (cpu); 0.0210634s (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 0.0158647s (cpu); 0.0172923s (thread); 0s (gc) │ │ │ + -- used 0.018488s (cpu); 0.0200899s (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 0.0360004s (cpu); 0.0375479s (thread); 0s (gc) │ │ │ + -- used 0.0472215s (cpu); 0.0447933s (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 0.00798565s (cpu); 0.00802576s (thread); 0s (gc) │ │ │ + -- used 0.0119736s (cpu); 0.00969539s (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 0.00457769s (cpu); 0.00651439s (thread); 0s (gc) │ │ │ + -- used 0.00759238s (cpu); 0.008202s (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 0.0183706s (cpu); 0.0201448s (thread); 0s (gc) │ │ │ + -- used 0.0205969s (cpu); 0.0213117s (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 0.0040224s (cpu); 0.00355676s (thread); 0s (gc) │ │ │ + -- used 0.00400873s (cpu); 0.00400396s (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 0.00399843s (cpu); 0.0024578s (thread); 0s (gc) │ │ │ + -- used 0.00397216s (cpu); 0.00421769s (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 0.00564656s (cpu); 0.00441136s (thread); 0s (gc) │ │ │ + -- used 0.00397811s (cpu); 0.00478977s (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 0.0246074s (cpu); 0.0276395s (thread); 0s (gc) │ │ │ + -- used 0.0280243s (cpu); 0.0311589s (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 0.102777s (cpu); 0.0396871s (thread); 0s (gc) │ │ │ + -- used 0.0929948s (cpu); 0.0483275s (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 0.153409s (cpu); 0.152472s (thread); 0s (gc) │ │ │ + -- used 0.180002s (cpu); 0.178937s (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 0.273835s (cpu); 0.165197s (thread); 0s (gc) │ │ │ + -- used 0.299947s (cpu); 0.198019s (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 0.0662252s (cpu); 0.0657386s (thread); 0s (gc) │ │ │ + -- used 0.0778175s (cpu); 0.0792761s (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 0.356787s (cpu); 0.306374s (thread); 0s (gc) │ │ │ + -- used 0.39215s (cpu); 0.336333s (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 0.62289s (cpu); 0.562436s (thread); 0s (gc) │ │ │ + -- used 0.699911s (cpu); 0.647579s (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 0.0199496s (cpu); 0.0221836s (thread); 0s (gc) │ │ │ + -- used 0.0240592s (cpu); 0.0253601s (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 2.62117s (cpu); 2.00344s (thread); 0s (gc) │ │ │ + -- used 2.27294s (cpu); 1.83811s (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 0.567125s (cpu); 0.415889s (thread); 0s (gc) │ │ │ + -- used 0.411202s (cpu); 0.3554s (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 0.0280074s (cpu); 0.0291793s (thread); 0s (gc) │ │ │ + -- used 0.0527059s (cpu); 0.0549977s (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 0.0549354s (cpu); 0.0585691s (thread); 0s (gc) │ │ │ + -- used 0.0746506s (cpu); 0.0731387s (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 0.104019s (cpu); 0.0414726s (thread); 0s (gc) │ │ │ + -- used 0.103964s (cpu); 0.053282s (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 0.0357097s (cpu); 0.0368356s (thread); 0s (gc) │ │ │ + -- used 0.0707466s (cpu); 0.0730226s (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 0.159171s (cpu); 0.0996025s (thread); 0s (gc) │ │ │ + -- used 0.165837s (cpu); 0.112999s (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 0.0439944s (cpu); 0.0463877s (thread); 0s (gc)
    │ │ │ + -- used 0.0599842s (cpu); 0.0574975s (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 0.127607s (cpu); 0.0769988s (thread); 0s (gc)
│ │ │ + -- used 0.122599s (cpu); 0.0758033s (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 0.0439944s (cpu); 0.0463877s (thread); 0s (gc) │ │ │ │ + -- used 0.0599842s (cpu); 0.0574975s (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 0.127607s (cpu); 0.0769988s (thread); 0s (gc) │ │ │ │ + -- used 0.122599s (cpu); 0.0758033s (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 0.156403s (cpu); 0.100372s (thread); 0s (gc) │ │ │ │ + -- used 0.156654s (cpu); 0.104704s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.157983s (cpu); 0.0925469s (thread); 0s (gc) │ │ │ │ + -- used 0.160015s (cpu); 0.106295s (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 0.156403s (cpu); 0.100372s (thread); 0s (gc)
    │ │ │ + -- used 0.156654s (cpu); 0.104704s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.157983s (cpu); 0.0925469s (thread); 0s (gc)
    │ │ │ + -- used 0.160015s (cpu); 0.106295s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.0399817s (cpu); 0.0401994s (thread); 0s (gc)
│ │ │ + -- used 0.0478629s (cpu); 0.0498197s (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 0.109535s (cpu); 0.0517928s (thread); 0s (gc)
│ │ │ + -- used 0.110749s (cpu); 0.0546887s (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 0.145885s (cpu); 0.093329s (thread); 0s (gc)
│ │ │ + -- used 0.154881s (cpu); 0.100139s (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 0.136507s (cpu); 0.0861143s (thread); 0s (gc)
│ │ │ + -- used 0.139949s (cpu); 0.0836923s (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 0.0399817s (cpu); 0.0401994s (thread); 0s (gc) │ │ │ │ + -- used 0.0478629s (cpu); 0.0498197s (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 0.109535s (cpu); 0.0517928s (thread); 0s (gc) │ │ │ │ + -- used 0.110749s (cpu); 0.0546887s (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 0.145885s (cpu); 0.093329s (thread); 0s (gc) │ │ │ │ + -- used 0.154881s (cpu); 0.100139s (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 0.136507s (cpu); 0.0861143s (thread); 0s (gc) │ │ │ │ + -- used 0.139949s (cpu); 0.0836923s (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 5.00704s (cpu); 4.62662s (thread); 0s (gc)
│ │ │ + -- used 5.04364s (cpu); 4.80582s (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 1.27577s (cpu); 1.09674s (thread); 0s (gc)
    │ │ │ + -- used 1.09673s (cpu); 1.0502s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.225281s (cpu); 0.16014s (thread); 0s (gc)
    │ │ │ + -- used 0.232004s (cpu); 0.184684s (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 0.31187s (cpu); 0.194538s (thread); 0s (gc)
    │ │ │ + -- used 0.229954s (cpu); 0.181059s (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 5.00704s (cpu); 4.62662s (thread); 0s (gc) │ │ │ │ + -- used 5.04364s (cpu); 4.80582s (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 1.27577s (cpu); 1.09674s (thread); 0s (gc) │ │ │ │ + -- used 1.09673s (cpu); 1.0502s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.225281s (cpu); 0.16014s (thread); 0s (gc) │ │ │ │ + -- used 0.232004s (cpu); 0.184684s (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 0.31187s (cpu); 0.194538s (thread); 0s (gc) │ │ │ │ + -- used 0.229954s (cpu); 0.181059s (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 0.00800121s (cpu); 0.00865969s (thread); 0s (gc)
│ │ │ + -- used 0.0120158s (cpu); 0.010025s (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 0.00741673s (cpu); 0.00950854s (thread); 0s (gc)
│ │ │ + -- used 0.0108181s (cpu); 0.0110375s (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 0.422407s (cpu); 0.420635s (thread); 0s (gc)
│ │ │ + -- used 0.415819s (cpu); 0.415159s (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 0.00800121s (cpu); 0.00865969s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120158s (cpu); 0.010025s (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 0.00741673s (cpu); 0.00950854s (thread); 0s (gc)
│ │ │ │ + -- used 0.0108181s (cpu); 0.0110375s (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 0.422407s (cpu); 0.420635s (thread); 0s (gc)
│ │ │ │ + -- used 0.415819s (cpu); 0.415159s (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 0.108129s (cpu); 0.106181s (thread); 0s (gc)
│ │ │ + -- used 0.136265s (cpu); 0.135585s (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 0.219998s (cpu); 0.157691s (thread); 0s (gc)
│ │ │ + -- used 0.2179s (cpu); 0.162679s (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 0.108129s (cpu); 0.106181s (thread); 0s (gc) │ │ │ │ + -- used 0.136265s (cpu); 0.135585s (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 0.219998s (cpu); 0.157691s (thread); 0s (gc) │ │ │ │ + -- used 0.2179s (cpu); 0.162679s (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 0.061731s (cpu); 0.0602097s (thread); 0s (gc) │ │ │ │ + -- used 0.0738155s (cpu); 0.0729647s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ │ 3331 0 9 │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ │ - -- used 0.627729s (cpu); 0.578975s (thread); 0s (gc) │ │ │ │ + -- used 0.740854s (cpu); 0.689631s (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 0.061731s (cpu); 0.0602097s (thread); 0s (gc)
│ │ │ + -- used 0.0738155s (cpu); 0.0729647s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.627729s (cpu); 0.578975s (thread); 0s (gc)
│ │ │ + -- used 0.740854s (cpu); 0.689631s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.0437722s (cpu); 0.0407153s (thread); 0s (gc)
│ │ │ + -- used 0.0518947s (cpu); 0.0499503s (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 0.119026s (cpu); 0.0608633s (thread); 0s (gc)
│ │ │ + -- used 0.110451s (cpu); 0.0596955s (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 0.0200484s (cpu); 0.0193455s (thread); 0s (gc)
│ │ │ + -- used 0.0199995s (cpu); 0.0210634s (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 0.0158647s (cpu); 0.0172923s (thread); 0s (gc)
│ │ │ + -- used 0.018488s (cpu); 0.0200899s (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 0.0437722s (cpu); 0.0407153s (thread); 0s (gc) │ │ │ │ + -- used 0.0518947s (cpu); 0.0499503s (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 0.119026s (cpu); 0.0608633s (thread); 0s (gc) │ │ │ │ + -- used 0.110451s (cpu); 0.0596955s (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 0.0200484s (cpu); 0.0193455s (thread); 0s (gc) │ │ │ │ + -- used 0.0199995s (cpu); 0.0210634s (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 0.0158647s (cpu); 0.0172923s (thread); 0s (gc) │ │ │ │ + -- used 0.018488s (cpu); 0.0200899s (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 0.0360004s (cpu); 0.0375479s (thread); 0s (gc)
│ │ │ + -- used 0.0472215s (cpu); 0.0447933s (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 0.0360004s (cpu); 0.0375479s (thread); 0s (gc)
│ │ │ │ + -- used 0.0472215s (cpu); 0.0447933s (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 0.00798565s (cpu); 0.00802576s (thread); 0s (gc)
│ │ │ + -- used 0.0119736s (cpu); 0.00969539s (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 0.00457769s (cpu); 0.00651439s (thread); 0s (gc)
│ │ │ + -- used 0.00759238s (cpu); 0.008202s (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 0.00798565s (cpu); 0.00802576s (thread); 0s (gc) │ │ │ │ + -- used 0.0119736s (cpu); 0.00969539s (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 0.00457769s (cpu); 0.00651439s (thread); 0s (gc) │ │ │ │ + -- used 0.00759238s (cpu); 0.008202s (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 0.0183706s (cpu); 0.0201448s (thread); 0s (gc)
│ │ │ + -- used 0.0205969s (cpu); 0.0213117s (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 0.0183706s (cpu); 0.0201448s (thread); 0s (gc) │ │ │ │ + -- used 0.0205969s (cpu); 0.0213117s (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 0.0040224s (cpu); 0.00355676s (thread); 0s (gc)
│ │ │ + -- used 0.00400873s (cpu); 0.00400396s (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 0.00399843s (cpu); 0.0024578s (thread); 0s (gc)
│ │ │ + -- used 0.00397216s (cpu); 0.00421769s (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 0.0040224s (cpu); 0.00355676s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400873s (cpu); 0.00400396s (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 0.00399843s (cpu); 0.0024578s (thread); 0s (gc)
│ │ │ │ + -- used 0.00397216s (cpu); 0.00421769s (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 0.00564656s (cpu); 0.00441136s (thread); 0s (gc)
│ │ │ + -- used 0.00397811s (cpu); 0.00478977s (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 0.0246074s (cpu); 0.0276395s (thread); 0s (gc)
│ │ │ + -- used 0.0280243s (cpu); 0.0311589s (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 0.102777s (cpu); 0.0396871s (thread); 0s (gc)
│ │ │ + -- used 0.0929948s (cpu); 0.0483275s (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 0.153409s (cpu); 0.152472s (thread); 0s (gc)
│ │ │ + -- used 0.180002s (cpu); 0.178937s (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 0.00564656s (cpu); 0.00441136s (thread); 0s (gc)
│ │ │ │ + -- used 0.00397811s (cpu); 0.00478977s (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 0.0246074s (cpu); 0.0276395s (thread); 0s (gc)
│ │ │ │ + -- used 0.0280243s (cpu); 0.0311589s (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 0.102777s (cpu); 0.0396871s (thread); 0s (gc)
│ │ │ │ + -- used 0.0929948s (cpu); 0.0483275s (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 0.153409s (cpu); 0.152472s (thread); 0s (gc)
│ │ │ │ + -- used 0.180002s (cpu); 0.178937s (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 0.273835s (cpu); 0.165197s (thread); 0s (gc)
│ │ │ + -- used 0.299947s (cpu); 0.198019s (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 0.0662252s (cpu); 0.0657386s (thread); 0s (gc)
│ │ │ + -- used 0.0778175s (cpu); 0.0792761s (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 0.356787s (cpu); 0.306374s (thread); 0s (gc)
│ │ │ + -- used 0.39215s (cpu); 0.336333s (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 0.62289s (cpu); 0.562436s (thread); 0s (gc)
│ │ │ + -- used 0.699911s (cpu); 0.647579s (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 0.273835s (cpu); 0.165197s (thread); 0s (gc)
│ │ │ │ + -- used 0.299947s (cpu); 0.198019s (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 0.0662252s (cpu); 0.0657386s (thread); 0s (gc)
│ │ │ │ + -- used 0.0778175s (cpu); 0.0792761s (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 0.356787s (cpu); 0.306374s (thread); 0s (gc)
│ │ │ │ + -- used 0.39215s (cpu); 0.336333s (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 0.62289s (cpu); 0.562436s (thread); 0s (gc)
│ │ │ │ + -- used 0.699911s (cpu); 0.647579s (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 0.0199496s (cpu); 0.0221836s (thread); 0s (gc)
│ │ │ + -- used 0.0240592s (cpu); 0.0253601s (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 2.62117s (cpu); 2.00344s (thread); 0s (gc)
│ │ │ + -- used 2.27294s (cpu); 1.83811s (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 0.567125s (cpu); 0.415889s (thread); 0s (gc)
│ │ │ + -- used 0.411202s (cpu); 0.3554s (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 0.0199496s (cpu); 0.0221836s (thread); 0s (gc) │ │ │ │ + -- used 0.0240592s (cpu); 0.0253601s (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 2.62117s (cpu); 2.00344s (thread); 0s (gc) │ │ │ │ + -- used 2.27294s (cpu); 1.83811s (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 0.567125s (cpu); 0.415889s (thread); 0s (gc) │ │ │ │ + -- used 0.411202s (cpu); 0.3554s (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 0.0280074s (cpu); 0.0291793s (thread); 0s (gc)
│ │ │ + -- used 0.0527059s (cpu); 0.0549977s (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 0.0549354s (cpu); 0.0585691s (thread); 0s (gc)
│ │ │ + -- used 0.0746506s (cpu); 0.0731387s (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 0.104019s (cpu); 0.0414726s (thread); 0s (gc)
│ │ │ + -- used 0.103964s (cpu); 0.053282s (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 0.0357097s (cpu); 0.0368356s (thread); 0s (gc)
│ │ │ + -- used 0.0707466s (cpu); 0.0730226s (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 0.159171s (cpu); 0.0996025s (thread); 0s (gc)
    │ │ │ + -- used 0.165837s (cpu); 0.112999s (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 0.0280074s (cpu); 0.0291793s (thread); 0s (gc) │ │ │ │ + -- used 0.0527059s (cpu); 0.0549977s (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 0.0549354s (cpu); 0.0585691s (thread); 0s (gc) │ │ │ │ + -- used 0.0746506s (cpu); 0.0731387s (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 0.104019s (cpu); 0.0414726s (thread); 0s (gc) │ │ │ │ + -- used 0.103964s (cpu); 0.053282s (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 0.0357097s (cpu); 0.0368356s (thread); 0s (gc) │ │ │ │ + -- used 0.0707466s (cpu); 0.0730226s (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 0.159171s (cpu); 0.0996025s (thread); 0s (gc) │ │ │ │ + -- used 0.165837s (cpu); 0.112999s (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 │ │ │ @@ -3,16 +3,16 @@ │ │ │ i1 : run("ulimit -a") │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ -locked memory(kbytes) 2047000 │ │ │ -process 63807 │ │ │ +locked memory(kbytes) 8192 │ │ │ +process 63521 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 2927978066455787395 │ │ │ │ │ │ i1 : fn=temporaryFileName()|".m2" │ │ │ │ │ │ -o1 = /tmp/M2-78820-0/0.m2 │ │ │ +o1 = /tmp/M2-136300-0/0.m2 │ │ │ │ │ │ i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-78820-0/1.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/1.m2" >"/tmp/M2-136300-0/1.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i7 : h │ │ │ │ │ │ o7 = HashTable{"answer file" => null} │ │ │ "exit code" => 0 │ │ │ "output file" => null │ │ │ @@ -33,105 +33,105 @@ │ │ │ o8 = true │ │ │ │ │ │ i9 : h#"exit code"===0 │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : h=runExternalM2(fn,"justexit",()); │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-78820-0/2.m2" >"/tmp/M2-78820-0/2.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/2.m2" >"/tmp/M2-136300-0/2.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i11 : h │ │ │ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-78820-0/2.ans} │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-136300-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-78820-0/2.out │ │ │ + "output file" => /tmp/M2-136300-0/2.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/M2-78820-0/3.m2" >"/tmp/M2-78820-0/3.out" 2>&1 )) │ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/3.m2" >"/tmp/M2-136300-0/3.out" 2>&1 )) │ │ │ Killed │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i15 : h │ │ │ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-78820-0/3.ans} │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-136300-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-78820-0/3.out │ │ │ + "output file" => /tmp/M2-136300-0/3.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/M2-78820-0/3.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-136300-0/3.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-78820-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-136300-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/3.ans",spin (10)); │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/3.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/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1') >"/tmp/M2-78820-0/4.stat" 2>&1 )) │ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out" 2>&1') >"/tmp/M2-136300-0/4.stat" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i19 : h#"statistics" │ │ │ │ │ │ -o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1" │ │ │ - User time (seconds): 5.45 │ │ │ - System time (seconds): 0.11 │ │ │ - Percent of CPU this job got: 87% │ │ │ - Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.36 │ │ │ +o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out" 2>&1" │ │ │ + User time (seconds): 4.45 │ │ │ + System time (seconds): 0.25 │ │ │ + Percent of CPU this job got: 115% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.06 │ │ │ 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): 252732 │ │ │ + Maximum resident set size (kbytes): 338680 │ │ │ Average resident set size (kbytes): 0 │ │ │ Major (requiring I/O) page faults: 0 │ │ │ - Minor (reclaiming a frame) page faults: 9011 │ │ │ - Voluntary context switches: 1708 │ │ │ - Involuntary context switches: 4014 │ │ │ + Minor (reclaiming a frame) page faults: 10729 │ │ │ + Voluntary context switches: 5522 │ │ │ + Involuntary context switches: 895 │ │ │ Swaps: 0 │ │ │ File system inputs: 0 │ │ │ - File system outputs: 0 │ │ │ + File system outputs: 16 │ │ │ Socket messages sent: 0 │ │ │ Socket messages received: 0 │ │ │ Signals delivered: 0 │ │ │ Page size (bytes): 4096 │ │ │ Exit status: 0 │ │ │ │ │ │ │ │ │ i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///; │ │ │ │ │ │ i21 : (runExternalM2(fn,identity,v))#value===v │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-78820-0/6.m2" >"/tmp/M2-78820-0/6.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/6.m2" >"/tmp/M2-136300-0/6.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/M2-78820-0/7.m2" >"/tmp/M2-78820-0/7.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/7.m2" >"/tmp/M2-136300-0/7.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-78820-0/7.ans} │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-136300-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-78820-0/7.out │ │ │ + "output file" => /tmp/M2-136300-0/7.out │ │ │ "return code" => 256 │ │ │ "statistics" => null │ │ │ "time used" => 1 │ │ │ value => null │ │ │ │ │ │ o24 : HashTable │ │ │ │ │ │ i25 : get(h#"output file") │ │ │ │ │ │ o25 = │ │ │ - i1 : -- Script /tmp/M2-78820-0/7.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-136300-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-78820-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-136300-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/7.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: │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/7.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:4:75:(3):[1]: error: no method for binary operator ^ applied to objects: │ │ │ R (of class Symbol) │ │ │ ^ 2 (of class ZZ) │ │ │ │ │ │ │ │ │ i26 : fn<"/tmp/M2-78820-0/8.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/8.m2" >"/tmp/M2-136300-0/8.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/M2-78820-0/9.m2" >"/tmp/M2-78820-0/9.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-136300-0/9.m2" >"/tmp/M2-136300-0/9.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i30 : h#value │ │ │ │ │ │ o30 = QQ[x..y] │ │ │ │ │ │ o30 : PolynomialRing │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ @@ -74,16 +74,16 @@ │ │ │ 
      i1 : run("ulimit -a")
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │ -locked memory(kbytes) 2047000
│ │ │ -process              63807
│ │ │ +locked memory(kbytes) 8192
│ │ │ +process              63521
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
      │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -34,16 +34,16 @@ │ │ │ │ i1 : run("ulimit -a") │ │ │ │ time(seconds) 700 │ │ │ │ file(blocks) unlimited │ │ │ │ data(kbytes) unlimited │ │ │ │ stack(kbytes) 8192 │ │ │ │ coredump(blocks) unlimited │ │ │ │ memory(kbytes) 850000 │ │ │ │ -locked memory(kbytes) 2047000 │ │ │ │ -process 63807 │ │ │ │ +locked memory(kbytes) 8192 │ │ │ │ +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:

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -115,15 +115,15 @@ │ │ │
    │ │ │

    and then call them:

    │ │ │
    │ │ │
    │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-78820-0/0.m2
    │ │ │ +o1 = /tmp/M2-136300-0/0.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/M2-78820-0/1.m2" >"/tmp/M2-78820-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/1.m2" >"/tmp/M2-136300-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i7 : h
│ │ │  
│ │ │ @@ -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.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -350,21 +350,21 @@
│ │ │          
│ │ │          
    
│ │ │                
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-78820-0/2.m2" >"/tmp/M2-78820-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/2.m2" >"/tmp/M2-136300-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-78820-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-136300-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-78820-0/2.out
│ │ │ +                "output file" => /tmp/M2-136300-0/2.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/M2-78820-0/3.m2" >"/tmp/M2-78820-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/3.m2" >"/tmp/M2-136300-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-78820-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-136300-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-78820-0/3.out
│ │ │ +                "output file" => /tmp/M2-136300-0/3.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/M2-78820-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-136300-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-78820-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-136300-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/3.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/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1') >"/tmp/M2-78820-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out" 2>&1') >"/tmp/M2-136300-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.45
│ │ │ -              System time (seconds): 0.11
│ │ │ -              Percent of CPU this job got: 87%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.36
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.45
│ │ │ +              System time (seconds): 0.25
│ │ │ +              Percent of CPU this job got: 115%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.06
│ │ │                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): 252732
│ │ │ +              Maximum resident set size (kbytes): 338680
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 9011
│ │ │ -              Voluntary context switches: 1708
│ │ │ -              Involuntary context switches: 4014
│ │ │ +              Minor (reclaiming a frame) page faults: 10729
│ │ │ +              Voluntary context switches: 5522
│ │ │ +              Involuntary context switches: 895
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    
│ │ │              
    
│ │ │                
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    
│ │ │              
    
│ │ │                
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-78820-0/6.m2" >"/tmp/M2-78820-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/6.m2" >"/tmp/M2-136300-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -325,21 +325,21 @@
│ │ │                               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/M2-78820-0/7.m2" >"/tmp/M2-78820-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/7.m2" >"/tmp/M2-136300-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-78820-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-136300-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-78820-0/7.out
│ │ │ +                "output file" => /tmp/M2-136300-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-78820-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-136300-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-78820-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-136300-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/7.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:
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/7.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:4:75:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Keep in mind that the object you are passing must make sense in the context of the file containing your function! For instance, here we need to define the ring:
    
│ │ │ @@ -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/M2-78820-0/8.m2" >"/tmp/M2-78820-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/8.m2" >"/tmp/M2-136300-0/8.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/M2-78820-0/9.m2" >"/tmp/M2-78820-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-136300-0/9.m2" >"/tmp/M2-136300-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,25 +45,25 @@
│ │ │ │  the output file (unless it was deleted), the name of the answer file (unless it
│ │ │ │  was deleted), any statistics recorded about the resource usage, and the value
│ │ │ │  returned by the function func. If the child process terminates abnormally, then
│ │ │ │  usually the exit code is nonzero and the value returned is _n_u_l_l.
│ │ │ │  For example, we can write a few functions to a temporary file:
│ │ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-78820-0/0.m2
│ │ │ │ +o1 = /tmp/M2-136300-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-78820-0/1.out" 2>&1 ))
│ │ │ │ +M2-136300-0/1.m2" >"/tmp/M2-136300-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -79,167 +79,167 @@
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  An abnormal program exit will have a nonzero exit code; also, the value will be
│ │ │ │  null, the output file should exist, but the answer file may not exist unless
│ │ │ │  the routine finished successfully.
│ │ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-78820-0/2.m2" >"/tmp/M2-78820-0/2.out" 2>&1 ))
│ │ │ │ +M2-136300-0/2.m2" >"/tmp/M2-136300-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-78820-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-136300-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-78820-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-136300-0/2.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/M2-78820-0/3.m2" >"/tmp/M2-78820-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-136300-0/3.m2" >"/tmp/M2-136300-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-78820-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-136300-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-78820-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-136300-0/3.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/M2-78820-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-136300-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-78820-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-136300-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/3.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/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-78820-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out"
│ │ │ │ +2>&1') >"/tmp/M2-136300-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-78820-0/4.m2" >"/tmp/M2-78820-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 5.45
│ │ │ │ -              System time (seconds): 0.11
│ │ │ │ -              Percent of CPU this job got: 87%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.36
│ │ │ │ +--silent  -q  <"/tmp/M2-136300-0/4.m2" >"/tmp/M2-136300-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 4.45
│ │ │ │ +              System time (seconds): 0.25
│ │ │ │ +              Percent of CPU this job got: 115%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.06
│ │ │ │                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): 252732
│ │ │ │ +              Maximum resident set size (kbytes): 338680
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 9011
│ │ │ │ -              Voluntary context switches: 1708
│ │ │ │ -              Involuntary context switches: 4014
│ │ │ │ +              Minor (reclaiming a frame) page faults: 10729
│ │ │ │ +              Voluntary context switches: 5522
│ │ │ │ +              Involuntary context switches: 895
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 16
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-78820-0/6.m2" >"/tmp/M2-78820-0/6.out" 2>&1 ))
│ │ │ │ +M2-136300-0/6.m2" >"/tmp/M2-136300-0/6.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/
│ │ │ │ -M2-78820-0/7.m2" >"/tmp/M2-78820-0/7.out" 2>&1 ))
│ │ │ │ +M2-136300-0/7.m2" >"/tmp/M2-136300-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-78820-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-136300-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-78820-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-136300-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-78820-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-136300-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-78820-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-136300-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-78820-0/7.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
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-136300-0/7.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:4:75:(3):[1]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-78820-0/8.out" 2>&1 ))
│ │ │ │ +M2-136300-0/8.m2" >"/tmp/M2-136300-0/8.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/
│ │ │ │ -M2-78820-0/9.m2" >"/tmp/M2-78820-0/9.out" 2>&1 ))
│ │ │ │ +M2-136300-0/9.m2" >"/tmp/M2-136300-0/9.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 0.002138s (cpu); 0.000210425s (thread); 0s (gc)
│ │ │ + -- used 0.00360619s (cpu); 0.000272914s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.11949s (cpu); 3.63194s (thread); 0s (gc)
│ │ │ + -- used 3.78364s (cpu); 2.9076s (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 0.002138s (cpu); 0.000210425s (thread); 0s (gc)
│ │ │ + -- used 0.00360619s (cpu); 0.000272914s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : ZZ[y];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.11949s (cpu); 3.63194s (thread); 0s (gc)
│ │ │ + -- used 3.78364s (cpu); 2.9076s (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 0.002138s (cpu); 0.000210425s (thread); 0s (gc)
│ │ │ │ + -- used 0.00360619s (cpu); 0.000272914s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.11949s (cpu); 3.63194s (thread); 0s (gc)
│ │ │ │ + -- used 3.78364s (cpu); 2.9076s (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)
│ │ │ - -- .00735994s elapsed
│ │ │ + -- .00668055s 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;
│ │ │ - -- .0025963s elapsed
│ │ │ + -- .00289582s 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);
│ │ │ - -- .0056062s elapsed
│ │ │ + -- .00575379s 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)
│ │ │ - -- .00298678s elapsed
│ │ │ + -- .00326978s 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
│ │ │ - -- .000693988s elapsed
│ │ │ + -- .000739366s 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)
│ │ │ - -- .00128238s elapsed
│ │ │ + -- .00152101s 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)
│ │ │ - -- .00354641s elapsed
│ │ │ + -- .0043397s 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)
│ │ │ - -- .00287459s elapsed
│ │ │ + -- .00296703s 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)
│ │ │ - -- .00270204s elapsed
│ │ │ + -- .00291945s 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)
│ │ │ - -- .00735994s elapsed
│ │ │ + -- .00668055s 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;
│ │ │ - -- .0025963s elapsed
    │ │ │ + -- .00289582s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -212,15 +212,15 @@
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .0056062s elapsed
    │ │ │ + -- .00575379s 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) │ │ │ │ - -- .00735994s elapsed │ │ │ │ + -- .00668055s 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; │ │ │ │ - -- .0025963s elapsed │ │ │ │ + -- .00289582s 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); │ │ │ │ - -- .0056062s elapsed │ │ │ │ + -- .00575379s 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)
│ │ │ - -- .00298678s elapsed
│ │ │ + -- .00326978s 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
│ │ │ - -- .000693988s elapsed
│ │ │ + -- .000739366s 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)
│ │ │ - -- .00128238s elapsed
│ │ │ + -- .00152101s 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) │ │ │ │ - -- .00298678s elapsed │ │ │ │ + -- .00326978s 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 │ │ │ │ - -- .000693988s elapsed │ │ │ │ + -- .000739366s 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) │ │ │ │ - -- .00128238s elapsed │ │ │ │ + -- .00152101s 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)
│ │ │ - -- .00354641s elapsed
│ │ │ + -- .0043397s 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) │ │ │ │ - -- .00354641s elapsed │ │ │ │ + -- .0043397s 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)
│ │ │ - -- .00287459s elapsed
│ │ │ + -- .00296703s 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) │ │ │ │ - -- .00287459s elapsed │ │ │ │ + -- .00296703s 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)
│ │ │ - -- .00270204s elapsed
│ │ │ + -- .00291945s 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) │ │ │ │ - -- .00270204s elapsed │ │ │ │ + -- .00291945s 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 │ │ │ @@ -38,33 +38,33 @@ │ │ │ o5 : Ideal of S │ │ │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ - -- used 0.422335s (cpu); 0.363976s (thread); 0s (gc) │ │ │ + -- used 0.435242s (cpu); 0.36498s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.555104s (cpu); 0.485183s (thread); 0s (gc) │ │ │ + -- used 0.711718s (cpu); 0.657792s (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 0.0239982s (cpu); 0.0246267s (thread); 0s (gc) │ │ │ + -- used 0.0240064s (cpu); 0.0248119s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00653159s (cpu); 0.00838064s (thread); 0s (gc) │ │ │ + -- used 0.0063147s (cpu); 0.00906178s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -126,23 +126,23 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.422335s (cpu); 0.363976s (thread); 0s (gc)
│ │ │ + -- used 0.435242s (cpu); 0.36498s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.555104s (cpu); 0.485183s (thread); 0s (gc)
│ │ │ + -- used 0.711718s (cpu); 0.657792s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │ @@ -159,23 +159,23 @@ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0239982s (cpu); 0.0246267s (thread); 0s (gc)
│ │ │ + -- used 0.0240064s (cpu); 0.0248119s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00653159s (cpu); 0.00838064s (thread); 0s (gc)
│ │ │ + -- used 0.0063147s (cpu); 0.00906178s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -57,32 +57,32 @@ │ │ │ │ i5 : I = monomialCurveIdeal(S, 1..n-1); │ │ │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ │ - -- used 0.422335s (cpu); 0.363976s (thread); 0s (gc) │ │ │ │ + -- used 0.435242s (cpu); 0.36498s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.555104s (cpu); 0.485183s (thread); 0s (gc) │ │ │ │ + -- used 0.711718s (cpu); 0.657792s (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 0.0239982s (cpu); 0.0246267s (thread); 0s (gc) │ │ │ │ + -- used 0.0240064s (cpu); 0.0248119s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00653159s (cpu); 0.00838064s (thread); 0s (gc) │ │ │ │ + -- used 0.0063147s (cpu); 0.00906178s (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 0.00400125s (cpu); 0.00425118s (thread); 0s (gc) │ │ │ - -- used 0.00579092s (cpu); 0.00598973s (thread); 0s (gc) │ │ │ - -- used 0.00775557s (cpu); 0.0101601s (thread); 0s (gc) │ │ │ - -- used 0.0135813s (cpu); 0.0166916s (thread); 0s (gc) │ │ │ - -- used 0.0288746s (cpu); 0.0318281s (thread); 0s (gc) │ │ │ - -- used 0.0537256s (cpu); 0.0555003s (thread); 0s (gc) │ │ │ - -- used 0.0940662s (cpu); 0.095517s (thread); 0s (gc) │ │ │ - -- used 0.14605s (cpu); 0.149915s (thread); 0s (gc) │ │ │ - -- used 0.302334s (cpu); 0.254769s (thread); 0s (gc) │ │ │ + -- used 0.00403437s (cpu); 0.00551641s (thread); 0s (gc) │ │ │ + -- used 0.0068321s (cpu); 0.00700567s (thread); 0s (gc) │ │ │ + -- used 0.0079336s (cpu); 0.0118359s (thread); 0s (gc) │ │ │ + -- used 0.0182866s (cpu); 0.019593s (thread); 0s (gc) │ │ │ + -- used 0.0346282s (cpu); 0.0364903s (thread); 0s (gc) │ │ │ + -- used 0.0667395s (cpu); 0.069634s (thread); 0s (gc) │ │ │ + -- used 0.115994s (cpu); 0.116646s (thread); 0s (gc) │ │ │ + -- used 0.187009s (cpu); 0.188268s (thread); 0s (gc) │ │ │ + -- used 0.41342s (cpu); 0.34228s (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 0.00400125s (cpu); 0.00425118s (thread); 0s (gc)
│ │ │ - -- used 0.00579092s (cpu); 0.00598973s (thread); 0s (gc)
│ │ │ - -- used 0.00775557s (cpu); 0.0101601s (thread); 0s (gc)
│ │ │ - -- used 0.0135813s (cpu); 0.0166916s (thread); 0s (gc)
│ │ │ - -- used 0.0288746s (cpu); 0.0318281s (thread); 0s (gc)
│ │ │ - -- used 0.0537256s (cpu); 0.0555003s (thread); 0s (gc)
│ │ │ - -- used 0.0940662s (cpu); 0.095517s (thread); 0s (gc)
│ │ │ - -- used 0.14605s (cpu); 0.149915s (thread); 0s (gc)
│ │ │ - -- used 0.302334s (cpu); 0.254769s (thread); 0s (gc)
│ │ │ + -- used 0.00403437s (cpu); 0.00551641s (thread); 0s (gc)
│ │ │ + -- used 0.0068321s (cpu); 0.00700567s (thread); 0s (gc)
│ │ │ + -- used 0.0079336s (cpu); 0.0118359s (thread); 0s (gc)
│ │ │ + -- used 0.0182866s (cpu); 0.019593s (thread); 0s (gc)
│ │ │ + -- used 0.0346282s (cpu); 0.0364903s (thread); 0s (gc)
│ │ │ + -- used 0.0667395s (cpu); 0.069634s (thread); 0s (gc)
│ │ │ + -- used 0.115994s (cpu); 0.116646s (thread); 0s (gc)
│ │ │ + -- used 0.187009s (cpu); 0.188268s (thread); 0s (gc)
│ │ │ + -- used 0.41342s (cpu); 0.34228s (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 0.00400125s (cpu); 0.00425118s (thread); 0s (gc) │ │ │ │ - -- used 0.00579092s (cpu); 0.00598973s (thread); 0s (gc) │ │ │ │ - -- used 0.00775557s (cpu); 0.0101601s (thread); 0s (gc) │ │ │ │ - -- used 0.0135813s (cpu); 0.0166916s (thread); 0s (gc) │ │ │ │ - -- used 0.0288746s (cpu); 0.0318281s (thread); 0s (gc) │ │ │ │ - -- used 0.0537256s (cpu); 0.0555003s (thread); 0s (gc) │ │ │ │ - -- used 0.0940662s (cpu); 0.095517s (thread); 0s (gc) │ │ │ │ - -- used 0.14605s (cpu); 0.149915s (thread); 0s (gc) │ │ │ │ - -- used 0.302334s (cpu); 0.254769s (thread); 0s (gc) │ │ │ │ + -- used 0.00403437s (cpu); 0.00551641s (thread); 0s (gc) │ │ │ │ + -- used 0.0068321s (cpu); 0.00700567s (thread); 0s (gc) │ │ │ │ + -- used 0.0079336s (cpu); 0.0118359s (thread); 0s (gc) │ │ │ │ + -- used 0.0182866s (cpu); 0.019593s (thread); 0s (gc) │ │ │ │ + -- used 0.0346282s (cpu); 0.0364903s (thread); 0s (gc) │ │ │ │ + -- used 0.0667395s (cpu); 0.069634s (thread); 0s (gc) │ │ │ │ + -- used 0.115994s (cpu); 0.116646s (thread); 0s (gc) │ │ │ │ + -- used 0.187009s (cpu); 0.188268s (thread); 0s (gc) │ │ │ │ + -- used 0.41342s (cpu); 0.34228s (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/dump/rawdocumentation.dump │ │ │ @@ -1,8 +1,8 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:27 2025 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Aug 25 11:02:26 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=26 │ │ │ bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk= │ │ ├── ./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 4.70843s (cpu); 3.29414s (thread); 0s (gc) │ │ │ + -- used 6.84532s (cpu); 3.72155s (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 49.7451s (cpu); 45.9755s (thread); 0s (gc) │ │ │ + -- used 61.7206s (cpu); 57.1894s (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 0.332916s (cpu); 0.186701s (thread); 0s (gc) │ │ │ + -- used 0.488907s (cpu); 0.179058s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.293034s (cpu); 0.114663s (thread); 0s (gc) │ │ │ + -- used 0.462643s (cpu); 0.168342s (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 4.70843s (cpu); 3.29414s (thread); 0s (gc)
│ │ │ + -- used 6.84532s (cpu); 3.72155s (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 49.7451s (cpu); 45.9755s (thread); 0s (gc)
│ │ │ + -- used 61.7206s (cpu); 57.1894s (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 4.70843s (cpu); 3.29414s (thread); 0s (gc) │ │ │ │ + -- used 6.84532s (cpu); 3.72155s (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 49.7451s (cpu); 45.9755s (thread); 0s (gc) │ │ │ │ + -- used 61.7206s (cpu); 57.1894s (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 0.332916s (cpu); 0.186701s (thread); 0s (gc)
│ │ │ + -- used 0.488907s (cpu); 0.179058s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.293034s (cpu); 0.114663s (thread); 0s (gc)
│ │ │ + -- used 0.462643s (cpu); 0.168342s (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 0.332916s (cpu); 0.186701s (thread); 0s (gc) │ │ │ │ + -- used 0.488907s (cpu); 0.179058s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.293034s (cpu); 0.114663s (thread); 0s (gc) │ │ │ │ + -- used 0.462643s (cpu); 0.168342s (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") -- .198339s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .158411s 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")           -- .198339s elapsed
    │ │ │ + -- capturing check(0, "SimpleDoc") -- .158411s 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") -- .198339s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .158411s 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/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ @@ -9,14 +9,14 @@ │ │ │ │ │ │ i3 : (Y, T) = setOnesForest X; │ │ │ │ │ │ i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)}; │ │ │ │ │ │ i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1) │ │ │ │ │ │ - 2 2 2 2 2 2 2 2 2 2 │ │ │ -o5 = - x x x x x x + x x x x x x + x x x x x x x x │ │ │ - 1 4 6 7 10 11 2 3 5 8 10 11 2 3 5 6 7 8 9 12 │ │ │ + 2 2 2 2 2 2 2 2 2 2 │ │ │ +o5 = x x x x x x - x x x x x x + x x x x x x x x │ │ │ + 1 4 6 7 10 11 2 3 5 8 10 11 1 2 3 4 6 7 9 12 │ │ │ │ │ │ o5 : R │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_set__Ones__Forest.out │ │ │ @@ -14,20 +14,20 @@ │ │ │ │ │ │ 4 4 │ │ │ o2 : Matrix (QQ[x ..x ]) <-- (QQ[x ..x ]) │ │ │ 0 7 0 7 │ │ │ │ │ │ i3 : (Y, F) = setOnesForest X │ │ │ │ │ │ -o3 = (| 0 1 0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y , │ │ │ - | 1 0 0 x_3 | 1 4 3 4 0 5 2 │ │ │ - | 0 1 1 0 | "ring" => QQ[y ..y ] │ │ │ - | 1 0 1 0 | 0 7 │ │ │ +o3 = (| 0 1 0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y , │ │ │ + | 1 0 0 1 | 1 4 3 4 0 5 2 │ │ │ + | 0 x_4 1 0 | "ring" => QQ[y ..y ] │ │ │ + | 1 0 1 0 | 0 7 │ │ │ "vertices" => {y , y , y , y , y , y , y , y } │ │ │ 0 1 2 3 4 5 6 7 │ │ │ ------------------------------------------------------------------------ │ │ │ y }, {y , y }, {y , y }, {y , y }}}) │ │ │ - 5 2 6 3 6 0 7 │ │ │ + 6 3 6 0 7 1 7 │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ @@ -99,17 +99,17 @@ │ │ │ 
      i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
      │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -31,17 +31,17 @@ │ │ │ │ │ │ │ │ 6 5 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : (Y, T) = setOnesForest X; │ │ │ │ i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)}; │ │ │ │ i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1) │ │ │ │ │ │ │ │ - 2 2 2 2 2 2 2 2 2 2 │ │ │ │ -o5 = - x x x x x x + x x x x x x + x x x x x x x x │ │ │ │ - 1 4 6 7 10 11 2 3 5 8 10 11 2 3 5 6 7 8 9 12 │ │ │ │ + 2 2 2 2 2 2 2 2 2 2 │ │ │ │ +o5 = x x x x x x - x x x x x x + x x x x x x x x │ │ │ │ + 1 4 6 7 10 11 2 3 5 8 10 11 1 2 3 4 6 7 9 12 │ │ │ │ │ │ │ │ o5 : R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_e_t_O_n_e_s_F_o_r_e_s_t -- sets to 1 variables in a symbolic slack matrix which │ │ │ │ corresponding to edges of a spanning forest │ │ │ │ * _s_l_a_c_k_I_d_e_a_l -- computes the slack ideal │ │ │ │ * _s_y_m_b_o_l_i_c_S_l_a_c_k_M_a_t_r_i_x -- computes the symbolic slack matrix │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/html/_set__Ones__Forest.html │ │ │ @@ -95,23 +95,23 @@ │ │ │ 0 7 0 7 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : (Y, F) = setOnesForest X
│ │ │  
│ │ │ -o3 = (| 0 1 0 1   |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y ,
│ │ │ -      | 1 0 0 x_3 |                      1   4     3   4     0   5     2 
│ │ │ -      | 0 1 1 0   |        "ring" => QQ[y ..y ]
│ │ │ -      | 1 0 1 0   |                      0   7
│ │ │ +o3 = (| 0 1   0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y ,
│ │ │ +      | 1 0   0 1 |                      1   4     3   4     0   5     2 
│ │ │ +      | 0 x_4 1 0 |        "ring" => QQ[y ..y ]
│ │ │ +      | 1 0   1 0 |                      0   7
│ │ │                             "vertices" => {y , y , y , y , y , y , y , y }
│ │ │                                             0   1   2   3   4   5   6   7
│ │ │       ------------------------------------------------------------------------
│ │ │       y }, {y , y }, {y , y }, {y , y }}})
│ │ │ -      5     2   6     3   6     0   7
│ │ │ +      6     3   6     0   7     1   7
│ │ │  
│ │ │  o3 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,23 +36,23 @@ │ │ │ │ | x_6 0 x_7 0 | │ │ │ │ │ │ │ │ 4 4 │ │ │ │ o2 : Matrix (QQ[x ..x ]) <-- (QQ[x ..x ]) │ │ │ │ 0 7 0 7 │ │ │ │ i3 : (Y, F) = setOnesForest X │ │ │ │ │ │ │ │ -o3 = (| 0 1 0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y , │ │ │ │ - | 1 0 0 x_3 | 1 4 3 4 0 5 2 │ │ │ │ - | 0 1 1 0 | "ring" => QQ[y ..y ] │ │ │ │ - | 1 0 1 0 | 0 7 │ │ │ │ +o3 = (| 0 1 0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y , │ │ │ │ + | 1 0 0 1 | 1 4 3 4 0 5 2 │ │ │ │ + | 0 x_4 1 0 | "ring" => QQ[y ..y ] │ │ │ │ + | 1 0 1 0 | 0 7 │ │ │ │ "vertices" => {y , y , y , y , y , y , y , y } │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ y }, {y , y }, {y , y }, {y , y }}}) │ │ │ │ - 5 2 6 3 6 0 7 │ │ │ │ + 6 3 6 0 7 1 7 │ │ │ │ │ │ │ │ o3 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_r_a_p_h_F_r_o_m_S_l_a_c_k_M_a_t_r_i_x -- creates the vertex-edge incidence matrix for the │ │ │ │ bipartite non-incidence graph with adjacency matrix the given slack │ │ │ │ matrix │ │ │ │ * _s_l_a_c_k_I_d_e_a_l -- computes the slack ideal │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time degreeDeterminant n │ │ │ - -- used 0.00373186s (cpu); 6.8739e-05s (thread); 0s (gc) │ │ │ + -- used 0.00131975s (cpu); 6.4227e-05s (thread); 0s (gc) │ │ │ │ │ │ o2 = 6 │ │ │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ 0,0,0 1,2,1 │ │ │ │ │ │ i4 : time degree determinant M │ │ │ - -- used 0.181205s (cpu); 0.128803s (thread); 0s (gc) │ │ │ + -- used 0.282467s (cpu); 0.160176s (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 0.381139s (cpu); 0.268551s (thread); 0s (gc) │ │ │ + -- used 0.297348s (cpu); 0.201739s (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 0.0859612s (cpu); 0.0861238s (thread); 0s (gc) │ │ │ + -- used 0.108071s (cpu); 0.109538s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.303166s (cpu); 0.257137s (thread); 0s (gc) │ │ │ + -- used 0.383799s (cpu); 0.329861s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.348798s (cpu); 0.298015s (thread); 0s (gc) │ │ │ + -- used 0.36s (cpu); 0.349301s (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 0.148082s (cpu); 0.0957958s (thread); 0s (gc) │ │ │ + -- used 0.381939s (cpu); 0.132644s (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 0.48576s (cpu); 0.413779s (thread); 0s (gc) │ │ │ + -- used 0.551493s (cpu); 0.45391s (thread); 0s (gc) │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ 9257139493926586400187927813888 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Skew__Multidimensional__Matrix.out │ │ │ @@ -1,56 +1,56 @@ │ │ │ -- -*- M2-comint -*- hash: 6013064849487134497 │ │ │ │ │ │ i1 : genericSkewMultidimensionalMatrix(3,4) │ │ │ │ │ │ o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ - 3 2 3 0 2 0 │ │ │ + 1 0 1 2 0 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ - 3 2 3 1 2 1 3 │ │ │ + 1 0 1 3 0 3 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ - 0 3 1 0 1 2 0 │ │ │ + 2 1 3 2 3 0 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 2 1 0 1 │ │ │ + 0 3 2 3 │ │ │ │ │ │ o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ] │ │ │ 0 3 │ │ │ │ │ │ i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101) │ │ │ │ │ │ o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ - 1 0 1 2 0 2 │ │ │ + 3 2 3 0 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ - 1 0 1 3 0 3 1 │ │ │ + 3 2 3 1 2 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ - 2 1 3 2 3 0 2 │ │ │ + 0 3 1 0 1 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 0 3 2 3 │ │ │ + 2 1 0 1 │ │ │ │ │ │ ZZ │ │ │ o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ] │ │ │ 101 0 3 │ │ │ │ │ │ i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101,Variable=>"b") │ │ │ │ │ │ o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0, │ │ │ - 3 2 3 0 2 0 │ │ │ + 1 0 1 2 0 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b , │ │ │ - 3 2 3 1 2 1 3 │ │ │ + 1 0 1 3 0 3 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , │ │ │ - 0 3 1 0 1 2 0 │ │ │ + 2 1 3 2 3 0 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 2 1 0 1 │ │ │ + 0 3 2 3 │ │ │ │ │ │ ZZ │ │ │ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] │ │ │ 101 0 3 │ │ │ │ │ │ i4 : │ │ ├── ./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 2.2323s (cpu); 1.9812s (thread); 0s (gc) │ │ │ + -- used 2.37598s (cpu); 2.19176s (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 0.489848s (cpu); 0.431304s (thread); 0s (gc) │ │ │ + -- used 0.621184s (cpu); 0.548199s (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 0.0159836s (cpu); 0.0162134s (thread); 0s (gc) │ │ │ + -- used 0.0120254s (cpu); 0.0107307s (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 0.032098s (cpu); 0.0308024s (thread); 0s (gc) │ │ │ + -- used 0.101044s (cpu); 0.0609009s (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 0.0040004s (cpu); 0.00314361s (thread); 0s (gc) │ │ │ + -- used 0.00399765s (cpu); 0.00369965s (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 0.544959s (cpu); 0.487048s (thread); 0s (gc) │ │ │ + -- used 0.609335s (cpu); 0.499874s (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 0.00373186s (cpu); 6.8739e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00131975s (cpu); 6.4227e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -92,15 +92,15 @@
│ │ │  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 0.181205s (cpu); 0.128803s (thread); 0s (gc)
│ │ │ + -- used 0.282467s (cpu); 0.160176s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -15,23 +15,23 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : time degreeDeterminant n │ │ │ │ - -- used 0.00373186s (cpu); 6.8739e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00131975s (cpu); 6.4227e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 6 │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ │ 0,0,0 1,2,1 │ │ │ │ i4 : time degree determinant M │ │ │ │ - -- used 0.181205s (cpu); 0.128803s (thread); 0s (gc) │ │ │ │ + -- used 0.282467s (cpu); 0.160176s (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 0.381139s (cpu); 0.268551s (thread); 0s (gc)
│ │ │ + -- used 0.297348s (cpu); 0.201739s (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 0.381139s (cpu); 0.268551s (thread); 0s (gc) │ │ │ │ + -- used 0.297348s (cpu); 0.201739s (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 0.0859612s (cpu); 0.0861238s (thread); 0s (gc)
    │ │ │ + -- used 0.108071s (cpu); 0.109538s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.303166s (cpu); 0.257137s (thread); 0s (gc)
    │ │ │ + -- used 0.383799s (cpu); 0.329861s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.348798s (cpu); 0.298015s (thread); 0s (gc)
    │ │ │ + -- used 0.36s (cpu); 0.349301s (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 0.0859612s (cpu); 0.0861238s (thread); 0s (gc) │ │ │ │ + -- used 0.108071s (cpu); 0.109538s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.303166s (cpu); 0.257137s (thread); 0s (gc) │ │ │ │ + -- used 0.383799s (cpu); 0.329861s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.348798s (cpu); 0.298015s (thread); 0s (gc) │ │ │ │ + -- used 0.36s (cpu); 0.349301s (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 0.148082s (cpu); 0.0957958s (thread); 0s (gc)
│ │ │ + -- used 0.381939s (cpu); 0.132644s (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 0.48576s (cpu); 0.413779s (thread); 0s (gc)
│ │ │ + -- used 0.551493s (cpu); 0.45391s (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 0.148082s (cpu); 0.0957958s (thread); 0s (gc) │ │ │ │ + -- used 0.381939s (cpu); 0.132644s (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 0.48576s (cpu); 0.413779s (thread); 0s (gc) │ │ │ │ + -- used 0.551493s (cpu); 0.45391s (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/_generic__Skew__Multidimensional__Matrix.html │ │ │ @@ -75,65 +75,65 @@ │ │ │

    An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of the set $\{0,\ldots,n-1\}$ we have permute(M,s) == sign(s)*M.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : genericSkewMultidimensionalMatrix(3,4)
│ │ │  
│ │ │  o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              3    2        3       0        2   0           
│ │ │ +                              1    0        1       2        0   2           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         3   2                    3          1      2      1              3 
│ │ │ +         1   0                    1          3      0      3              1 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         0     3         1                    0    1                 2    0 
│ │ │ +         2     1         3                    2    3                 0    2 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           2       1        0   1
│ │ │ +           0       3        2   3
│ │ │  
│ │ │  o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ]
│ │ │                                                        0   3
    │ │ │
    │ │ │ 
    i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101)
│ │ │  
│ │ │  o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │                                                     ZZ
│ │ │  o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ]
│ │ │                                                    101  0   3
    │ │ │
    │ │ │ 
    i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101,Variable=>"b")
│ │ │  
│ │ │  o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0,
│ │ │ -                              3    2        3       0        2   0           
│ │ │ +                              1    0        1       2        0   2           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b ,
│ │ │ -         3   2                    3          1      2      1              3 
│ │ │ +         1   0                    1          3      0      3              1 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b ,
│ │ │ -         0     3         1                    0    1                 2    0 
│ │ │ +         2     1         3                    2    3                 0    2 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           2       1        0   1
│ │ │ +           0       3        2   3
│ │ │  
│ │ │                                                     ZZ
│ │ │  o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ]
│ │ │                                                    101  0   3
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,58 +17,58 @@ │ │ │ │ $d\times\cdots\times d$ ($n$ times). │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of │ │ │ │ the set $\{0,\ldots,n-1\}$ we have _p_e_r_m_u_t_e(M,s) == sign(s)*M. │ │ │ │ i1 : genericSkewMultidimensionalMatrix(3,4) │ │ │ │ │ │ │ │ o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ │ - 3 2 3 0 2 0 │ │ │ │ + 1 0 1 2 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ │ - 3 2 3 1 2 1 3 │ │ │ │ + 1 0 1 3 0 3 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ │ - 0 3 1 0 1 2 0 │ │ │ │ + 2 1 3 2 3 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 2 1 0 1 │ │ │ │ + 0 3 2 3 │ │ │ │ │ │ │ │ o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ] │ │ │ │ 0 3 │ │ │ │ i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101) │ │ │ │ │ │ │ │ o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ │ - 1 0 1 2 0 2 │ │ │ │ + 3 2 3 0 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ │ - 1 0 1 3 0 3 1 │ │ │ │ + 3 2 3 1 2 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ │ - 2 1 3 2 3 0 2 │ │ │ │ + 0 3 1 0 1 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 0 3 2 3 │ │ │ │ + 2 1 0 1 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ] │ │ │ │ 101 0 3 │ │ │ │ i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/ │ │ │ │ 101,Variable=>"b") │ │ │ │ │ │ │ │ o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0, │ │ │ │ - 3 2 3 0 2 0 │ │ │ │ + 1 0 1 2 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b , │ │ │ │ - 3 2 3 1 2 1 3 │ │ │ │ + 1 0 1 3 0 3 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , │ │ │ │ - 0 3 1 0 1 2 0 │ │ │ │ + 2 1 3 2 3 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 2 1 0 1 │ │ │ │ + 0 3 2 3 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] │ │ │ │ 101 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_i_c_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- make a generic multidimensional matrix │ │ │ │ of variables │ │ ├── ./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 2.2323s (cpu); 1.9812s (thread); 0s (gc)
│ │ │ + -- used 2.37598s (cpu); 2.19176s (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 2.2323s (cpu); 1.9812s (thread); 0s (gc)
│ │ │ │ + -- used 2.37598s (cpu); 2.19176s (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 0.489848s (cpu); 0.431304s (thread); 0s (gc)
│ │ │ + -- used 0.621184s (cpu); 0.548199s (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 0.0159836s (cpu); 0.0162134s (thread); 0s (gc)
│ │ │ + -- used 0.0120254s (cpu); 0.0107307s (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 0.00103605s (cpu); 0.00141134s (thread); 0s (gc)
│ │ │ │ + -- used 0.00394272s (cpu); 0.00142621s (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 0.00155363s (cpu); 0.00114943s (thread); 0s (gc)
│ │ │ │ + -- used 0.00266434s (cpu); 0.00149343s (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 0.00311517s (cpu); 0.0019859s (thread); 0s (gc)
│ │ │ │ + -- used 0.00383129s (cpu); 0.0029121s (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 0.032098s (cpu); 0.0308024s (thread); 0s (gc)
│ │ │ + -- used 0.101044s (cpu); 0.0609009s (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 0.0040004s (cpu); 0.00314361s (thread); 0s (gc)
│ │ │ + -- used 0.00399765s (cpu); 0.00369965s (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 0.544959s (cpu); 0.487048s (thread); 0s (gc)
    
│ │ │ + -- used 0.609335s (cpu); 0.499874s (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 0.489848s (cpu); 0.431304s (thread); 0s (gc)
│ │ │ │ + -- used 0.621184s (cpu); 0.548199s (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 0.0159836s (cpu); 0.0162134s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120254s (cpu); 0.0107307s (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 0.032098s (cpu); 0.0308024s (thread); 0s (gc)
│ │ │ │ + -- used 0.101044s (cpu); 0.0609009s (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 0.0040004s (cpu); 0.00314361s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399765s (cpu); 0.00369965s (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 0.544959s (cpu); 0.487048s (thread); 0s (gc)
│ │ │ │ + -- used 0.609335s (cpu); 0.499874s (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 0.00103605s (cpu); 0.00141134s (thread); 0s (gc)
│ │ │ + -- used 0.00394272s (cpu); 0.00142621s (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 0.00155363s (cpu); 0.00114943s (thread); 0s (gc)
│ │ │ + -- used 0.00266434s (cpu); 0.00149343s (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 0.00311517s (cpu); 0.0019859s (thread); 0s (gc)
│ │ │ + -- used 0.00383129s (cpu); 0.0029121s (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 0.29506s (cpu); 0.237494s (thread); 0s (gc)
│ │ │ + -- used 0.29387s (cpu); 0.247778s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}
│ │ │  
│ │ │  o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time seco = secondaryInvariants(genList,R);
│ │ │ - -- used 0.611783s (cpu); 0.446463s (thread); 0s (gc)
│ │ │ + -- used 0.697406s (cpu); 0.535393s (thread); 0s (gc)
│ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │              
    
│ │ │                
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00103605s (cpu); 0.00141134s (thread); 0s (gc)
│ │ │ + -- used 0.00394272s (cpu); 0.00142621s (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 0.00155363s (cpu); 0.00114943s (thread); 0s (gc)
│ │ │ + -- used 0.00266434s (cpu); 0.00149343s (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 0.00311517s (cpu); 0.0019859s (thread); 0s (gc)
│ │ │ + -- used 0.00383129s (cpu); 0.0029121s (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 0.29506s (cpu); 0.237494s (thread); 0s (gc)
│ │ │ + -- used 0.29387s (cpu); 0.247778s (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 0.29506s (cpu); 0.237494s (thread); 0s (gc)
│ │ │ │ + -- used 0.29387s (cpu); 0.247778s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time seco = secondaryInvariants(genList,R);
│ │ │ - -- used 0.611783s (cpu); 0.446463s (thread); 0s (gc)
│ │ │ + -- used 0.697406s (cpu); 0.535393s (thread); 0s (gc)
│ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}
│ │ │ │  
│ │ │ │  o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time seco = secondaryInvariants(genList,R);
│ │ │ │ - -- used 0.611783s (cpu); 0.446463s (thread); 0s (gc)
│ │ │ │ + -- used 0.697406s (cpu); 0.535393s (thread); 0s (gc)
│ │ │ │  (Partition{6}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │  (Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/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 2.80338s (cpu); 1.1067s (thread); 0s (gc)
│ │ │ + -- used 2.60786s (cpu); 1.16879s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 2.24131s (cpu); 1.07276s (thread); 0s (gc)
│ │ │ + -- used 2.62512s (cpu); 1.12447s (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 7.60177s (cpu); 4.0919s (thread); 0s (gc)
│ │ │ + -- used 8.16174s (cpu); 4.75768s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 4.34987s (cpu); 2.1208s (thread); 0s (gc)
│ │ │ + -- used 3.71089s (cpu); 1.89414s (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 0.472787s (cpu); 0.279498s (thread); 0s (gc)
│ │ │ + -- used 0.488348s (cpu); 0.327165s (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 14.2087s (cpu); 6.87568s (thread); 0s (gc)
│ │ │ + -- used 18.2275s (cpu); 7.30077s (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 0.548599s (cpu); 0.239649s (thread); 0s (gc)
│ │ │ + -- used 0.458649s (cpu); 0.269982s (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 0.41255s (cpu); 0.112853s (thread); 0s (gc)
│ │ │ + -- used 0.604095s (cpu); 0.231588s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1730220932418738713
│ │ │  
│ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 1.02407s (cpu); 0.461853s (thread); 0s (gc)
│ │ │ + -- used 0.989367s (cpu); 0.490447s (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 0.303761s (cpu); 0.216641s (thread); 0s (gc)
│ │ │ + -- used 0.342473s (cpu); 0.240442s (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 0.66827s (cpu); 0.380481s (thread); 0s (gc)
│ │ │ + -- used 0.662203s (cpu); 0.415964s (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 3.9881s (cpu); 2.4405s (thread); 0s (gc)
│ │ │ + -- used 4.02334s (cpu); 2.95181s (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 1.71935s (cpu); 0.721299s (thread); 0s (gc)
│ │ │ + -- used 1.61106s (cpu); 0.745611s (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 0.0119997s (cpu); 0.00917689s (thread); 0s (gc)
│ │ │ + -- used 0.0081382s (cpu); 0.00942669s (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 0.613775s (cpu); 0.18104s (thread); 0s (gc)
│ │ │ + -- used 0.930137s (cpu); 0.253666s (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 2.44435s (cpu); 1.66347s (thread); 0s (gc)
│ │ │ + -- used 1.91991s (cpu); 1.58814s (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 5.44337s (cpu); 2.88105s (thread); 0s (gc)
│ │ │ + -- used 6.3038s (cpu); 3.49591s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = specialGushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.19991s (cpu); 2.91461s (thread); 0s (gc)
│ │ │ + -- used 5.47963s (cpu); 3.04116s (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 5.59171s (cpu); 2.88459s (thread); 0s (gc)
│ │ │ + -- used 5.35286s (cpu); 2.98139s (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 1.17984s (cpu); 0.53319s (thread); 0s (gc)
│ │ │ + -- used 1.18995s (cpu); 0.641027s (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 2.80338s (cpu); 1.1067s (thread); 0s (gc)
│ │ │ + -- used 2.60786s (cpu); 1.16879s (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 2.80338s (cpu); 1.1067s (thread); 0s (gc)
│ │ │ │ + -- used 2.60786s (cpu); 1.16879s (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 2.24131s (cpu); 1.07276s (thread); 0s (gc)
│ │ │ + -- used 2.62512s (cpu); 1.12447s (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 2.24131s (cpu); 1.07276s (thread); 0s (gc)
│ │ │ │ + -- used 2.62512s (cpu); 1.12447s (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 7.60177s (cpu); 4.0919s (thread); 0s (gc)
│ │ │ + -- used 8.16174s (cpu); 4.75768s (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 7.60177s (cpu); 4.0919s (thread); 0s (gc)
│ │ │ │ + -- used 8.16174s (cpu); 4.75768s (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 4.34987s (cpu); 2.1208s (thread); 0s (gc)
│ │ │ + -- used 3.71089s (cpu); 1.89414s (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 0.472787s (cpu); 0.279498s (thread); 0s (gc)
│ │ │ + -- used 0.488348s (cpu); 0.327165s (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 4.34987s (cpu); 2.1208s (thread); 0s (gc)
│ │ │ │ + -- used 3.71089s (cpu); 1.89414s (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 0.472787s (cpu); 0.279498s (thread); 0s (gc)
│ │ │ │ + -- used 0.488348s (cpu); 0.327165s (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 14.2087s (cpu); 6.87568s (thread); 0s (gc)
│ │ │ + -- used 18.2275s (cpu); 7.30077s (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 0.548599s (cpu); 0.239649s (thread); 0s (gc)
│ │ │ + -- used 0.458649s (cpu); 0.269982s (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 14.2087s (cpu); 6.87568s (thread); 0s (gc)
│ │ │ │ + -- used 18.2275s (cpu); 7.30077s (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 0.548599s (cpu); 0.239649s (thread); 0s (gc)
│ │ │ │ + -- used 0.458649s (cpu); 0.269982s (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 0.41255s (cpu); 0.112853s (thread); 0s (gc)
│ │ │ + -- used 0.604095s (cpu); 0.231588s (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 0.41255s (cpu); 0.112853s (thread); 0s (gc) │ │ │ │ + -- used 0.604095s (cpu); 0.231588s (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 1.02407s (cpu); 0.461853s (thread); 0s (gc)
│ │ │ + -- used 0.989367s (cpu); 0.490447s (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 1.02407s (cpu); 0.461853s (thread); 0s (gc) │ │ │ │ + -- used 0.989367s (cpu); 0.490447s (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 0.303761s (cpu); 0.216641s (thread); 0s (gc)
│ │ │ + -- used 0.342473s (cpu); 0.240442s (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 0.303761s (cpu); 0.216641s (thread); 0s (gc)
│ │ │ │ + -- used 0.342473s (cpu); 0.240442s (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 0.66827s (cpu); 0.380481s (thread); 0s (gc)
│ │ │ + -- used 0.662203s (cpu); 0.415964s (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 0.66827s (cpu); 0.380481s (thread); 0s (gc)
│ │ │ │ + -- used 0.662203s (cpu); 0.415964s (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 3.9881s (cpu); 2.4405s (thread); 0s (gc)
│ │ │ + -- used 4.02334s (cpu); 2.95181s (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 3.9881s (cpu); 2.4405s (thread); 0s (gc)
│ │ │ │ + -- used 4.02334s (cpu); 2.95181s (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 1.71935s (cpu); 0.721299s (thread); 0s (gc)
│ │ │ + -- used 1.61106s (cpu); 0.745611s (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 1.71935s (cpu); 0.721299s (thread); 0s (gc)
│ │ │ │ + -- used 1.61106s (cpu); 0.745611s (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 0.0119997s (cpu); 0.00917689s (thread); 0s (gc)
│ │ │ + -- used 0.0081382s (cpu); 0.00942669s (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 0.613775s (cpu); 0.18104s (thread); 0s (gc)
│ │ │ + -- used 0.930137s (cpu); 0.253666s (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 0.0119997s (cpu); 0.00917689s (thread); 0s (gc) │ │ │ │ + -- used 0.0081382s (cpu); 0.00942669s (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 0.613775s (cpu); 0.18104s (thread); 0s (gc) │ │ │ │ + -- used 0.930137s (cpu); 0.253666s (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 2.44435s (cpu); 1.66347s (thread); 0s (gc)
│ │ │ + -- used 1.91991s (cpu); 1.58814s (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 5.44337s (cpu); 2.88105s (thread); 0s (gc)
│ │ │ + -- used 6.3038s (cpu); 3.49591s (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 2.44435s (cpu); 1.66347s (thread); 0s (gc) │ │ │ │ + -- used 1.91991s (cpu); 1.58814s (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 5.44337s (cpu); 2.88105s (thread); 0s (gc) │ │ │ │ + -- used 6.3038s (cpu); 3.49591s (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 5.19991s (cpu); 2.91461s (thread); 0s (gc)
│ │ │ + -- used 5.47963s (cpu); 3.04116s (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 5.19991s (cpu); 2.91461s (thread); 0s (gc) │ │ │ │ + -- used 5.47963s (cpu); 3.04116s (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 5.59171s (cpu); 2.88459s (thread); 0s (gc)
│ │ │ + -- used 5.35286s (cpu); 2.98139s (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 5.59171s (cpu); 2.88459s (thread); 0s (gc) │ │ │ │ + -- used 5.35286s (cpu); 2.98139s (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 1.17984s (cpu); 0.53319s (thread); 0s (gc)
│ │ │ + -- used 1.18995s (cpu); 0.641027s (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 1.17984s (cpu); 0.53319s (thread); 0s (gc)
│ │ │ │ + -- used 1.18995s (cpu); 0.641027s (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 = {Digraph, Graph, Bigraph}
│ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  o2 : HashTable
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Digraph, Graph, Bigraph}
│ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │  
│ │ │  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 = {Digraph, Graph, Bigraph}
│ │ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │ │  
│ │ │ │  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/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10276-0/0
│ │ │ +o1 = /tmp/M2-10316-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10276-0/0.in
│ │ │ +o2 = /tmp/M2-10316-0/0.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/M2-10276-0/0
│ │ │ + -- generating /tmp/M2-10316-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.06. 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/M2-10276-0/0
    │ │ │ +o1 = /tmp/M2-10316-0/0 │ │ │
    │ │ │ 
    i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10276-0/0.in
    │ │ │ +o2 = /tmp/M2-10316-0/0.in │ │ │
    │ │ │ 
    i3 : template << "@M2BANNER@" << endl << endl;
    │ │ │
    │ │ │ 
    i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10276-0/0
    │ │ │ + -- generating /tmp/M2-10316-0/0 │ │ │
    │ │ │ 
    i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.06. 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/M2-10276-0/0
│ │ │ │ +o1 = /tmp/M2-10316-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10276-0/0.in
│ │ │ │ +o2 = /tmp/M2-10316-0/0.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/M2-10276-0/0
│ │ │ │ + -- generating /tmp/M2-10316-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.25.06. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./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 0.328526s (cpu); 0.210081s (thread); 0s (gc)
│ │ │ + -- used 0.283245s (cpu); 0.183829s (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 0.0614469s (cpu); 0.0598898s (thread); 0s (gc)
│ │ │ + -- used 0.0399929s (cpu); 0.0386639s (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 0.328526s (cpu); 0.210081s (thread); 0s (gc)
│ │ │ + -- used 0.283245s (cpu); 0.183829s (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 0.0614469s (cpu); 0.0598898s (thread); 0s (gc)
│ │ │ + -- used 0.0399929s (cpu); 0.0386639s (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 0.328526s (cpu); 0.210081s (thread); 0s (gc) │ │ │ │ + -- used 0.283245s (cpu); 0.183829s (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 0.0614469s (cpu); 0.0598898s (thread); 0s (gc) │ │ │ │ + -- used 0.0399929s (cpu); 0.0386639s (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 0.266395s (cpu); 0.168731s (thread); 0s (gc) │ │ │ + -- used 0.286797s (cpu); 0.165516s (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 0.266395s (cpu); 0.168731s (thread); 0s (gc)
    │ │ │ + -- used 0.286797s (cpu); 0.165516s (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 0.266395s (cpu); 0.168731s (thread); 0s (gc)
│ │ │ │ + -- used 0.286797s (cpu); 0.165516s (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 0.7587s (cpu); 0.604654s (thread); 0s (gc)
│ │ │ + -- used 0.860494s (cpu); 0.705608s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.39203s (cpu); 2.02854s (thread); 0s (gc)
│ │ │ + -- used 2.6853s (cpu); 2.29791s (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 0.00399968s (cpu); 0.0019853s (thread); 0s (gc)
│ │ │ + -- used 0.00402112s (cpu); 0.00219978s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00361386s (cpu); 0.00459849s (thread); 0s (gc)
│ │ │ + -- used 0.00205639s (cpu); 0.00527305s (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 0.128753s (cpu); 0.0735572s (thread); 0s (gc)
│ │ │ + -- used 0.126868s (cpu); 0.0606938s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.18033s (cpu); 1.26071s (thread); 0s (gc)
│ │ │ + -- used 2.45205s (cpu); 1.40901s (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 0.0687823s (cpu); 0.0683784s (thread); 0s (gc)
│ │ │ + -- used 0.0837647s (cpu); 0.0836872s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.152167s (cpu); 0.0866231s (thread); 0s (gc)
│ │ │ + -- used 0.161178s (cpu); 0.101474s (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 0.924719s (cpu); 0.726837s (thread); 0s (gc)
│ │ │ + -- used 1.05813s (cpu); 0.876717s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.367014s (cpu); 0.246319s (thread); 0s (gc)
│ │ │ + -- used 0.397651s (cpu); 0.272125s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -79,20 +79,20 @@
│ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ - -- used 0.120814s (cpu); 0.0560466s (thread); 0s (gc)
│ │ │ + -- used 0.153867s (cpu); 0.0837357s (thread); 0s (gc)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.216312s (cpu); 0.163117s (thread); 0s (gc)
│ │ │ + -- used 0.27573s (cpu); 0.206983s (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 0.273002s (cpu); 0.148856s (thread); 0s (gc)
│ │ │ + -- used 0.343884s (cpu); 0.222019s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.411962s (cpu); 0.233772s (thread); 0s (gc)
│ │ │ + -- used 0.52849s (cpu); 0.315012s (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 0.7587s (cpu); 0.604654s (thread); 0s (gc)
│ │ │ + -- used 0.860494s (cpu); 0.705608s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.39203s (cpu); 2.02854s (thread); 0s (gc)
│ │ │ + -- used 2.6853s (cpu); 2.29791s (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 0.7587s (cpu); 0.604654s (thread); 0s (gc)
│ │ │ │ + -- used 0.860494s (cpu); 0.705608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.39203s (cpu); 2.02854s (thread); 0s (gc)
│ │ │ │ + -- used 2.6853s (cpu); 2.29791s (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 0.00399968s (cpu); 0.0019853s (thread); 0s (gc)
│ │ │ + -- used 0.00402112s (cpu); 0.00219978s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00361386s (cpu); 0.00459849s (thread); 0s (gc)
│ │ │ + -- used 0.00205639s (cpu); 0.00527305s (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 0.00399968s (cpu); 0.0019853s (thread); 0s (gc)
│ │ │ │ + -- used 0.00402112s (cpu); 0.00219978s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00361386s (cpu); 0.00459849s (thread); 0s (gc)
│ │ │ │ + -- used 0.00205639s (cpu); 0.00527305s (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 0.128753s (cpu); 0.0735572s (thread); 0s (gc)
│ │ │ + -- used 0.126868s (cpu); 0.0606938s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.18033s (cpu); 1.26071s (thread); 0s (gc)
│ │ │ + -- used 2.45205s (cpu); 1.40901s (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 0.0687823s (cpu); 0.0683784s (thread); 0s (gc)
│ │ │ + -- used 0.0837647s (cpu); 0.0836872s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.152167s (cpu); 0.0866231s (thread); 0s (gc)
│ │ │ + -- used 0.161178s (cpu); 0.101474s (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 0.924719s (cpu); 0.726837s (thread); 0s (gc)
│ │ │ + -- used 1.05813s (cpu); 0.876717s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.367014s (cpu); 0.246319s (thread); 0s (gc)
│ │ │ + -- used 0.397651s (cpu); 0.272125s (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 0.128753s (cpu); 0.0735572s (thread); 0s (gc)
│ │ │ │ + -- used 0.126868s (cpu); 0.0606938s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.18033s (cpu); 1.26071s (thread); 0s (gc)
│ │ │ │ + -- used 2.45205s (cpu); 1.40901s (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 0.0687823s (cpu); 0.0683784s (thread); 0s (gc)
│ │ │ │ + -- used 0.0837647s (cpu); 0.0836872s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.152167s (cpu); 0.0866231s (thread); 0s (gc)
│ │ │ │ + -- used 0.161178s (cpu); 0.101474s (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 0.924719s (cpu); 0.726837s (thread); 0s (gc)
│ │ │ │ + -- used 1.05813s (cpu); 0.876717s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.367014s (cpu); 0.246319s (thread); 0s (gc)
│ │ │ │ + -- used 0.397651s (cpu); 0.272125s (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
│ │ │ @@ -272,24 +272,24 @@
│ │ │              
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ - -- used 0.120814s (cpu); 0.0560466s (thread); 0s (gc)
│ │ │ + -- used 0.153867s (cpu); 0.0837357s (thread); 0s (gc)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.216312s (cpu); 0.163117s (thread); 0s (gc)
│ │ │ + -- used 0.27573s (cpu); 0.206983s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : debugLevel = 0;
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -112,21 +112,21 @@
│ │ │ │  also use the option DepthOfSearch to increase the depth of search.
│ │ │ │  i24 : S = ZZ/7[x,y,z,u,v,w];
│ │ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : debugLevel = 1;
│ │ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ │ - -- used 0.120814s (cpu); 0.0560466s (thread); 0s (gc)
│ │ │ │ + -- used 0.153867s (cpu); 0.0837357s (thread); 0s (gc)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.216312s (cpu); 0.163117s (thread); 0s (gc)
│ │ │ │ + -- used 0.27573s (cpu); 0.206983s (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 0.273002s (cpu); 0.148856s (thread); 0s (gc)
│ │ │ + -- used 0.343884s (cpu); 0.222019s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.411962s (cpu); 0.233772s (thread); 0s (gc)
│ │ │ + -- used 0.52849s (cpu); 0.315012s (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 0.273002s (cpu); 0.148856s (thread); 0s (gc)
│ │ │ │ + -- used 0.343884s (cpu); 0.222019s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.411962s (cpu); 0.233772s (thread); 0s (gc)
│ │ │ │ + -- used 0.52849s (cpu); 0.315012s (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
│ │ │ @@ -17,18 +17,16 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -39,16 +37,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o4 : LineageTable
│ │ │  
│ │ │  i5 : minimize T
│ │ │  
│ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o5 = 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/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -2,17 +2,15 @@
│ │ │  
│ │ │  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
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = 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/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -4,16 +4,14 @@
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │                                     3
│ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ -                                    2
│ │ │ -                  ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -25,15 +23,14 @@
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │  o4 = 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/_tgb.out
│ │ │ @@ -6,52 +6,48 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                                         2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), (0, 1)), 2), 1) => -31y z      }
│ │ │ -                                                       2 8
│ │ │ -                  (((((0, 1), 2), (0, 1)), 2), 2) => 9y z
│ │ │ -                                                    3 6     2 12
│ │ │ -                  ((((0, 1), 2), (0, 1)), 2) => - 8y z  + 9y z
│ │ │ -                                                2 11      2 10
│ │ │ -                  ((((0, 1), 2), 3), 1) => - 15y z   + 17y z
│ │ │ -                                             2 11      2 10
│ │ │ -                  ((((0, 1), 2), 3), 2) => 9y z   + 28y z
│ │ │ -                                              2 10      2 9
│ │ │ -                  ((((0, 1), 2), 3), 3) => 28y z   - 25y z
│ │ │ -                                              3 10      3 6
│ │ │ -                  (((0, 1), 2), (0, 1)) => 22y z   - 25y z
│ │ │ -                                           3 9     3 6
│ │ │ -                  (((0, 1), 2), 3) => - 29y z  - 9y z
│ │ │ -                                                    2 12      2 9
│ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => - 2y z   - 25y z
│ │ │ -                                           2 13     2 9
│ │ │ -                  (((0, 1), 3), 2) => - 16y z   + 9y z
│ │ │ -                                   4 4     3 7
│ │ │ -                  ((0, 1), 2) => 9y z  - 6y z
│ │ │ -                                     3 7      3 6
│ │ │ -                  ((0, 1), 3) => - 6y z  + 27y z
│ │ │ -                                                              2 5
│ │ │ -                  ((0, 2), ((((0, 1), 2), (0, 1)), 2)) => -13y z
│ │ │ -                                               3 4
│ │ │ -                  ((0, 2), ((0, 1), 2)) => -13y z
│ │ │ -                                   2 7
│ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ +                                         3 4      2 4
│ │ │ +o4 = LineageTable{((0, 2), (0, 3)) => 39y z  + 22y z }
│ │ │ +                                      3 6    3 5
│ │ │ +                  ((0, 2), 1) => - 11y z  - y z
│ │ │ +                                    3 7      2 7
│ │ │ +                  ((0, 2), 3) => 27y z  + 23y z
│ │ │ +                                    4 7      3 5
│ │ │ +                  ((0, 3), 1) => 14y z  - 11y z
│ │ │ +                                   4 4      2 6
│ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ +                                     2 4
│ │ │ +                  ((1, 2), 2) => -47y z
│ │ │ +                                         2 6      2 4
│ │ │ +                  ((2, 3), (0, 2)) => 36y z  - 22y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((2, 3), (0, 3)) => 36y z  + 40y z
│ │ │ +                                      3 5      2 7
│ │ │ +                  ((2, 3), 1) => - 36y z  - 41y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((2, 3), 2) => 24y z  + 19y z
│ │ │ +                                    2 5      2 4
│ │ │ +                  ((2, 3), 3) => 24y z  + 48y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5
│ │ │ -                  (0, 3) => 28y z
│ │ │ -                               2 4
│ │ │ -                  (2, 3) => -9y 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
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │ +                               5      2 4
│ │ │ +                  (2, 3) => 25y z - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -96,18 +96,16 @@
│ │ │            
    By default, the option is false. The basis can also be minimized after the distributed computation is finished:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -121,16 +119,15 @@
│ │ │  o4 : LineageTable
    
│ │ │              
    
│ │ │                
    i5 : minimize T
│ │ │  
│ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o5 = 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 {}
│ │ │ │ @@ -28,18 +28,16 @@
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  By default, the option is false. The basis can also be minimized after the
│ │ │ │  distributed computation is finished:
│ │ │ │  i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   3
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -49,16 +47,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o4 : LineageTable
│ │ │ │  i5 : minimize T
│ │ │ │  
│ │ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o5 = 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/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -86,17 +86,15 @@
│ │ │                
    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
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = 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 {}
│ │ │ │ @@ -19,17 +19,15 @@
│ │ │ │  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
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │ +o3 = 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/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -84,16 +84,14 @@
│ │ │            
    
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │                                     3
│ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ -                                    2
│ │ │ -                  ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -108,15 +106,14 @@
│ │ │              
    
│ │ │                
    i4 : minimize T
│ │ │  
│ │ │  o4 = 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 {}
│ │ │ │ @@ -21,16 +21,14 @@
│ │ │ │  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, 2), 0) => -c      }
│ │ │ │ -                                    2
│ │ │ │ -                  ((0, 2), 1) => a*c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,15 +39,14 @@
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │  o4 = 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/_tgb.html
│ │ │ @@ -95,52 +95,48 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
    
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                                         2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), (0, 1)), 2), 1) => -31y z      }
│ │ │ -                                                       2 8
│ │ │ -                  (((((0, 1), 2), (0, 1)), 2), 2) => 9y z
│ │ │ -                                                    3 6     2 12
│ │ │ -                  ((((0, 1), 2), (0, 1)), 2) => - 8y z  + 9y z
│ │ │ -                                                2 11      2 10
│ │ │ -                  ((((0, 1), 2), 3), 1) => - 15y z   + 17y z
│ │ │ -                                             2 11      2 10
│ │ │ -                  ((((0, 1), 2), 3), 2) => 9y z   + 28y z
│ │ │ -                                              2 10      2 9
│ │ │ -                  ((((0, 1), 2), 3), 3) => 28y z   - 25y z
│ │ │ -                                              3 10      3 6
│ │ │ -                  (((0, 1), 2), (0, 1)) => 22y z   - 25y z
│ │ │ -                                           3 9     3 6
│ │ │ -                  (((0, 1), 2), 3) => - 29y z  - 9y z
│ │ │ -                                                    2 12      2 9
│ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => - 2y z   - 25y z
│ │ │ -                                           2 13     2 9
│ │ │ -                  (((0, 1), 3), 2) => - 16y z   + 9y z
│ │ │ -                                   4 4     3 7
│ │ │ -                  ((0, 1), 2) => 9y z  - 6y z
│ │ │ -                                     3 7      3 6
│ │ │ -                  ((0, 1), 3) => - 6y z  + 27y z
│ │ │ -                                                              2 5
│ │ │ -                  ((0, 2), ((((0, 1), 2), (0, 1)), 2)) => -13y z
│ │ │ -                                               3 4
│ │ │ -                  ((0, 2), ((0, 1), 2)) => -13y z
│ │ │ -                                   2 7
│ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ +                                         3 4      2 4
│ │ │ +o4 = LineageTable{((0, 2), (0, 3)) => 39y z  + 22y z }
│ │ │ +                                      3 6    3 5
│ │ │ +                  ((0, 2), 1) => - 11y z  - y z
│ │ │ +                                    3 7      2 7
│ │ │ +                  ((0, 2), 3) => 27y z  + 23y z
│ │ │ +                                    4 7      3 5
│ │ │ +                  ((0, 3), 1) => 14y z  - 11y z
│ │ │ +                                   4 4      2 6
│ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ +                                     2 4
│ │ │ +                  ((1, 2), 2) => -47y z
│ │ │ +                                         2 6      2 4
│ │ │ +                  ((2, 3), (0, 2)) => 36y z  - 22y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((2, 3), (0, 3)) => 36y z  + 40y z
│ │ │ +                                      3 5      2 7
│ │ │ +                  ((2, 3), 1) => - 36y z  - 41y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((2, 3), 2) => 24y z  + 19y z
│ │ │ +                                    2 5      2 4
│ │ │ +                  ((2, 3), 3) => 24y z  + 48y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5
│ │ │ -                  (0, 3) => 28y z
│ │ │ -                               2 4
│ │ │ -                  (2, 3) => -9y 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
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │ +                               5      2 4
│ │ │ +                  (2, 3) => 25y z - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,52 +26,48 @@
│ │ │ │  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 9
│ │ │ │ -o4 = LineageTable{(((((0, 1), 2), (0, 1)), 2), 1) => -31y z      }
│ │ │ │ -                                                       2 8
│ │ │ │ -                  (((((0, 1), 2), (0, 1)), 2), 2) => 9y z
│ │ │ │ -                                                    3 6     2 12
│ │ │ │ -                  ((((0, 1), 2), (0, 1)), 2) => - 8y z  + 9y z
│ │ │ │ -                                                2 11      2 10
│ │ │ │ -                  ((((0, 1), 2), 3), 1) => - 15y z   + 17y z
│ │ │ │ -                                             2 11      2 10
│ │ │ │ -                  ((((0, 1), 2), 3), 2) => 9y z   + 28y z
│ │ │ │ -                                              2 10      2 9
│ │ │ │ -                  ((((0, 1), 2), 3), 3) => 28y z   - 25y z
│ │ │ │ -                                              3 10      3 6
│ │ │ │ -                  (((0, 1), 2), (0, 1)) => 22y z   - 25y z
│ │ │ │ -                                           3 9     3 6
│ │ │ │ -                  (((0, 1), 2), 3) => - 29y z  - 9y z
│ │ │ │ -                                                    2 12      2 9
│ │ │ │ -                  (((0, 1), 3), ((0, 1), 2)) => - 2y z   - 25y z
│ │ │ │ -                                           2 13     2 9
│ │ │ │ -                  (((0, 1), 3), 2) => - 16y z   + 9y z
│ │ │ │ -                                   4 4     3 7
│ │ │ │ -                  ((0, 1), 2) => 9y z  - 6y z
│ │ │ │ -                                     3 7      3 6
│ │ │ │ -                  ((0, 1), 3) => - 6y z  + 27y z
│ │ │ │ -                                                              2 5
│ │ │ │ -                  ((0, 2), ((((0, 1), 2), (0, 1)), 2)) => -13y z
│ │ │ │ -                                               3 4
│ │ │ │ -                  ((0, 2), ((0, 1), 2)) => -13y z
│ │ │ │ -                                   2 7
│ │ │ │ -                  ((0, 2), 2) => 9y z
│ │ │ │ +                                         3 4      2 4
│ │ │ │ +o4 = LineageTable{((0, 2), (0, 3)) => 39y z  + 22y z }
│ │ │ │ +                                      3 6    3 5
│ │ │ │ +                  ((0, 2), 1) => - 11y z  - y z
│ │ │ │ +                                    3 7      2 7
│ │ │ │ +                  ((0, 2), 3) => 27y z  + 23y z
│ │ │ │ +                                    4 7      3 5
│ │ │ │ +                  ((0, 3), 1) => 14y z  - 11y z
│ │ │ │ +                                   4 4      2 6
│ │ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ │ +                                     2 4
│ │ │ │ +                  ((1, 2), 2) => -47y z
│ │ │ │ +                                         2 6      2 4
│ │ │ │ +                  ((2, 3), (0, 2)) => 36y z  - 22y z
│ │ │ │ +                                         2 5      2 4
│ │ │ │ +                  ((2, 3), (0, 3)) => 36y z  + 40y z
│ │ │ │ +                                      3 5      2 7
│ │ │ │ +                  ((2, 3), 1) => - 36y z  - 41y z
│ │ │ │ +                                    2 6      2 4
│ │ │ │ +                  ((2, 3), 2) => 24y z  + 19y z
│ │ │ │ +                                    2 5      2 4
│ │ │ │ +                  ((2, 3), 3) => 24y z  + 48y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ -                               5
│ │ │ │ -                  (0, 3) => 28y z
│ │ │ │ -                               2 4
│ │ │ │ -                  (2, 3) => -9y 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
│ │ │ │ +                                 7       2 6
│ │ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │ │ +                               5      2 4
│ │ │ │ +                  (2, 3) => 25y z - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ │ │                    2 => 5x*y z  + 9x*z
│ │ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │       | 0 1 2 0 2 0 |
│ │ │       | 1 1 1 1 1 1 |
│ │ │  
│ │ │                4       6
│ │ │  o3 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i4 : time edDeg(A)
│ │ │ - -- used 1.12614s (cpu); 0.750032s (thread); 0s (gc)
│ │ │ + -- used 1.25375s (cpu); 0.881484s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │ @@ -47,15 +47,15 @@
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ - -- used 4.85991s (cpu); 3.15661s (thread); 0s (gc)
│ │ │ + -- used 4.7789s (cpu); 3.28475s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │                4       6
│ │ │  o3 : Matrix ZZ  <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time edDeg(A)
│ │ │ - -- used 1.12614s (cpu); 0.750032s (thread); 0s (gc)
│ │ │ + -- used 1.25375s (cpu); 0.881484s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
    
│ │ │                
    i5 : time edDeg(A,ForceAmat=>true)
│ │ │ - -- used 4.85991s (cpu); 3.15661s (thread); 0s (gc)
│ │ │ + -- used 4.7789s (cpu); 3.28475s (thread); 0s (gc)
│ │ │  
│ │ │  The toric variety has degree = 28
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,30 +57,30 @@
│ │ │ │       | 3 5 0 2 1 3 |
│ │ │ │       | 0 1 2 0 2 0 |
│ │ │ │       | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │                4       6
│ │ │ │  o3 : Matrix ZZ  <-- ZZ
│ │ │ │  i4 : time edDeg(A)
│ │ │ │ - -- used 1.12614s (cpu); 0.750032s (thread); 0s (gc)
│ │ │ │ + -- used 1.25375s (cpu); 0.881484s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │ - -- used 4.85991s (cpu); 3.15661s (thread); 0s (gc)
│ │ │ │ + -- used 4.7789s (cpu); 3.28475s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/example-output/___Triangular__Sets.out
│ │ │ @@ -4,16 +4,16 @@
│ │ │  
│ │ │  i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : triangularize I
│ │ │  
│ │ │ -o3 = {{c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c,
│ │ │ +o3 = {{a*d - b*c, e, f} / d, {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     d, g, h}, {c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c,
│ │ │ +     {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     c*f - d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h}
│ │ │ +     f}, {b, d, f, h}, {c, d, g, h}, {c, d, f, h}, {b, d, e, f}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/example-output/_triangularize.out
│ │ │ @@ -2,25 +2,25 @@
│ │ │  
│ │ │  i1 : R = QQ[a..h, MonomialOrder=>Lex];
│ │ │  
│ │ │  i2 : F = {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │  
│ │ │  i3 : TT = triangularize(R,F,{})
│ │ │  
│ │ │ -o3 = {{c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f -
│ │ │ +o3 = {{c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d -
│ │ │ +     d, g, h}, {c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c, d, g, h}}
│ │ │ +     c*f - d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : first TT
│ │ │  
│ │ │ -o4 = {c, d, f, h}
│ │ │ +o4 = {c, d, e, f}
│ │ │  
│ │ │  o4 : TriaSystem
│ │ │  
│ │ │  i5 : H = {b,d};
│ │ │  
│ │ │  i6 : triangularize(R,F,H)
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/html/_triangularize.html
│ │ │ @@ -92,28 +92,28 @@
│ │ │                
    i2 : F = {a*d - b*c, c*f - d*e, e*h - f*g};
    
│ │ │              
    
│ │ │                
    i3 : TT = triangularize(R,F,{})
│ │ │  
│ │ │ -o3 = {{c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f -
│ │ │ +o3 = {{c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d -
│ │ │ +     d, g, h}, {c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c, d, g, h}}
│ │ │ +     c*f - d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : first TT
│ │ │  
│ │ │ -o4 = {c, d, f, h}
│ │ │ +o4 = {c, d, e, f}
│ │ │  
│ │ │  o4 : TriaSystem
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We now include some inequations.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,24 +30,24 @@
│ │ │ │  U_1)\cup\cdots\cup Z(T_r/U_r).$$ These simpler sets, called _t_r_i_a_n_g_u_l_a_r_ _s_y_s_t_e_m_s,
│ │ │ │  have very nice algorithmic properties.
│ │ │ │  As a first example we consider a case without inequations ($H=\emptyset$).
│ │ │ │  i1 : R = QQ[a..h, MonomialOrder=>Lex];
│ │ │ │  i2 : F = {a*d - b*c, c*f - d*e, e*h - f*g};
│ │ │ │  i3 : TT = triangularize(R,F,{})
│ │ │ │  
│ │ │ │ -o3 = {{c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f -
│ │ │ │ +o3 = {{c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d -
│ │ │ │ +     d, g, h}, {c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c, d, g, h}}
│ │ │ │ +     c*f - d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : first TT
│ │ │ │  
│ │ │ │ -o4 = {c, d, f, h}
│ │ │ │ +o4 = {c, d, e, f}
│ │ │ │  
│ │ │ │  o4 : TriaSystem
│ │ │ │  We now include some inequations.
│ │ │ │  i5 : H = {b,d};
│ │ │ │  i6 : triangularize(R,F,H)
│ │ │ │  
│ │ │ │  o6 = {{a*d - b*c, c*f - d*e, g, h} / {b, d, f}, {a*d - b*c, c*f - d*e, e*h -
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/html/index.html
│ │ │ @@ -64,19 +64,19 @@
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : triangularize I
│ │ │  
│ │ │ -o3 = {{c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c,
│ │ │ +o3 = {{a*d - b*c, e, f} / d, {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     d, g, h}, {c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c,
│ │ │ +     {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d - b*c, c*f - d*e, g, h} / {d,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     c*f - d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h}
│ │ │ +     f}, {b, d, f, h}, {c, d, g, h}, {c, d, f, h}, {b, d, e, f}}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    The method triangularize is implemented in M2 only for monomial and binomial ideals. For the general case we interface to Maple.

    This package also provides methods for manipulating triangular sets:
    │ │ │
    │ │ │
    │ │ │

    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 = {| 2 |, | -1 |, | 1 |} │ │ │ │ - | -1 | | 2 | | 1 | │ │ │ │ +o2 = {| 1 |, | 2 |, | -1 |} │ │ │ │ + | 1 | | -1 | | 2 | │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : v=weight(R,{1,2}) │ │ │ │ │ │ │ │ o3 = | 1 | │ │ │ │ | 2 | │ │ │ │ │ │ │ │ 2 │ │ │ │ o3 : ZZ │ │ │ │ i4 : eval(R,v,L#0) │ │ │ │ │ │ │ │ -o4 = 1 │ │ │ │ +o4 = 3 │ │ │ │ i5 : eval(R,v,L#1) │ │ │ │ │ │ │ │ -o5 = 2 │ │ │ │ +o5 = 1 │ │ │ │ i6 : eval(R,v,L#2) │ │ │ │ │ │ │ │ -o6 = 3 │ │ │ │ +o6 = 2 │ │ │ │ ********** 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- │ │ │ │ 1.25.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2488:0. │ │ ├── ./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, | -1 |}, {1, |  0 |}, {2, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}, {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 |}, {{1, |  2 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{2, |  2 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        |  2 |       | -1 |       |  1 |                                              | -1 |        |  1 |       | -1 |       | -1 |                                            | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | -1 |       |  2 |       | -1 |                                              |  2 |        |  1 |       |  0 |       |  2 |                                            |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                              |  2 |        |  0 |       |  2 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        |  0 |       | -1 |                                              | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │ +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
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 2}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}}, {{, {{, 1}, {, 3}}}, {, {{, 0}, {, 1}}}, {, {{, 1}, {, 2}}}, {, {{, 0}, {, 2}}}, {, {{, 2}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +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
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    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, {{2, 0}, {3, 1}, {121, 2}}}}, {{1213, {{232, 0}, {1, 2}, {3, 4}}}, {2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}}, {{213, {{1, 1}, {3, 3}}}, {232, {{3, 0}, {2, 1}}}, {123, {{12321, 1}, {3, 2}}}, {132, {{121, 0}, {232, 2}}}, {121, {{1, 2}, {2, 3}}}}, {{32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}, {21, {{1, 0}}}}, {{2, {}}}}
│ │ │ +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
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -33,74 +33,74 @@ │ │ │ │ o3 = WeylGroupElement{RootSystem{...8...}, | -1 |} │ │ │ │ | -2 | │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o3 : WeylGroupElement │ │ │ │ i4 : myInterval=intervalBruhat(w1,w2) │ │ │ │ │ │ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | - │ │ │ │ -1 |}, {1, | 0 |}, {2, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ -2 |}, {{0, | -1 |}, {2, | 2 |}, {4, | 0 |}}}, {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 |}, {{1, | 2 |}, {3, | │ │ │ │ -0 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{0, | 0 |}, {1, | - │ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | 0 │ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {2, | - │ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{2, | 2 |}, {3, | - │ │ │ │ -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | -1 |}}}, │ │ │ │ +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...}, | 1 |}, {{0, | 2 |}}}}, { │ │ │ │ +{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} │ │ │ │ - | -2 | | 2 | │ │ │ │ -| -1 | | 1 | | -1 | │ │ │ │ -| 1 | | -1 | | -1 | | - │ │ │ │ -3 | | 2 | | 1 | | 0 | | 1 | │ │ │ │ + | -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 | | │ │ │ │ --1 | | 2 | | 1 | | │ │ │ │ -1 | | -2 | | -1 | │ │ │ │ -| 1 | | 1 | | -2 | | │ │ │ │ --1 | | -1 | │ │ │ │ - | 1 | | -1 | │ │ │ │ -| 2 | | -1 | | 2 | │ │ │ │ -| 1 | | 0 | | 2 | | │ │ │ │ -1 | | -1 | | -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 | | 0 | | 2 | | │ │ │ │ +| 2 | | 1 | | 0 | | 2 | │ │ │ │ +| 2 | | 2 | | 0 | | │ │ │ │ -1 | | 2 | | -1 | | 1 │ │ │ │ | | 1 | | 2 | | -2 | │ │ │ │ | -1 | | 1 | | 3 | | │ │ │ │ -0 | | -1 | | -2 | | │ │ │ │ -1 | | 1 | | 2 | │ │ │ │ -| 2 | | -1 | | 3 | | │ │ │ │ -0 | | 2 | │ │ │ │ +-1 | | 0 | | 3 | | │ │ │ │ +0 | | -2 | | 1 | │ │ │ │ +| 1 | | 2 | | 2 | | │ │ │ │ +-1 | | 2 | │ │ │ │ │ │ │ │ o4 : HasseDiagram │ │ │ │ i5 : hasseDiagramToGraph(myInterval) │ │ │ │ │ │ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 2}, {, 4}}}, {, {{, │ │ │ │ -0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}}, {{, {{, 1}, {, 3}}}, {, │ │ │ │ -{{, 0}, {, 1}}}, {, {{, 1}, {, 2}}}, {, {{, 0}, {, 2}}}, {, {{, 2}, {, 3}}}}, { │ │ │ │ +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 │ │ │ │ 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, {{2, 0}, {3, 1}, {121, 2}}}}, {{1213, {{232, 0}, {1, │ │ │ │ -2}, {3, 4}}}, {2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, │ │ │ │ -1}, {2, 2}, {3, 3}}}}, {{213, {{1, 1}, {3, 3}}}, {232, {{3, 0}, {2, 1}}}, {123, │ │ │ │ -{{12321, 1}, {3, 2}}}, {132, {{121, 0}, {232, 2}}}, {121, {{1, 2}, {2, 3}}}}, { │ │ │ │ -{32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}, {21, {{1, 0}}}}, {{2, {}}}} │ │ │ │ +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 │ │ │ │ ********** 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 │ │ │ @@ -110,35 +110,35 @@ │ │ │ o4 : WeylGroupElement │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        |  1 |       | -1 |                                            |  3 |        |  0 |       | -1 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            | -2 |        |  1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        |  0 |       | -1 |                                              | -2 |        |  1 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1
│ │ │                                               |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ +                                             | -2 |        | -1 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │       |                                            |  3 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │       | -1 |}}}}
│ │ │       |  2 |
│ │ │       | -1 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,43 +41,43 @@
│ │ │ │                                             | -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 |}, {{0, | -1 |},
│ │ │ │ -{2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |},
│ │ │ │ -{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {
│ │ │ │ +{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |},
│ │ │ │ +{1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |},
│ │ │ │ +{2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -
│ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}},
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -3 |        | 0 |
│ │ │ │  |  1 |                                              |  3 |        |  1 |
│ │ │ │  |  1 |                                            | -1 |        | -1 |       |
│ │ │ │  2 |                                              |  1 |        |  1 |
│ │ │ │ -| -2 |        | -1 |                                            |  1 |        |
│ │ │ │ -1 |                                              | -1 |
│ │ │ │ +|  1 |        |  1 |                                            | -2 |        |
│ │ │ │ +-1 |                                              | -1 |
│ │ │ │                                                            |  1 |        | 1 |
│ │ │ │ -|  1 |                                              | -2 |        |  1 |
│ │ │ │ -| -1 |                                            |  3 |        |  0 |       |
│ │ │ │ --1 |                                              |  2 |        | -1 |
│ │ │ │ -|  3 |        |  0 |                                            | -2 |        |
│ │ │ │ -1 |                                              |  2 |
│ │ │ │ +|  1 |                                              | -2 |        | -1 |
│ │ │ │ +|  1 |                                            |  3 |        |  0 |       |
│ │ │ │ +-1 |                                              | -2 |        |  1 |
│ │ │ │ +|  2 |        | -1 |                                            |  3 |        |
│ │ │ │ +0 |                                              |  2 |
│ │ │ │  
│ │ │ │  o5 : HasseDiagram
│ │ │ │  Each row of the Hasse diagram contains the elements of a certain length
│ │ │ │  together with their links to the next row.
│ │ │ │  i6 : myInterval#1
│ │ │ │  
│ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1
│ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}, {1, | -1
│ │ │ │                                               |  3 |        |  1 |       |  1
│ │ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ │ +                                             | -2 |        | -1 |       |  1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1,
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2,
│ │ │ │       |                                            | -1 |        | -1 |
│ │ │ │       |                                            |  3 |        |  0 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       | -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, |  2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {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 |        | 1 |       |  2 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │ +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
    │ │ │
    │ │ │ 
    i3 : ZZ[x]
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,24 +20,24 @@
│ │ │ │  
│ │ │ │  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, |
│ │ │ │ -2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1
│ │ │ │ -|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -
│ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {
│ │ │ │ +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 |        | 1 |       |  2
│ │ │ │ -|                                              | -1 |        |  2 |
│ │ │ │ -|  2 |        | -1 |                                              | 1 |
│ │ │ │ +| -1 |                                              | -2 |        | 1 |       |
│ │ │ │ +-1 |                                            |  1 |        |  2 |       | 1
│ │ │ │ +|                                              |  2 |        | -1 |
│ │ │ │ +| -1 |        |  2 |                                              | 1 |
│ │ │ │  
│ │ │ │  o2 : HasseDiagram
│ │ │ │  i3 : ZZ[x]
│ │ │ │  
│ │ │ │  o3 = ZZ[x]
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ ├── ./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...}, |  2 |}, {{1, | -1 |}, {2, | -1
│ │ │ -                                             | -1 |        |  1 |       |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             | -1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1,
│ │ │ -     |                                            |  2 |        |  1 |      
│ │ │ -     |                                            | -1 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2,
│ │ │ +     |                                            |  2 |        | -1 |      
│ │ │ +     |                                            | -3 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |},
│ │ │ -     | -1 |                                            | -1 |        |  2 |  
│ │ │ -     |  0 |                                            |  2 |        | -1 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |},
│ │ │ +     |  1 |                                            | -3 |        | -1 |  
│ │ │ +     |  1 |                                            |  1 |        |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ -         | -1 |                                            | -3 |        |  2
│ │ │ +     {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            | -1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0,
│ │ │ -     |       |  1 |                                            | -3 |       
│ │ │ -     |       |  1 |                                            |  1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0,
│ │ │ +     |       |  2 |                                            |  2 |       
│ │ │ +     |       | -1 |                                            | -1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  1 |}, {1, |  2 |}}}}
│ │ │ +     |  1 |       | -1 |
│ │ │ +     | -1 |       |  0 |
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,37 +48,37 @@ │ │ │ │ WeylGroupElement{RootSystem{...8...}, | 1 |}} │ │ │ │ | -2 | │ │ │ │ | -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : underBruhat(L1) │ │ │ │ │ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, | -1 │ │ │ │ - | -1 | | 1 | | 2 │ │ │ │ - | -2 | | 1 | | -1 │ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, | 1 │ │ │ │ + | -1 | | 2 | | 1 │ │ │ │ + | 2 | | -1 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, │ │ │ │ - | | 2 | | 1 | │ │ │ │ - | | -1 | | -1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 0 |}, {2, │ │ │ │ + | | 2 | | -1 | │ │ │ │ + | | -3 | | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, │ │ │ │ - | -1 | | -1 | | 2 | │ │ │ │ - | 0 | | 2 | | -1 | │ │ │ │ + | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 |}, │ │ │ │ + | 1 | | -3 | | -1 | │ │ │ │ + | 1 | | 1 | | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 0 │ │ │ │ - | 1 | | 2 | | -1 │ │ │ │ - | -1 | | -3 | | 2 │ │ │ │ + {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 │ │ │ │ + | -1 | | -1 | | 1 │ │ │ │ + | 0 | | -2 | | 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, │ │ │ │ - | | 1 | | -3 | │ │ │ │ - | | 1 | | 1 | │ │ │ │ + |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, │ │ │ │ + | | 2 | | 2 | │ │ │ │ + | | -1 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 0 |}, {2, | 2 |}}}} │ │ │ │ - | -1 | | -1 | │ │ │ │ - | 2 | | 0 | │ │ │ │ + | 1 |}, {1, | 2 |}}}} │ │ │ │ + | 1 | | -1 | │ │ │ │ + | -1 | | 0 | │ │ │ │ │ │ │ │ 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 │ │ │ @@ -90,41 +90,41 @@ │ │ │ i22 : peek last ms │ │ │ │ │ │ o22 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly") │ │ │ - -- used 1.75476s (cpu); 1.07299s (thread); 0s (gc) │ │ │ + -- used 1.99753s (cpu); 1.22694s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ i24 : peek last ms │ │ │ │ │ │ o24 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most") │ │ │ - -- used 4.51087s (cpu); 2.67611s (thread); 0s (gc) │ │ │ + -- used 6.62324s (cpu); 3.07855s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ i26 : peek last ms │ │ │ │ │ │ o26 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all") │ │ │ - -- used 5.40142s (cpu); 3.44402s (thread); 0s (gc) │ │ │ + -- used 7.69087s (cpu); 3.59146s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ i28 : peek last ms │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html │ │ │ @@ -260,15 +260,15 @@ │ │ │
    │ │ │

    Finally we remark that the option: StratsToFind, may be used with this function, but should only be used with care. The default setting is StratsToFind=>"all", and this is the only value of the option which is guaranteed to compute the complete stratification, the other options may fail to find all strata but are provided to allow the user to obtain partial information on larger examples which may take too long to run on the default "all" setting. The other possible values are StratsToFind=>"singularOnly", and StratsToFind=>"most". The option StratsToFind=>"singularOnly" is the fastest, but also the most likely to return incomplete answers, and hence the output of this command should be treated as a partial answer only. The option StratsToFind=>"most" will most often get the full answer, but can miss strata, so again the output should be treated as a partial answer. In the example below all options return the complete answer, but only the output with StratsToFind=>"all" should be considered complete; StratsToFind=>"all" is run when no option is given.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -279,15 +279,15 @@ │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -298,15 +298,15 @@ │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ 2 => {ideal 0} │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -148,37 +148,37 @@ │ │ │ │ this command should be treated as a partial answer only. The option │ │ │ │ StratsToFind=>"most" will most often get the full answer, but can miss strata, │ │ │ │ so again the output should be treated as a partial answer. In the example below │ │ │ │ all options return the complete answer, but only the output with │ │ │ │ StratsToFind=>"all" should be considered complete; StratsToFind=>"all" is run │ │ │ │ when no option is given. │ │ │ │ i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly") │ │ │ │ - -- used 1.75476s (cpu); 1.07299s (thread); 0s (gc) │ │ │ │ + -- used 1.99753s (cpu); 1.22694s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o23 : List │ │ │ │ i24 : peek last ms │ │ │ │ │ │ │ │ o24 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ │ 2 => {ideal 0} │ │ │ │ i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most") │ │ │ │ - -- used 4.51087s (cpu); 2.67611s (thread); 0s (gc) │ │ │ │ + -- used 6.62324s (cpu); 3.07855s (thread); 0s (gc) │ │ │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o25 : List │ │ │ │ i26 : peek last ms │ │ │ │ │ │ │ │ o26 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ 1 => {ideal P, ideal M1, ideal(4M1 - P)} │ │ │ │ 2 => {ideal 0} │ │ │ │ i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all") │ │ │ │ - -- used 5.40142s (cpu); 3.44402s (thread); 0s (gc) │ │ │ │ + -- used 7.69087s (cpu); 3.59146s (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/M2-37698-0/172 │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-61144-0/172 │ │ │ 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/M2-37698-0/172 │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37698-0/174 │ │ │ +using temporary file /tmp/M2-61144-0/172 │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-61144-0/174 │ │ │ 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/M2-37698-0/174 │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37698-0/176 │ │ │ +using temporary file /tmp/M2-61144-0/174 │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-61144-0/176 │ │ │ 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/M2-37698-0/176 │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37698-0/178 │ │ │ +using temporary file /tmp/M2-61144-0/176 │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-61144-0/178 │ │ │ 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/M2-37698-0/178 │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37698-0/180 │ │ │ +using temporary file /tmp/M2-61144-0/178 │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-61144-0/180 │ │ │ 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/M2-37698-0/180 │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37698-0/182 │ │ │ +using temporary file /tmp/M2-61144-0/180 │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-61144-0/182 │ │ │ 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/M2-37698-0/182 │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37698-0/184 │ │ │ +using temporary file /tmp/M2-61144-0/182 │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-61144-0/184 │ │ │ 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/M2-37698-0/184 │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37698-0/186 │ │ │ +using temporary file /tmp/M2-61144-0/184 │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-61144-0/186 │ │ │ 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/M2-37698-0/186 │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37698-0/188 │ │ │ +using temporary file /tmp/M2-61144-0/186 │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-61144-0/188 │ │ │ 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/M2-37698-0/188 │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37698-0/190 │ │ │ +using temporary file /tmp/M2-61144-0/188 │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-61144-0/190 │ │ │ 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/M2-37698-0/190 │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37698-0/192 │ │ │ +using temporary file /tmp/M2-61144-0/190 │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-61144-0/192 │ │ │ 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/M2-37698-0/192 │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37698-0/194 │ │ │ +using temporary file /tmp/M2-61144-0/192 │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-61144-0/194 │ │ │ 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/M2-37698-0/194 │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37698-0/196 │ │ │ +using temporary file /tmp/M2-61144-0/194 │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-61144-0/196 │ │ │ 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/M2-37698-0/196 │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37698-0/198 │ │ │ +using temporary file /tmp/M2-61144-0/196 │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-61144-0/198 │ │ │ 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/M2-37698-0/198 │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37698-0/200 │ │ │ +using temporary file /tmp/M2-61144-0/198 │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-61144-0/200 │ │ │ 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/M2-37698-0/200 │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37698-0/202 │ │ │ +using temporary file /tmp/M2-61144-0/200 │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-61144-0/202 │ │ │ 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/M2-37698-0/202 │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37698-0/204 │ │ │ +using temporary file /tmp/M2-61144-0/202 │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-61144-0/204 │ │ │ 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/M2-37698-0/204 │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37698-0/206 │ │ │ +using temporary file /tmp/M2-61144-0/204 │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-61144-0/206 │ │ │ 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/M2-37698-0/206 │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37698-0/208 │ │ │ +using temporary file /tmp/M2-61144-0/206 │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-61144-0/208 │ │ │ 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/M2-37698-0/208 │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37698-0/210 │ │ │ +using temporary file /tmp/M2-61144-0/208 │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-61144-0/210 │ │ │ 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/M2-37698-0/210 │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37698-0/212 │ │ │ +using temporary file /tmp/M2-61144-0/210 │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-61144-0/212 │ │ │ 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/M2-37698-0/212 │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37698-0/214 │ │ │ +using temporary file /tmp/M2-61144-0/212 │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-61144-0/214 │ │ │ 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/M2-37698-0/214 │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37698-0/216 │ │ │ +using temporary file /tmp/M2-61144-0/214 │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-61144-0/216 │ │ │ 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/M2-37698-0/216 │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37698-0/218 │ │ │ +using temporary file /tmp/M2-61144-0/216 │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-61144-0/218 │ │ │ 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/M2-37698-0/218 │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37698-0/220 │ │ │ +using temporary file /tmp/M2-61144-0/218 │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-61144-0/220 │ │ │ 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/M2-37698-0/220 │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37698-0/222 │ │ │ +using temporary file /tmp/M2-61144-0/220 │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-61144-0/222 │ │ │ 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/M2-37698-0/222 │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37698-0/224 │ │ │ +using temporary file /tmp/M2-61144-0/222 │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-61144-0/224 │ │ │ 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/M2-37698-0/224 │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37698-0/226 │ │ │ +using temporary file /tmp/M2-61144-0/224 │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-61144-0/226 │ │ │ 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/M2-37698-0/226 │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37698-0/228 │ │ │ +using temporary file /tmp/M2-61144-0/226 │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-61144-0/228 │ │ │ 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/M2-37698-0/228 │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37698-0/230 │ │ │ +using temporary file /tmp/M2-61144-0/228 │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-61144-0/230 │ │ │ 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/M2-37698-0/230 │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37698-0/232 │ │ │ +using temporary file /tmp/M2-61144-0/230 │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-61144-0/232 │ │ │ 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/M2-37698-0/232 │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37698-0/234 │ │ │ +using temporary file /tmp/M2-61144-0/232 │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-61144-0/234 │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-37698-0/234 │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37698-0/236 │ │ │ +using temporary file /tmp/M2-61144-0/234 │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-61144-0/236 │ │ │ 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/M2-37698-0/236 │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37698-0/238 │ │ │ +using temporary file /tmp/M2-61144-0/236 │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-61144-0/238 │ │ │ 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/M2-37698-0/238 │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37698-0/240 │ │ │ +using temporary file /tmp/M2-61144-0/238 │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-61144-0/240 │ │ │ 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/M2-37698-0/240 │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37698-0/242 │ │ │ +using temporary file /tmp/M2-61144-0/240 │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-61144-0/242 │ │ │ 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/M2-37698-0/242 │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37698-0/244 │ │ │ +using temporary file /tmp/M2-61144-0/242 │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-61144-0/244 │ │ │ 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/M2-37698-0/244 │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37698-0/246 │ │ │ +using temporary file /tmp/M2-61144-0/244 │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-61144-0/246 │ │ │ This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-37698-0/246 │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37698-0/248 │ │ │ +using temporary file /tmp/M2-61144-0/246 │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-61144-0/248 │ │ │ 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/M2-37698-0/248 │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37698-0/250 │ │ │ +using temporary file /tmp/M2-61144-0/248 │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-61144-0/250 │ │ │ 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/M2-37698-0/250 │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37698-0/252 │ │ │ +using temporary file /tmp/M2-61144-0/250 │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-61144-0/252 │ │ │ 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/M2-37698-0/252 │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37698-0/254 │ │ │ +using temporary file /tmp/M2-61144-0/252 │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-61144-0/254 │ │ │ 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/M2-37698-0/254 │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37698-0/256 │ │ │ +using temporary file /tmp/M2-61144-0/254 │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-61144-0/256 │ │ │ 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/M2-37698-0/256 │ │ │ +using temporary file /tmp/M2-61144-0/256 │ │ │ │ │ │ i6 : QQ[x,y]; │ │ │ │ │ │ i7 : gfan {x,y}; │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37698-0/258 │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-61144-0/258 │ │ │ Q[x1,x2] │ │ │ {{ │ │ │ x2, │ │ │ x1} │ │ │ } │ │ │ -using temporary file /tmp/M2-37698-0/258 │ │ │ +using temporary file /tmp/M2-61144-0/258 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html │ │ │ @@ -109,15 +109,15 @@ │ │ │

    │ │ │ │ │ │
    │ │ │ 
    i23 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")
│ │ │ - -- used 1.75476s (cpu); 1.07299s (thread); 0s (gc)
│ │ │ + -- used 1.99753s (cpu); 1.22694s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
    │ │ │
    │ │ │ 
    i25 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")
│ │ │ - -- used 4.51087s (cpu); 2.67611s (thread); 0s (gc)
│ │ │ + -- used 6.62324s (cpu); 3.07855s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
    │ │ │
    │ │ │ 
    i27 : time ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")
│ │ │ - -- used 5.40142s (cpu); 3.44402s (thread); 0s (gc)
│ │ │ + -- used 7.69087s (cpu); 3.59146s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    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/M2-37698-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-61144-0/172
│ │ │  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/M2-37698-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37698-0/174
│ │ │ +using temporary file /tmp/M2-61144-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-61144-0/174
│ │ │  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/M2-37698-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37698-0/176
│ │ │ +using temporary file /tmp/M2-61144-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-61144-0/176
│ │ │  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/M2-37698-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37698-0/178
│ │ │ +using temporary file /tmp/M2-61144-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-61144-0/178
│ │ │  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/M2-37698-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37698-0/180
│ │ │ +using temporary file /tmp/M2-61144-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-61144-0/180
│ │ │  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/M2-37698-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37698-0/182
│ │ │ +using temporary file /tmp/M2-61144-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-61144-0/182
│ │ │  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/M2-37698-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37698-0/184
│ │ │ +using temporary file /tmp/M2-61144-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-61144-0/184
│ │ │  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/M2-37698-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37698-0/186
│ │ │ +using temporary file /tmp/M2-61144-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-61144-0/186
│ │ │  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/M2-37698-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37698-0/188
│ │ │ +using temporary file /tmp/M2-61144-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-61144-0/188
│ │ │  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/M2-37698-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37698-0/190
│ │ │ +using temporary file /tmp/M2-61144-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-61144-0/190
│ │ │  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/M2-37698-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37698-0/192
│ │ │ +using temporary file /tmp/M2-61144-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-61144-0/192
│ │ │  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/M2-37698-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37698-0/194
│ │ │ +using temporary file /tmp/M2-61144-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-61144-0/194
│ │ │  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/M2-37698-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37698-0/196
│ │ │ +using temporary file /tmp/M2-61144-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-61144-0/196
│ │ │  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/M2-37698-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37698-0/198
│ │ │ +using temporary file /tmp/M2-61144-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-61144-0/198
│ │ │  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/M2-37698-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37698-0/200
│ │ │ +using temporary file /tmp/M2-61144-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-61144-0/200
│ │ │  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/M2-37698-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37698-0/202
│ │ │ +using temporary file /tmp/M2-61144-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-61144-0/202
│ │ │  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/M2-37698-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37698-0/204
│ │ │ +using temporary file /tmp/M2-61144-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-61144-0/204
│ │ │  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/M2-37698-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37698-0/206
│ │ │ +using temporary file /tmp/M2-61144-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-61144-0/206
│ │ │  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/M2-37698-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37698-0/208
│ │ │ +using temporary file /tmp/M2-61144-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-61144-0/208
│ │ │  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/M2-37698-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37698-0/210
│ │ │ +using temporary file /tmp/M2-61144-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-61144-0/210
│ │ │  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/M2-37698-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37698-0/212
│ │ │ +using temporary file /tmp/M2-61144-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-61144-0/212
│ │ │  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/M2-37698-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37698-0/214
│ │ │ +using temporary file /tmp/M2-61144-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-61144-0/214
│ │ │  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/M2-37698-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37698-0/216
│ │ │ +using temporary file /tmp/M2-61144-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-61144-0/216
│ │ │  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/M2-37698-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37698-0/218
│ │ │ +using temporary file /tmp/M2-61144-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-61144-0/218
│ │ │  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/M2-37698-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37698-0/220
│ │ │ +using temporary file /tmp/M2-61144-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-61144-0/220
│ │ │  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/M2-37698-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37698-0/222
│ │ │ +using temporary file /tmp/M2-61144-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-61144-0/222
│ │ │  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/M2-37698-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37698-0/224
│ │ │ +using temporary file /tmp/M2-61144-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-61144-0/224
│ │ │  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/M2-37698-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37698-0/226
│ │ │ +using temporary file /tmp/M2-61144-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-61144-0/226
│ │ │  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/M2-37698-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37698-0/228
│ │ │ +using temporary file /tmp/M2-61144-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-61144-0/228
│ │ │  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/M2-37698-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37698-0/230
│ │ │ +using temporary file /tmp/M2-61144-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-61144-0/230
│ │ │  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/M2-37698-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37698-0/232
│ │ │ +using temporary file /tmp/M2-61144-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-61144-0/232
│ │ │  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/M2-37698-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37698-0/234
│ │ │ +using temporary file /tmp/M2-61144-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-61144-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37698-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37698-0/236
│ │ │ +using temporary file /tmp/M2-61144-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-61144-0/236
│ │ │  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/M2-37698-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37698-0/238
│ │ │ +using temporary file /tmp/M2-61144-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-61144-0/238
│ │ │  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/M2-37698-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37698-0/240
│ │ │ +using temporary file /tmp/M2-61144-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-61144-0/240
│ │ │  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/M2-37698-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37698-0/242
│ │ │ +using temporary file /tmp/M2-61144-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-61144-0/242
│ │ │  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/M2-37698-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37698-0/244
│ │ │ +using temporary file /tmp/M2-61144-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-61144-0/244
│ │ │  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/M2-37698-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37698-0/246
│ │ │ +using temporary file /tmp/M2-61144-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-61144-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-37698-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37698-0/248
│ │ │ +using temporary file /tmp/M2-61144-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-61144-0/248
│ │ │  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/M2-37698-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37698-0/250
│ │ │ +using temporary file /tmp/M2-61144-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-61144-0/250
│ │ │  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/M2-37698-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37698-0/252
│ │ │ +using temporary file /tmp/M2-61144-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-61144-0/252
│ │ │  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/M2-37698-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37698-0/254
│ │ │ +using temporary file /tmp/M2-61144-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-61144-0/254
│ │ │  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/M2-37698-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37698-0/256
│ │ │ +using temporary file /tmp/M2-61144-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-61144-0/256
│ │ │  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/M2-37698-0/256
    │ │ │ +using temporary file /tmp/M2-61144-0/256 │ │ │
    │ │ │ 
    i6 : QQ[x,y];
    │ │ │
    │ │ │ 
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37698-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-61144-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-37698-0/258
    │ │ │ +using temporary file /tmp/M2-61144-0/258 │ │ │
    │ │ │
    │ │ │

    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/M2-37698-0/172 │ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-61144-0/172 │ │ │ │ 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/M2-37698-0/172 │ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-37698-0/174 │ │ │ │ +using temporary file /tmp/M2-61144-0/172 │ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-61144-0/174 │ │ │ │ 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/M2-37698-0/174 │ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-37698-0/176 │ │ │ │ +using temporary file /tmp/M2-61144-0/174 │ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-61144-0/176 │ │ │ │ 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/M2-37698-0/176 │ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-37698-0/178 │ │ │ │ +using temporary file /tmp/M2-61144-0/176 │ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-61144-0/178 │ │ │ │ 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/M2-37698-0/178 │ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-37698-0/180 │ │ │ │ +using temporary file /tmp/M2-61144-0/178 │ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-61144-0/180 │ │ │ │ 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/M2-37698-0/180 │ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-37698-0/182 │ │ │ │ +using temporary file /tmp/M2-61144-0/180 │ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-61144-0/182 │ │ │ │ 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/M2-37698-0/182 │ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-37698-0/184 │ │ │ │ +using temporary file /tmp/M2-61144-0/182 │ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-61144-0/184 │ │ │ │ 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/M2-37698-0/184 │ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-37698-0/186 │ │ │ │ +using temporary file /tmp/M2-61144-0/184 │ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-61144-0/186 │ │ │ │ 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/M2-37698-0/186 │ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-37698-0/188 │ │ │ │ +using temporary file /tmp/M2-61144-0/186 │ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-61144-0/188 │ │ │ │ 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/M2-37698-0/188 │ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-37698-0/190 │ │ │ │ +using temporary file /tmp/M2-61144-0/188 │ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-61144-0/190 │ │ │ │ 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/M2-37698-0/190 │ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-37698-0/192 │ │ │ │ +using temporary file /tmp/M2-61144-0/190 │ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-61144-0/192 │ │ │ │ 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/M2-37698-0/192 │ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-37698-0/194 │ │ │ │ +using temporary file /tmp/M2-61144-0/192 │ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-61144-0/194 │ │ │ │ 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/M2-37698-0/194 │ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-37698-0/196 │ │ │ │ +using temporary file /tmp/M2-61144-0/194 │ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-61144-0/196 │ │ │ │ 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/M2-37698-0/196 │ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-37698-0/198 │ │ │ │ +using temporary file /tmp/M2-61144-0/196 │ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-61144-0/198 │ │ │ │ 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/M2-37698-0/198 │ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-37698-0/200 │ │ │ │ +using temporary file /tmp/M2-61144-0/198 │ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-61144-0/200 │ │ │ │ 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/M2-37698-0/200 │ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-37698-0/202 │ │ │ │ +using temporary file /tmp/M2-61144-0/200 │ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-61144-0/202 │ │ │ │ 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/M2-37698-0/202 │ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-37698-0/204 │ │ │ │ +using temporary file /tmp/M2-61144-0/202 │ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-61144-0/204 │ │ │ │ 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/M2-37698-0/204 │ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-37698-0/206 │ │ │ │ +using temporary file /tmp/M2-61144-0/204 │ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-61144-0/206 │ │ │ │ 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/M2-37698-0/206 │ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-37698-0/208 │ │ │ │ +using temporary file /tmp/M2-61144-0/206 │ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-61144-0/208 │ │ │ │ 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/M2-37698-0/208 │ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-37698-0/210 │ │ │ │ +using temporary file /tmp/M2-61144-0/208 │ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-61144-0/210 │ │ │ │ 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/M2-37698-0/210 │ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-37698-0/212 │ │ │ │ +using temporary file /tmp/M2-61144-0/210 │ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-61144-0/212 │ │ │ │ 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/M2-37698-0/212 │ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-37698-0/214 │ │ │ │ +using temporary file /tmp/M2-61144-0/212 │ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-61144-0/214 │ │ │ │ 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/M2-37698-0/214 │ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-37698-0/216 │ │ │ │ +using temporary file /tmp/M2-61144-0/214 │ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-61144-0/216 │ │ │ │ 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/M2-37698-0/216 │ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-37698-0/218 │ │ │ │ +using temporary file /tmp/M2-61144-0/216 │ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-61144-0/218 │ │ │ │ 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/M2-37698-0/218 │ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-37698-0/220 │ │ │ │ +using temporary file /tmp/M2-61144-0/218 │ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-61144-0/220 │ │ │ │ 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/M2-37698-0/220 │ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-37698-0/222 │ │ │ │ +using temporary file /tmp/M2-61144-0/220 │ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-61144-0/222 │ │ │ │ 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/M2-37698-0/222 │ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-37698-0/224 │ │ │ │ +using temporary file /tmp/M2-61144-0/222 │ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-61144-0/224 │ │ │ │ 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/M2-37698-0/224 │ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-37698-0/226 │ │ │ │ +using temporary file /tmp/M2-61144-0/224 │ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-61144-0/226 │ │ │ │ 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/M2-37698-0/226 │ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-37698-0/228 │ │ │ │ +using temporary file /tmp/M2-61144-0/226 │ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-61144-0/228 │ │ │ │ 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/M2-37698-0/228 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-37698-0/230 │ │ │ │ +using temporary file /tmp/M2-61144-0/228 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-61144-0/230 │ │ │ │ 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/M2-37698-0/230 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-37698-0/232 │ │ │ │ +using temporary file /tmp/M2-61144-0/230 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-61144-0/232 │ │ │ │ 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/M2-37698-0/232 │ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-37698-0/234 │ │ │ │ +using temporary file /tmp/M2-61144-0/232 │ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-61144-0/234 │ │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-37698-0/234 │ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-37698-0/236 │ │ │ │ +using temporary file /tmp/M2-61144-0/234 │ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-61144-0/236 │ │ │ │ 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/M2-37698-0/236 │ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-37698-0/238 │ │ │ │ +using temporary file /tmp/M2-61144-0/236 │ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-61144-0/238 │ │ │ │ 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/M2-37698-0/238 │ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-37698-0/240 │ │ │ │ +using temporary file /tmp/M2-61144-0/238 │ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-61144-0/240 │ │ │ │ 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/M2-37698-0/240 │ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-37698-0/242 │ │ │ │ +using temporary file /tmp/M2-61144-0/240 │ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-61144-0/242 │ │ │ │ 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/M2-37698-0/242 │ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-37698-0/244 │ │ │ │ +using temporary file /tmp/M2-61144-0/242 │ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-61144-0/244 │ │ │ │ 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/M2-37698-0/244 │ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-37698-0/246 │ │ │ │ +using temporary file /tmp/M2-61144-0/244 │ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-61144-0/246 │ │ │ │ This program computes the multiplicity of a tropical cone given a marked │ │ │ │ reduced Groebner basis for its initial ideal. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-37698-0/246 │ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-37698-0/248 │ │ │ │ +using temporary file /tmp/M2-61144-0/246 │ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-61144-0/248 │ │ │ │ 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/M2-37698-0/248 │ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-37698-0/250 │ │ │ │ +using temporary file /tmp/M2-61144-0/248 │ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-61144-0/250 │ │ │ │ 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/M2-37698-0/250 │ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-37698-0/252 │ │ │ │ +using temporary file /tmp/M2-61144-0/250 │ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-61144-0/252 │ │ │ │ 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/M2-37698-0/252 │ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-37698-0/254 │ │ │ │ +using temporary file /tmp/M2-61144-0/252 │ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-61144-0/254 │ │ │ │ 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/M2-37698-0/254 │ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-37698-0/256 │ │ │ │ +using temporary file /tmp/M2-61144-0/254 │ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-61144-0/256 │ │ │ │ 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/M2-37698-0/256 │ │ │ │ +using temporary file /tmp/M2-61144-0/256 │ │ │ │ i6 : QQ[x,y]; │ │ │ │ i7 : gfan {x,y}; │ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37698-0/258 │ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-61144-0/258 │ │ │ │ Q[x1,x2] │ │ │ │ {{ │ │ │ │ x2, │ │ │ │ x1} │ │ │ │ } │ │ │ │ -using temporary file /tmp/M2-37698-0/258 │ │ │ │ +using temporary file /tmp/M2-61144-0/258 │ │ │ │ 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- │ │ │ │ 1.25.06+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 2031 2e36 3634 |.| -- 1.664 │ │ │ │ -00017fc0: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ +00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e33 3731 |.| -- 1.371 │ │ │ │ +00017fc0: 3638 7320 656c 6170 7365 6420 2020 2020 68s 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 2031 2e39 3339 3031 7320 656c 6170 -- 1.93901s elap │ │ │ │ +00018260: 2d2d 2031 2e35 3732 3834 7320 656c 6170 -- 1.57284s 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 3836 3931 |.| -- .08691 │ │ │ │ -00002ec0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3336 3634 |.| -- .03664 │ │ │ │ +00002ec0: 3331 7320 656c 6170 7365 6420 2020 2020 31s 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: 2e36 3134 3533 3173 2065 6c61 7073 6564 .614531s elapsed │ │ │ │ +00006420: 2e35 3139 3939 3873 2065 6c61 7073 6564 .519998s 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 │ │ │ │ @@ -1651,16 +1651,16 @@ │ │ │ │ 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 2036 2e32 3239 3336 7320 656c | -- 6.22936s el │ │ │ │ -000067a0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00006790: 7c20 2d2d 2034 2e37 3437 3373 2065 6c61 | -- 4.7473s ela │ │ │ │ +000067a0: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 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 │ │ │ │ 00006810: 6c79 2061 7070 6c79 2862 6173 6550 7473 ly apply(basePts │ │ │ │ @@ -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: 3336 3539 3273 2065 6c61 7073 6564 2020 36592s elapsed │ │ │ │ +0000a360: 3133 3234 3837 7320 656c 6170 7365 6420 132487s 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 3739 3330 3373 |.| -- .0579303s │ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3036 3936 3673 |.| -- .0606966s │ │ │ │ 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: 2031 2e38 3335 3136 7320 656c 6170 7365 1.83516s elapse │ │ │ │ +0000a920: 2031 2e34 3734 3236 7320 656c 6170 7365 1.47426s elapse │ │ │ │ 0000a930: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 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 3433 3833 3733 7320 2863 7075 293b 0.438373s (cpu); │ │ │ │ -00010f90: 2030 2e33 3538 3334 3373 2028 7468 7265 0.358343s (thre │ │ │ │ +00010f80: 302e 3435 3639 3335 7320 2863 7075 293b 0.456935s (cpu); │ │ │ │ +00010f90: 2030 2e33 3833 3132 3573 2028 7468 7265 0.383125s (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 │ │ │ │ @@ -4398,954 +4398,952 @@ │ │ │ │ 000112d0: 206f 6620 7661 7269 6162 6c65 7320 696e of variables in │ │ │ │ 000112e0: 0a41 2074 6869 7320 7275 6e73 206d 7563 .A this runs muc │ │ │ │ 000112f0: 6820 6661 7374 6572 2074 6861 6e20 7461 h faster than ta │ │ │ │ 00011300: 6b69 6e67 2061 2072 616e 646f 6d20 6d61 king a random ma │ │ │ │ 00011310: 7472 6978 204d 2e0a 0a2b 2d2d 2d2d 2d2d trix M...+------ │ │ │ │ 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 2e35 3834 3334 3673 2028 6370 ed 0.584346s (cp │ │ │ │ -000113a0: 7529 3b20 302e 3432 3030 3234 7320 2874 u); 0.420024s (t │ │ │ │ -000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ -000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00011350: 7c69 3135 203a 2074 696d 6520 6265 7474 |i15 : time bett │ │ │ │ +00011360: 6920 2846 203d 2070 7572 6552 6573 6f6c i (F = pureResol │ │ │ │ +00011370: 7574 696f 6e28 3131 2c34 2c7b 302c 322c ution(11,4,{0,2, │ │ │ │ +00011380: 347d 2929 207c 0a7c 202d 2d20 7573 6564 4})) |.| -- used │ │ │ │ +00011390: 2030 2e36 3332 3133 3573 2028 6370 7529 0.632135s (cpu) │ │ │ │ +000113a0: 3b20 302e 3437 3937 3973 2028 7468 7265 ; 0.47979s (thre │ │ │ │ +000113b0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ +000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000113f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011400: 2020 2020 2020 2030 2031 2032 2020 2020 0 1 2 │ │ │ │ +000113f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00011400: 2020 2030 2031 2032 2020 2020 2020 2020 0 1 2 │ │ │ │ 00011410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011430: 0a7c 6f31 3520 3d20 746f 7461 6c3a 2033 .|o15 = total: 3 │ │ │ │ -00011440: 2036 2033 2020 2020 2020 2020 2020 2020 6 3 │ │ │ │ +00011420: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ +00011430: 203d 2074 6f74 616c 3a20 3320 3620 3320 = total: 3 6 3 │ │ │ │ +00011440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011470: 2020 2020 303a 2033 202e 202e 2020 2020 0: 3 . . │ │ │ │ +00011460: 207c 0a7c 2020 2020 2020 2020 2020 303a |.| 0: │ │ │ │ +00011470: 2033 202e 202e 2020 2020 2020 2020 2020 3 . . │ │ │ │ 00011480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000114a0: 0a7c 2020 2020 2020 2020 2020 313a 202e .| 1: . │ │ │ │ -000114b0: 2036 202e 2020 2020 2020 2020 2020 2020 6 . │ │ │ │ -000114c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000114d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000114e0: 2020 2020 323a 202e 202e 2033 2020 2020 2: . . 3 │ │ │ │ +00011490: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000114a0: 2020 2020 2031 3a20 2e20 3620 2e20 2020 1: . 6 . │ │ │ │ +000114b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000114d0: 0a7c 2020 2020 2020 2020 2020 323a 202e .| 2: . │ │ │ │ +000114e0: 202e 2033 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ 000114f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011510: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011500: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00011510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011540: 2020 2020 2020 207c 0a7c 6f31 3520 3a20 |.|o15 : │ │ │ │ -00011550: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +00011530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00011540: 6f31 3520 3a20 4265 7474 6954 616c 6c79 o15 : BettiTally │ │ │ │ +00011550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011570: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00011570: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00011580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ -000115c0: 7269 6e67 2046 2020 2020 2020 2020 2020 ring F │ │ │ │ +000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000115b0: 3620 3a20 7269 6e67 2046 2020 2020 2020 6 : ring F │ │ │ │ +000115c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000115d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000115e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000115f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000115e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000115f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011620: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011630: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00011610: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011620: 2020 5a5a 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00011630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011660: 0a7c 6f31 3620 3d20 2d2d 5b61 202e 2e61 .|o16 = --[a ..a │ │ │ │ -00011670: 2020 5d20 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00011680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000116a0: 3131 2020 3020 2020 3135 2020 2020 2020 11 0 15 │ │ │ │ -000116b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000116d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011650: 7c0a 7c6f 3136 203d 202d 2d5b 6120 2e2e |.|o16 = --[a .. │ │ │ │ +00011660: 6120 205d 2020 2020 2020 2020 2020 2020 a ] │ │ │ │ +00011670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011680: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00011690: 3131 2020 3020 2020 3135 2020 2020 2020 11 0 15 │ │ │ │ +000116a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000116c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000116d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000116e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011700: 2020 2020 2020 207c 0a7c 6f31 3620 3a20 |.|o16 : │ │ │ │ -00011710: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -00011720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011740: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000116f0: 2020 2020 207c 0a7c 6f31 3620 3a20 506f |.|o16 : Po │ │ │ │ +00011700: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020 lynomialRing │ │ │ │ +00011710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011720: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00011730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011770: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00011780: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00011790: 202a 6e6f 7465 2064 6972 6563 7449 6d61 *note directIma │ │ │ │ -000117a0: 6765 436f 6d70 6c65 783a 2064 6972 6563 geComplex: direc │ │ │ │ -000117b0: 7449 6d61 6765 436f 6d70 6c65 782c 202d tImageComplex, - │ │ │ │ -000117c0: 2d20 6469 7265 6374 2069 6d61 6765 2063 - direct image c │ │ │ │ -000117d0: 6f6d 706c 6578 0a20 202a 202a 6e6f 7465 omplex. * *note │ │ │ │ +00011760: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +00011770: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00011780: 7465 2064 6972 6563 7449 6d61 6765 436f te directImageCo │ │ │ │ +00011790: 6d70 6c65 783a 2064 6972 6563 7449 6d61 mplex: directIma │ │ │ │ +000117a0: 6765 436f 6d70 6c65 782c 202d 2d20 6469 geComplex, -- di │ │ │ │ +000117b0: 7265 6374 2069 6d61 6765 2063 6f6d 706c rect image compl │ │ │ │ +000117c0: 6578 0a20 202a 202a 6e6f 7465 2075 6e69 ex. * *note uni │ │ │ │ +000117d0: 7665 7273 616c 4578 7465 6e73 696f 6e3a versalExtension: │ │ │ │ 000117e0: 2075 6e69 7665 7273 616c 4578 7465 6e73 universalExtens │ │ │ │ -000117f0: 696f 6e3a 2075 6e69 7665 7273 616c 4578 ion: universalEx │ │ │ │ -00011800: 7465 6e73 696f 6e2c 202d 2d20 556e 6976 tension, -- Univ │ │ │ │ -00011810: 6572 7361 6c20 6578 7465 6e73 696f 6e20 ersal extension │ │ │ │ -00011820: 6f66 0a20 2020 2076 6563 746f 7220 6275 of. vector bu │ │ │ │ -00011830: 6e64 6c65 7320 6f6e 2050 5e31 0a0a 5761 ndles on P^1..Wa │ │ │ │ -00011840: 7973 2074 6f20 7573 6520 7075 7265 5265 ys to use pureRe │ │ │ │ -00011850: 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d 3d3d solution:.====== │ │ │ │ -00011860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011870: 3d3d 3d3d 3d0a 0a20 202a 2022 7075 7265 =====.. * "pure │ │ │ │ -00011880: 5265 736f 6c75 7469 6f6e 284d 6174 7269 Resolution(Matri │ │ │ │ -00011890: 782c 4c69 7374 2922 0a20 202a 2022 7075 x,List)". * "pu │ │ │ │ -000118a0: 7265 5265 736f 6c75 7469 6f6e 2852 696e reResolution(Rin │ │ │ │ -000118b0: 672c 4c69 7374 2922 0a20 202a 2022 7075 g,List)". * "pu │ │ │ │ -000118c0: 7265 5265 736f 6c75 7469 6f6e 285a 5a2c reResolution(ZZ, │ │ │ │ -000118d0: 4c69 7374 2922 0a20 202a 2022 7075 7265 List)". * "pure │ │ │ │ -000118e0: 5265 736f 6c75 7469 6f6e 285a 5a2c 5a5a Resolution(ZZ,ZZ │ │ │ │ -000118f0: 2c4c 6973 7429 220a 0a46 6f72 2074 6865 ,List)"..For the │ │ │ │ -00011900: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00011910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00011920: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00011930: 2070 7572 6552 6573 6f6c 7574 696f 6e3a pureResolution: │ │ │ │ -00011940: 2070 7572 6552 6573 6f6c 7574 696f 6e2c pureResolution, │ │ │ │ -00011950: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -00011960: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61 od function:.(Ma │ │ │ │ -00011970: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00011980: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +000117f0: 696f 6e2c 202d 2d20 556e 6976 6572 7361 ion, -- Universa │ │ │ │ +00011800: 6c20 6578 7465 6e73 696f 6e20 6f66 0a20 l extension of. │ │ │ │ +00011810: 2020 2076 6563 746f 7220 6275 6e64 6c65 vector bundle │ │ │ │ +00011820: 7320 6f6e 2050 5e31 0a0a 5761 7973 2074 s on P^1..Ways t │ │ │ │ +00011830: 6f20 7573 6520 7075 7265 5265 736f 6c75 o use pureResolu │ │ │ │ +00011840: 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d tion:.========== │ │ │ │ +00011850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011860: 3d0a 0a20 202a 2022 7075 7265 5265 736f =.. * "pureReso │ │ │ │ +00011870: 6c75 7469 6f6e 284d 6174 7269 782c 4c69 lution(Matrix,Li │ │ │ │ +00011880: 7374 2922 0a20 202a 2022 7075 7265 5265 st)". * "pureRe │ │ │ │ +00011890: 736f 6c75 7469 6f6e 2852 696e 672c 4c69 solution(Ring,Li │ │ │ │ +000118a0: 7374 2922 0a20 202a 2022 7075 7265 5265 st)". * "pureRe │ │ │ │ +000118b0: 736f 6c75 7469 6f6e 285a 5a2c 4c69 7374 solution(ZZ,List │ │ │ │ +000118c0: 2922 0a20 202a 2022 7075 7265 5265 736f )". * "pureReso │ │ │ │ +000118d0: 6c75 7469 6f6e 285a 5a2c 5a5a 2c4c 6973 lution(ZZ,ZZ,Lis │ │ │ │ +000118e0: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ +000118f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00011900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00011910: 6f62 6a65 6374 202a 6e6f 7465 2070 7572 object *note pur │ │ │ │ +00011920: 6552 6573 6f6c 7574 696f 6e3a 2070 7572 eResolution: pur │ │ │ │ +00011930: 6552 6573 6f6c 7574 696f 6e2c 2069 7320 eResolution, is │ │ │ │ +00011940: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +00011950: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +00011960: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00011970: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00011980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -000119e0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000119f0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011a00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011a10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011a20: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ -00011a30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011a40: 6765 732f 4247 472e 6d32 3a31 3232 360a ges/BGG.m2:1226. │ │ │ │ -00011a50: 3a30 2e0a 1f0a 4669 6c65 3a20 4247 472e :0....File: BGG. │ │ │ │ -00011a60: 696e 666f 2c20 4e6f 6465 3a20 5265 6775 info, Node: Regu │ │ │ │ -00011a70: 6c61 7269 7479 2c20 4e65 7874 3a20 7379 larity, Next: sy │ │ │ │ -00011a80: 6d45 7874 2c20 5072 6576 3a20 7075 7265 mExt, Prev: pure │ │ │ │ -00011a90: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20 Resolution, Up: │ │ │ │ -00011aa0: 546f 700a 0a52 6567 756c 6172 6974 7920 Top..Regularity │ │ │ │ -00011ab0: 2d2d 204f 7074 696f 6e20 666f 7220 6469 -- Option for di │ │ │ │ -00011ac0: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ -00011ad0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00011ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4361 ************..Ca │ │ │ │ -00011b00: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a43 7572 veat.======..Cur │ │ │ │ -00011b10: 7265 6e74 6c79 206e 6f74 2073 7570 706f rently not suppo │ │ │ │ -00011b20: 7274 6564 0a0a 4675 6e63 7469 6f6e 7320 rted..Functions │ │ │ │ -00011b30: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ -00011b40: 6775 6d65 6e74 206e 616d 6564 2052 6567 gument named Reg │ │ │ │ -00011b50: 756c 6172 6974 793a 0a3d 3d3d 3d3d 3d3d ularity:.======= │ │ │ │ +000119c0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000119d0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000119e0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000119f0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00011a00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00011a10: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ +00011a20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00011a30: 4247 472e 6d32 3a31 3232 360a 3a30 2e0a BGG.m2:1226.:0.. │ │ │ │ +00011a40: 1f0a 4669 6c65 3a20 4247 472e 696e 666f ..File: BGG.info │ │ │ │ +00011a50: 2c20 4e6f 6465 3a20 5265 6775 6c61 7269 , Node: Regulari │ │ │ │ +00011a60: 7479 2c20 4e65 7874 3a20 7379 6d45 7874 ty, Next: symExt │ │ │ │ +00011a70: 2c20 5072 6576 3a20 7075 7265 5265 736f , Prev: pureReso │ │ │ │ +00011a80: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a lution, Up: Top. │ │ │ │ +00011a90: 0a52 6567 756c 6172 6974 7920 2d2d 204f .Regularity -- O │ │ │ │ +00011aa0: 7074 696f 6e20 666f 7220 6469 7265 6374 ption for direct │ │ │ │ +00011ab0: 496d 6167 6543 6f6d 706c 6578 0a2a 2a2a ImageComplex.*** │ │ │ │ +00011ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ae0: 2a2a 2a2a 2a2a 2a2a 0a0a 4361 7665 6174 ********..Caveat │ │ │ │ +00011af0: 0a3d 3d3d 3d3d 3d0a 0a43 7572 7265 6e74 .======..Current │ │ │ │ +00011b00: 6c79 206e 6f74 2073 7570 706f 7274 6564 ly not supported │ │ │ │ +00011b10: 0a0a 4675 6e63 7469 6f6e 7320 7769 7468 ..Functions with │ │ │ │ +00011b20: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ +00011b30: 6e74 206e 616d 6564 2052 6567 756c 6172 nt named Regular │ │ │ │ +00011b40: 6974 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ity:.=========== │ │ │ │ +00011b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00011b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00011b90: 2022 6469 7265 6374 496d 6167 6543 6f6d "directImageCom │ │ │ │ -00011ba0: 706c 6578 282e 2e2e 2c52 6567 756c 6172 plex(...,Regular │ │ │ │ -00011bb0: 6974 793d 3e2e 2e2e 2922 202d 2d20 7365 ity=>...)" -- se │ │ │ │ -00011bc0: 6520 2a6e 6f74 6520 6469 7265 6374 496d e *note directIm │ │ │ │ -00011bd0: 6167 6543 6f6d 706c 6578 3a0a 2020 2020 ageComplex:. │ │ │ │ -00011be0: 6469 7265 6374 496d 6167 6543 6f6d 706c directImageCompl │ │ │ │ -00011bf0: 6578 2c20 2d2d 2064 6972 6563 7420 696d ex, -- direct im │ │ │ │ -00011c00: 6167 6520 636f 6d70 6c65 780a 0a46 6f72 age complex..For │ │ │ │ -00011c10: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00011c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011c30: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00011c40: 6e6f 7465 2052 6567 756c 6172 6974 793a note Regularity: │ │ │ │ -00011c50: 2028 5265 6775 6c61 7269 7479 2954 6f70 (Regularity)Top │ │ │ │ -00011c60: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ -00011c70: 626f 6c3a 0a28 4d61 6361 756c 6179 3244 bol:.(Macaulay2D │ │ │ │ -00011c80: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +00011b70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6469 =======.. * "di │ │ │ │ +00011b80: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ +00011b90: 282e 2e2e 2c52 6567 756c 6172 6974 793d (...,Regularity= │ │ │ │ +00011ba0: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +00011bb0: 6f74 6520 6469 7265 6374 496d 6167 6543 ote directImageC │ │ │ │ +00011bc0: 6f6d 706c 6578 3a0a 2020 2020 6469 7265 omplex:. dire │ │ │ │ +00011bd0: 6374 496d 6167 6543 6f6d 706c 6578 2c20 ctImageComplex, │ │ │ │ +00011be0: 2d2d 2064 6972 6563 7420 696d 6167 6520 -- direct image │ │ │ │ +00011bf0: 636f 6d70 6c65 780a 0a46 6f72 2074 6865 complex..For the │ │ │ │ +00011c00: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +00011c10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00011c20: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +00011c30: 2052 6567 756c 6172 6974 793a 2028 5265 Regularity: (Re │ │ │ │ +00011c40: 6775 6c61 7269 7479 2954 6f70 2c20 6973 gularity)Top, is │ │ │ │ +00011c50: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ +00011c60: 0a28 4d61 6361 756c 6179 3244 6f63 2953 .(Macaulay2Doc)S │ │ │ │ +00011c70: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00011ce0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00011cf0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011d00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011d10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011d20: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ -00011d30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011d40: 6765 732f 4247 472e 6d32 3a38 3831 3a0a ges/BGG.m2:881:. │ │ │ │ -00011d50: 302e 0a1f 0a46 696c 653a 2042 4747 2e69 0....File: BGG.i │ │ │ │ -00011d60: 6e66 6f2c 204e 6f64 653a 2073 796d 4578 nfo, Node: symEx │ │ │ │ -00011d70: 742c 204e 6578 743a 2074 6174 6552 6573 t, Next: tateRes │ │ │ │ -00011d80: 6f6c 7574 696f 6e2c 2050 7265 763a 2052 olution, Prev: R │ │ │ │ -00011d90: 6567 756c 6172 6974 792c 2055 703a 2054 egularity, Up: T │ │ │ │ -00011da0: 6f70 0a0a 7379 6d45 7874 202d 2d20 7468 op..symExt -- th │ │ │ │ -00011db0: 6520 6669 7273 7420 6469 6666 6572 656e e first differen │ │ │ │ -00011dc0: 7469 616c 206f 6620 7468 6520 636f 6d70 tial of the comp │ │ │ │ -00011dd0: 6c65 7820 5228 4d29 0a2a 2a2a 2a2a 2a2a lex R(M).******* │ │ │ │ +00011cc0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00011cd0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00011ce0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00011cf0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00011d00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00011d10: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ +00011d20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00011d30: 4247 472e 6d32 3a38 3831 3a0a 302e 0a1f BGG.m2:881:.0... │ │ │ │ +00011d40: 0a46 696c 653a 2042 4747 2e69 6e66 6f2c .File: BGG.info, │ │ │ │ +00011d50: 204e 6f64 653a 2073 796d 4578 742c 204e Node: symExt, N │ │ │ │ +00011d60: 6578 743a 2074 6174 6552 6573 6f6c 7574 ext: tateResolut │ │ │ │ +00011d70: 696f 6e2c 2050 7265 763a 2052 6567 756c ion, Prev: Regul │ │ │ │ +00011d80: 6172 6974 792c 2055 703a 2054 6f70 0a0a arity, Up: Top.. │ │ │ │ +00011d90: 7379 6d45 7874 202d 2d20 7468 6520 6669 symExt -- the fi │ │ │ │ +00011da0: 7273 7420 6469 6666 6572 656e 7469 616c rst differential │ │ │ │ +00011db0: 206f 6620 7468 6520 636f 6d70 6c65 7820 of the complex │ │ │ │ +00011dc0: 5228 4d29 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a R(M).*********** │ │ │ │ +00011dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00011de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00011e10: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00011e20: 2020 2073 796d 4578 7428 6d2c 4529 0a20 symExt(m,E). │ │ │ │ -00011e30: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00011e40: 202a 206d 2c20 6120 2a6e 6f74 6520 6d61 * m, a *note ma │ │ │ │ -00011e50: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ -00011e60: 446f 6329 4d61 7472 6978 2c2c 2061 2070 Doc)Matrix,, a p │ │ │ │ -00011e70: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00011e80: 6978 2066 6f72 2061 0a20 2020 2020 2020 ix for a. │ │ │ │ -00011e90: 2070 6f73 6974 6976 656c 7920 6772 6164 positively grad │ │ │ │ -00011ea0: 6564 206d 6f64 756c 6520 4d20 6f76 6572 ed module M over │ │ │ │ -00011eb0: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269 a polynomial ri │ │ │ │ -00011ec0: 6e67 0a20 2020 2020 202a 2045 2c20 6120 ng. * E, a │ │ │ │ -00011ed0: 2a6e 6f74 6520 706f 6c79 6e6f 6d69 616c *note polynomial │ │ │ │ -00011ee0: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00011ef0: 3244 6f63 2950 6f6c 796e 6f6d 6961 6c52 2Doc)PolynomialR │ │ │ │ -00011f00: 696e 672c 2c20 6578 7465 7269 6f72 0a20 ing,, exterior. │ │ │ │ -00011f10: 2020 2020 2020 2061 6c67 6562 7261 0a20 algebra. │ │ │ │ -00011f20: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00011f30: 2020 2a20 6120 2a6e 6f74 6520 6d61 7472 * a *note matr │ │ │ │ -00011f40: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00011f50: 6329 4d61 7472 6978 2c2c 2061 206d 6174 c)Matrix,, a mat │ │ │ │ -00011f60: 7269 7820 7265 7072 6573 656e 7469 6e67 rix representing │ │ │ │ -00011f70: 2074 6865 206d 6170 0a20 2020 2020 2020 the map. │ │ │ │ -00011f80: 204d 5f31 202a 2a20 6f6d 6567 615f 4520 M_1 ** omega_E │ │ │ │ -00011f90: 3c2d 2d20 4d5f 3020 2a2a 206f 6d65 6761 <-- M_0 ** omega │ │ │ │ -00011fa0: 5f45 0a0a 4465 7363 7269 7074 696f 6e0a _E..Description. │ │ │ │ -00011fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00011fc0: 7320 6675 6e63 7469 6f6e 2074 616b 6573 s function takes │ │ │ │ -00011fd0: 2061 7320 696e 7075 7420 6120 6d61 7472 as input a matr │ │ │ │ -00011fe0: 6978 206d 2077 6974 6820 6c69 6e65 6172 ix m with linear │ │ │ │ -00011ff0: 2065 6e74 7269 6573 2c20 7768 6963 6820 entries, which │ │ │ │ -00012000: 7765 2074 6869 6e6b 206f 660a 6173 2061 we think of.as a │ │ │ │ -00012010: 2070 7265 7365 6e74 6174 696f 6e20 6d61 presentation ma │ │ │ │ -00012020: 7472 6978 2066 6f72 2061 2070 6f73 6974 trix for a posit │ │ │ │ -00012030: 6976 656c 7920 6772 6164 6564 2053 2d6d ively graded S-m │ │ │ │ -00012040: 6f64 756c 6520 4d20 6d61 7472 6978 2072 odule M matrix r │ │ │ │ -00012050: 6570 7265 7365 6e74 696e 670a 7468 6520 epresenting.the │ │ │ │ -00012060: 6d61 7020 4d5f 3120 2a2a 206f 6d65 6761 map M_1 ** omega │ │ │ │ -00012070: 5f45 203c 2d2d 204d 5f30 202a 2a20 6f6d _E <-- M_0 ** om │ │ │ │ -00012080: 6567 615f 4520 7768 6963 6820 6973 2074 ega_E which is t │ │ │ │ -00012090: 6865 2066 6972 7374 2064 6966 6665 7265 he first differe │ │ │ │ -000120a0: 6e74 6961 6c20 6f66 0a74 6865 2063 6f6d ntial of.the com │ │ │ │ -000120b0: 706c 6578 2052 284d 292e 0a2b 2d2d 2d2d plex R(M)..+---- │ │ │ │ +00011df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00011e00: 7361 6765 3a20 0a20 2020 2020 2020 2073 sage: . s │ │ │ │ +00011e10: 796d 4578 7428 6d2c 4529 0a20 202a 2049 ymExt(m,E). * I │ │ │ │ +00011e20: 6e70 7574 733a 0a20 2020 2020 202a 206d nputs:. * m │ │ │ │ +00011e30: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +00011e40: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00011e50: 4d61 7472 6978 2c2c 2061 2070 7265 7365 Matrix,, a prese │ │ │ │ +00011e60: 6e74 6174 696f 6e20 6d61 7472 6978 2066 ntation matrix f │ │ │ │ +00011e70: 6f72 2061 0a20 2020 2020 2020 2070 6f73 or a. pos │ │ │ │ +00011e80: 6974 6976 656c 7920 6772 6164 6564 206d itively graded m │ │ │ │ +00011e90: 6f64 756c 6520 4d20 6f76 6572 2061 2070 odule M over a p │ │ │ │ +00011ea0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ +00011eb0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ +00011ec0: 6520 706f 6c79 6e6f 6d69 616c 2072 696e e polynomial rin │ │ │ │ +00011ed0: 673a 2028 4d61 6361 756c 6179 3244 6f63 g: (Macaulay2Doc │ │ │ │ +00011ee0: 2950 6f6c 796e 6f6d 6961 6c52 696e 672c )PolynomialRing, │ │ │ │ +00011ef0: 2c20 6578 7465 7269 6f72 0a20 2020 2020 , exterior. │ │ │ │ +00011f00: 2020 2061 6c67 6562 7261 0a20 202a 204f algebra. * O │ │ │ │ +00011f10: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00011f20: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00011f30: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00011f40: 7472 6978 2c2c 2061 206d 6174 7269 7820 trix,, a matrix │ │ │ │ +00011f50: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +00011f60: 206d 6170 0a20 2020 2020 2020 204d 5f31 map. M_1 │ │ │ │ +00011f70: 202a 2a20 6f6d 6567 615f 4520 3c2d 2d20 ** omega_E <-- │ │ │ │ +00011f80: 4d5f 3020 2a2a 206f 6d65 6761 5f45 0a0a M_0 ** omega_E.. │ │ │ │ +00011f90: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00011fa0: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675 =======..This fu │ │ │ │ +00011fb0: 6e63 7469 6f6e 2074 616b 6573 2061 7320 nction takes as │ │ │ │ +00011fc0: 696e 7075 7420 6120 6d61 7472 6978 206d input a matrix m │ │ │ │ +00011fd0: 2077 6974 6820 6c69 6e65 6172 2065 6e74 with linear ent │ │ │ │ +00011fe0: 7269 6573 2c20 7768 6963 6820 7765 2074 ries, which we t │ │ │ │ +00011ff0: 6869 6e6b 206f 660a 6173 2061 2070 7265 hink of.as a pre │ │ │ │ +00012000: 7365 6e74 6174 696f 6e20 6d61 7472 6978 sentation matrix │ │ │ │ +00012010: 2066 6f72 2061 2070 6f73 6974 6976 656c for a positivel │ │ │ │ +00012020: 7920 6772 6164 6564 2053 2d6d 6f64 756c y graded S-modul │ │ │ │ +00012030: 6520 4d20 6d61 7472 6978 2072 6570 7265 e M matrix repre │ │ │ │ +00012040: 7365 6e74 696e 670a 7468 6520 6d61 7020 senting.the map │ │ │ │ +00012050: 4d5f 3120 2a2a 206f 6d65 6761 5f45 203c M_1 ** omega_E < │ │ │ │ +00012060: 2d2d 204d 5f30 202a 2a20 6f6d 6567 615f -- M_0 ** omega_ │ │ │ │ +00012070: 4520 7768 6963 6820 6973 2074 6865 2066 E which is the f │ │ │ │ +00012080: 6972 7374 2064 6966 6665 7265 6e74 6961 irst differentia │ │ │ │ +00012090: 6c20 6f66 0a74 6865 2063 6f6d 706c 6578 l of.the complex │ │ │ │ +000120a0: 2052 284d 292e 0a2b 2d2d 2d2d 2d2d 2d2d R(M)..+-------- │ │ │ │ +000120b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000120c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000120d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000120e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000120f0: 0a7c 6931 203a 2053 203d 205a 5a2f 3332 .|i1 : S = ZZ/32 │ │ │ │ -00012100: 3030 335b 785f 302e 2e78 5f32 5d3b 2020 003[x_0..x_2]; │ │ │ │ -00012110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012120: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000120d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000120e0: 203a 2053 203d 205a 5a2f 3332 3030 335b : S = ZZ/32003[ │ │ │ │ +000120f0: 785f 302e 2e78 5f32 5d3b 2020 2020 2020 x_0..x_2]; │ │ │ │ +00012100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012110: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00012120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00012160: 203a 2045 203d 205a 5a2f 3332 3030 335b : E = ZZ/32003[ │ │ │ │ -00012170: 655f 302e 2e65 5f32 2c20 536b 6577 436f e_0..e_2, SkewCo │ │ │ │ -00012180: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ -00012190: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +00012140: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 -------+.|i2 : E │ │ │ │ +00012150: 203d 205a 5a2f 3332 3030 335b 655f 302e = ZZ/32003[e_0. │ │ │ │ +00012160: 2e65 5f32 2c20 536b 6577 436f 6d6d 7574 .e_2, SkewCommut │ │ │ │ +00012170: 6174 6976 653d 3e74 7275 655d 3b7c 0a2b ative=>true];|.+ │ │ │ │ +00012180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000121a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -000121d0: 203d 2063 6f6b 6572 206d 6174 7269 7820 = coker matrix │ │ │ │ -000121e0: 7b7b 785f 305e 322c 2078 5f31 5e32 7d7d {{x_0^2, x_1^2}} │ │ │ │ -000121f0: 3b20 2020 2020 2020 2020 2020 207c 0a2b ; |.+ │ │ │ │ +000121b0: 2d2d 2d2b 0a7c 6933 203a 204d 203d 2063 ---+.|i3 : M = c │ │ │ │ +000121c0: 6f6b 6572 206d 6174 7269 7820 7b7b 785f oker matrix {{x_ │ │ │ │ +000121d0: 305e 322c 2078 5f31 5e32 7d7d 3b20 2020 0^2, x_1^2}}; │ │ │ │ +000121e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000121f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012230: 2d2d 2d2b 0a7c 6934 203a 206d 203d 2070 ---+.|i4 : m = p │ │ │ │ -00012240: 7265 7365 6e74 6174 696f 6e20 7472 756e resentation trun │ │ │ │ -00012250: 6361 7465 2872 6567 756c 6172 6974 7920 cate(regularity │ │ │ │ -00012260: 4d2c 4d29 3b20 2020 207c 0a7c 2020 2020 M,M); |.| │ │ │ │ +00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012220: 0a7c 6934 203a 206d 203d 2070 7265 7365 .|i4 : m = prese │ │ │ │ +00012230: 6e74 6174 696f 6e20 7472 756e 6361 7465 ntation truncate │ │ │ │ +00012240: 2872 6567 756c 6172 6974 7920 4d2c 4d29 (regularity M,M) │ │ │ │ +00012250: 3b20 2020 207c 0a7c 2020 2020 2020 2020 ; |.| │ │ │ │ +00012260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000122a0: 0a7c 2020 2020 2020 2020 2020 2020 2034 .| 4 │ │ │ │ -000122b0: 2020 2020 2020 3820 2020 2020 2020 2020 8 │ │ │ │ -000122c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000122d0: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ -000122e0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ -000122f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012280: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012290: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ +000122a0: 2020 3820 2020 2020 2020 2020 2020 2020 8 │ │ │ │ +000122b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122c0: 207c 0a7c 6f34 203a 204d 6174 7269 7820 |.|o4 : Matrix │ │ │ │ +000122d0: 5320 203c 2d2d 2053 2020 2020 2020 2020 S <-- S │ │ │ │ +000122e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012340: 2d2b 0a7c 6935 203a 2073 796d 4578 7428 -+.|i5 : symExt( │ │ │ │ -00012350: 6d2c 4529 2020 2020 2020 2020 2020 2020 m,E) │ │ │ │ -00012360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012370: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00012330: 6935 203a 2073 796d 4578 7428 6d2c 4529 i5 : symExt(m,E) │ │ │ │ +00012340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012360: 2020 207c 0a7c 2020 2020 2020 2020 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 2020 2020 2020 │ │ │ │ -000123a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000123b0: 6f35 203d 207b 2d31 7d20 7c20 655f 3220 o5 = {-1} | e_2 │ │ │ │ -000123c0: 3020 2020 3020 2020 3020 2020 7c20 2020 0 0 0 | │ │ │ │ -000123d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123e0: 2020 207c 0a7c 2020 2020 207b 2d31 7d20 |.| {-1} │ │ │ │ -000123f0: 7c20 655f 3120 655f 3220 3020 2020 3020 | e_1 e_2 0 0 │ │ │ │ -00012400: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00012410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00012420: 207b 2d31 7d20 7c20 655f 3020 3020 2020 {-1} | e_0 0 │ │ │ │ -00012430: 655f 3220 3020 2020 7c20 2020 2020 2020 e_2 0 | │ │ │ │ -00012440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012450: 0a7c 2020 2020 207b 2d31 7d20 7c20 3020 .| {-1} | 0 │ │ │ │ -00012460: 2020 655f 3020 655f 3120 655f 3220 7c20 e_0 e_1 e_2 | │ │ │ │ -00012470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012480: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00012390: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +000123a0: 207b 2d31 7d20 7c20 655f 3220 3020 2020 {-1} | e_2 0 │ │ │ │ +000123b0: 3020 2020 3020 2020 7c20 2020 2020 2020 0 0 | │ │ │ │ +000123c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000123d0: 0a7c 2020 2020 207b 2d31 7d20 7c20 655f .| {-1} | e_ │ │ │ │ +000123e0: 3120 655f 3220 3020 2020 3020 2020 7c20 1 e_2 0 0 | │ │ │ │ +000123f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012400: 2020 2020 207c 0a7c 2020 2020 207b 2d31 |.| {-1 │ │ │ │ +00012410: 7d20 7c20 655f 3020 3020 2020 655f 3220 } | e_0 0 e_2 │ │ │ │ +00012420: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ +00012430: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012440: 2020 207b 2d31 7d20 7c20 3020 2020 655f {-1} | 0 e_ │ │ │ │ +00012450: 3020 655f 3120 655f 3220 7c20 2020 2020 0 e_1 e_2 | │ │ │ │ +00012460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000124c0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ -000124d0: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000124e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124f0: 207c 0a7c 6f35 203a 204d 6174 7269 7820 |.|o5 : Matrix │ │ │ │ -00012500: 4520 203c 2d2d 2045 2020 2020 2020 2020 E <-- E │ │ │ │ -00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012520: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000124a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000124b0: 2020 2020 2020 2034 2020 2020 2020 3420 4 4 │ │ │ │ +000124c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000124d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000124e0: 6f35 203a 204d 6174 7269 7820 4520 203c o5 : Matrix E < │ │ │ │ +000124f0: 2d2d 2045 2020 2020 2020 2020 2020 2020 -- E │ │ │ │ +00012500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012510: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00012520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00012560: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 Caveat.======..T │ │ │ │ -00012570: 6869 7320 6675 6e63 7469 6f6e 2069 7320 his function is │ │ │ │ -00012580: 6120 7175 6963 6b2d 616e 642d 6469 7274 a quick-and-dirt │ │ │ │ -00012590: 7920 746f 6f6c 2077 6869 6368 2072 6571 y tool which req │ │ │ │ -000125a0: 7569 7265 7320 6c69 7474 6c65 2063 6f6d uires little com │ │ │ │ -000125b0: 7075 7461 7469 6f6e 2e0a 486f 7765 7665 putation..Howeve │ │ │ │ -000125c0: 7220 6966 2069 7420 6973 2063 616c 6c65 r if it is calle │ │ │ │ -000125d0: 6420 6f6e 2074 776f 2073 7563 6365 7373 d on two success │ │ │ │ -000125e0: 6976 6520 7472 756e 6361 7469 6f6e 7320 ive truncations │ │ │ │ -000125f0: 6f66 2061 206d 6f64 756c 652c 2074 6865 of a module, the │ │ │ │ -00012600: 6e20 7468 650a 6d61 7073 2069 7420 7072 n the.maps it pr │ │ │ │ -00012610: 6f64 7563 6573 206d 6179 204e 4f54 2063 oduces may NOT c │ │ │ │ -00012620: 6f6d 706f 7365 2074 6f20 7a65 726f 2062 ompose to zero b │ │ │ │ -00012630: 6563 6175 7365 2074 6865 2063 686f 6963 ecause the choic │ │ │ │ -00012640: 6520 6f66 2062 6173 6573 2069 7320 6e6f e of bases is no │ │ │ │ -00012650: 740a 636f 6e73 6973 7465 6e74 2e0a 0a53 t.consistent...S │ │ │ │ -00012660: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00012670: 0a0a 2020 2a20 2a6e 6f74 6520 6267 673a .. * *note bgg: │ │ │ │ -00012680: 2062 6767 2c20 2d2d 2074 6865 2069 7468 bgg, -- the ith │ │ │ │ -00012690: 2064 6966 6665 7265 6e74 6961 6c20 6f66 differential of │ │ │ │ -000126a0: 2074 6865 2063 6f6d 706c 6578 2052 284d the complex R(M │ │ │ │ -000126b0: 290a 0a57 6179 7320 746f 2075 7365 2073 )..Ways to use s │ │ │ │ -000126c0: 796d 4578 743a 0a3d 3d3d 3d3d 3d3d 3d3d ymExt:.========= │ │ │ │ -000126d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -000126e0: 2273 796d 4578 7428 4d61 7472 6978 2c50 "symExt(Matrix,P │ │ │ │ -000126f0: 6f6c 796e 6f6d 6961 6c52 696e 6729 220a olynomialRing)". │ │ │ │ -00012700: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00012710: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00012720: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00012730: 6374 202a 6e6f 7465 2073 796d 4578 743a ct *note symExt: │ │ │ │ -00012740: 2073 796d 4578 742c 2069 7320 6120 2a6e symExt, is a *n │ │ │ │ -00012750: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00012760: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00012770: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00012780: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00012540: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ +00012550: 6174 0a3d 3d3d 3d3d 3d0a 0a54 6869 7320 at.======..This │ │ │ │ +00012560: 6675 6e63 7469 6f6e 2069 7320 6120 7175 function is a qu │ │ │ │ +00012570: 6963 6b2d 616e 642d 6469 7274 7920 746f ick-and-dirty to │ │ │ │ +00012580: 6f6c 2077 6869 6368 2072 6571 7569 7265 ol which require │ │ │ │ +00012590: 7320 6c69 7474 6c65 2063 6f6d 7075 7461 s little computa │ │ │ │ +000125a0: 7469 6f6e 2e0a 486f 7765 7665 7220 6966 tion..However if │ │ │ │ +000125b0: 2069 7420 6973 2063 616c 6c65 6420 6f6e it is called on │ │ │ │ +000125c0: 2074 776f 2073 7563 6365 7373 6976 6520 two successive │ │ │ │ +000125d0: 7472 756e 6361 7469 6f6e 7320 6f66 2061 truncations of a │ │ │ │ +000125e0: 206d 6f64 756c 652c 2074 6865 6e20 7468 module, then th │ │ │ │ +000125f0: 650a 6d61 7073 2069 7420 7072 6f64 7563 e.maps it produc │ │ │ │ +00012600: 6573 206d 6179 204e 4f54 2063 6f6d 706f es may NOT compo │ │ │ │ +00012610: 7365 2074 6f20 7a65 726f 2062 6563 6175 se to zero becau │ │ │ │ +00012620: 7365 2074 6865 2063 686f 6963 6520 6f66 se the choice of │ │ │ │ +00012630: 2062 6173 6573 2069 7320 6e6f 740a 636f bases is not.co │ │ │ │ +00012640: 6e73 6973 7465 6e74 2e0a 0a53 6565 2061 nsistent...See a │ │ │ │ +00012650: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00012660: 2a20 2a6e 6f74 6520 6267 673a 2062 6767 * *note bgg: bgg │ │ │ │ +00012670: 2c20 2d2d 2074 6865 2069 7468 2064 6966 , -- the ith dif │ │ │ │ +00012680: 6665 7265 6e74 6961 6c20 6f66 2074 6865 ferential of the │ │ │ │ +00012690: 2063 6f6d 706c 6578 2052 284d 290a 0a57 complex R(M)..W │ │ │ │ +000126a0: 6179 7320 746f 2075 7365 2073 796d 4578 ays to use symEx │ │ │ │ +000126b0: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d t:.============= │ │ │ │ +000126c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273 796d ======.. * "sym │ │ │ │ +000126d0: 4578 7428 4d61 7472 6978 2c50 6f6c 796e Ext(Matrix,Polyn │ │ │ │ +000126e0: 6f6d 6961 6c52 696e 6729 220a 0a46 6f72 omialRing)"..For │ │ │ │ +000126f0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00012700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00012710: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00012720: 6e6f 7465 2073 796d 4578 743a 2073 796d note symExt: sym │ │ │ │ +00012730: 4578 742c 2069 7320 6120 2a6e 6f74 6520 Ext, is a *note │ │ │ │ +00012740: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00012750: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00012760: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00012770: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00012780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000127a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000127b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127d0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -000127e0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -000127f0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00012800: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00012810: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -00012820: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00012830: 7932 2f70 6163 6b61 6765 732f 4247 472e y2/packages/BGG. │ │ │ │ -00012840: 6d32 3a36 3937 3a0a 302e 0a1f 0a46 696c m2:697:.0....Fil │ │ │ │ -00012850: 653a 2042 4747 2e69 6e66 6f2c 204e 6f64 e: BGG.info, Nod │ │ │ │ -00012860: 653a 2074 6174 6552 6573 6f6c 7574 696f e: tateResolutio │ │ │ │ -00012870: 6e2c 204e 6578 743a 2075 6e69 7665 7273 n, Next: univers │ │ │ │ -00012880: 616c 4578 7465 6e73 696f 6e2c 2050 7265 alExtension, Pre │ │ │ │ -00012890: 763a 2073 796d 4578 742c 2055 703a 2054 v: symExt, Up: T │ │ │ │ -000128a0: 6f70 0a0a 7461 7465 5265 736f 6c75 7469 op..tateResoluti │ │ │ │ -000128b0: 6f6e 202d 2d20 6669 6e69 7465 2070 6965 on -- finite pie │ │ │ │ -000128c0: 6365 206f 6620 7468 6520 5461 7465 2072 ce of the Tate r │ │ │ │ -000128d0: 6573 6f6c 7574 696f 6e0a 2a2a 2a2a 2a2a esolution.****** │ │ │ │ +000127c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000127d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000127e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000127f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00012800: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +00012810: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00012820: 6163 6b61 6765 732f 4247 472e 6d32 3a36 ackages/BGG.m2:6 │ │ │ │ +00012830: 3937 3a0a 302e 0a1f 0a46 696c 653a 2042 97:.0....File: B │ │ │ │ +00012840: 4747 2e69 6e66 6f2c 204e 6f64 653a 2074 GG.info, Node: t │ │ │ │ +00012850: 6174 6552 6573 6f6c 7574 696f 6e2c 204e ateResolution, N │ │ │ │ +00012860: 6578 743a 2075 6e69 7665 7273 616c 4578 ext: universalEx │ │ │ │ +00012870: 7465 6e73 696f 6e2c 2050 7265 763a 2073 tension, Prev: s │ │ │ │ +00012880: 796d 4578 742c 2055 703a 2054 6f70 0a0a ymExt, Up: Top.. │ │ │ │ +00012890: 7461 7465 5265 736f 6c75 7469 6f6e 202d tateResolution - │ │ │ │ +000128a0: 2d20 6669 6e69 7465 2070 6965 6365 206f - finite piece o │ │ │ │ +000128b0: 6620 7468 6520 5461 7465 2072 6573 6f6c f the Tate resol │ │ │ │ +000128c0: 7574 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ution.********** │ │ │ │ +000128d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000128e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000128f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00012900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00012910: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00012920: 2020 2020 2074 6174 6552 6573 6f6c 7574 tateResolut │ │ │ │ -00012930: 696f 6e28 6d2c 452c 6c2c 6829 0a20 202a ion(m,E,l,h). * │ │ │ │ -00012940: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00012950: 206d 2c20 6120 2a6e 6f74 6520 6d61 7472 m, a *note matr │ │ │ │ -00012960: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00012970: 6329 4d61 7472 6978 2c2c 2061 2070 7265 c)Matrix,, a pre │ │ │ │ -00012980: 7365 6e74 6174 696f 6e20 6d61 7472 6978 sentation matrix │ │ │ │ -00012990: 2066 6f72 2061 0a20 2020 2020 2020 206d for a. m │ │ │ │ -000129a0: 6f64 756c 650a 2020 2020 2020 2a20 452c odule. * E, │ │ │ │ -000129b0: 2061 202a 6e6f 7465 2070 6f6c 796e 6f6d a *note polynom │ │ │ │ -000129c0: 6961 6c20 7269 6e67 3a20 284d 6163 6175 ial ring: (Macau │ │ │ │ -000129d0: 6c61 7932 446f 6329 506f 6c79 6e6f 6d69 lay2Doc)Polynomi │ │ │ │ -000129e0: 616c 5269 6e67 2c2c 2065 7874 6572 696f alRing,, exterio │ │ │ │ -000129f0: 720a 2020 2020 2020 2020 616c 6765 6272 r. algebr │ │ │ │ -00012a00: 610a 2020 2020 2020 2a20 6c2c 2061 6e20 a. * l, an │ │ │ │ -00012a10: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00012a20: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00012a30: 2c20 6c6f 7765 7220 636f 686f 6d6f 6c6f , lower cohomolo │ │ │ │ -00012a40: 6769 6361 6c20 6465 6772 6565 0a20 2020 gical degree. │ │ │ │ -00012a50: 2020 202a 2068 2c20 616e 202a 6e6f 7465 * h, an *note │ │ │ │ -00012a60: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -00012a70: 6c61 7932 446f 6329 5a5a 2c2c 2075 7070 lay2Doc)ZZ,, upp │ │ │ │ -00012a80: 6572 2062 6f75 6e64 206f 6e20 7468 650a er bound on the. │ │ │ │ -00012a90: 2020 2020 2020 2020 636f 686f 6d6f 6c6f cohomolo │ │ │ │ -00012aa0: 6769 6361 6c20 6465 6772 6565 0a20 202a gical degree. * │ │ │ │ -00012ab0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00012ac0: 2a20 6120 2a6e 6f74 6520 636f 6d70 6c65 * a *note comple │ │ │ │ -00012ad0: 783a 2028 436f 6d70 6c65 7865 7329 436f x: (Complexes)Co │ │ │ │ -00012ae0: 6d70 6c65 782c 2c20 6120 6669 6e69 7465 mplex,, a finite │ │ │ │ -00012af0: 2070 6965 6365 206f 6620 7468 6520 5461 piece of the Ta │ │ │ │ -00012b00: 7465 0a20 2020 2020 2020 2072 6573 6f6c te. resol │ │ │ │ -00012b10: 7574 696f 6e0a 0a44 6573 6372 6970 7469 ution..Descripti │ │ │ │ -00012b20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00012b30: 5468 6973 2066 756e 6374 696f 6e20 7461 This function ta │ │ │ │ -00012b40: 6b65 7320 6173 2069 6e70 7574 2061 2070 kes as input a p │ │ │ │ -00012b50: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00012b60: 6978 206d 206f 6620 6120 6669 6e69 7465 ix m of a finite │ │ │ │ -00012b70: 6c79 2067 656e 6572 6174 6564 0a67 7261 ly generated.gra │ │ │ │ -00012b80: 6465 6420 532d 6d6f 6475 6c65 204d 2061 ded S-module M a │ │ │ │ -00012b90: 6e20 6578 7465 7269 6f72 2061 6c67 6562 n exterior algeb │ │ │ │ -00012ba0: 7261 2045 2061 6e64 2074 776f 2069 6e74 ra E and two int │ │ │ │ -00012bb0: 6567 6572 7320 6c20 616e 6420 682e 2049 egers l and h. I │ │ │ │ -00012bc0: 6620 7220 6973 2074 6865 0a72 6567 756c f r is the.regul │ │ │ │ -00012bd0: 6172 6974 7920 6f66 204d 2c20 7468 656e arity of M, then │ │ │ │ -00012be0: 2074 6869 7320 6675 6e63 7469 6f6e 2063 this function c │ │ │ │ -00012bf0: 6f6d 7075 7465 7320 7468 6520 7069 6563 omputes the piec │ │ │ │ -00012c00: 6520 6f66 2074 6865 2054 6174 6520 7265 e of the Tate re │ │ │ │ -00012c10: 736f 6c75 7469 6f6e 0a66 726f 6d20 636f solution.from co │ │ │ │ -00012c20: 686f 6d6f 6c6f 6769 6361 6c20 6465 6772 homological degr │ │ │ │ -00012c30: 6565 206c 2074 6f20 636f 686f 6d6f 6c6f ee l to cohomolo │ │ │ │ -00012c40: 6769 6361 6c20 6465 6772 6565 206d 6178 gical degree max │ │ │ │ -00012c50: 2872 2b32 2c68 292e 2046 6f72 2069 6e73 (r+2,h). For ins │ │ │ │ -00012c60: 7461 6e63 652c 0a66 6f72 2074 6865 2068 tance,.for the h │ │ │ │ -00012c70: 6f6d 6f67 656e 656f 7573 2063 6f6f 7264 omogeneous coord │ │ │ │ -00012c80: 696e 6174 6520 7269 6e67 206f 6620 6120 inate ring of a │ │ │ │ -00012c90: 706f 696e 7420 696e 2074 6865 2070 726f point in the pro │ │ │ │ -00012ca0: 6a65 6374 6976 6520 706c 616e 653a 0a2b jective plane:.+ │ │ │ │ +000128f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00012900: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00012910: 2074 6174 6552 6573 6f6c 7574 696f 6e28 tateResolution( │ │ │ │ +00012920: 6d2c 452c 6c2c 6829 0a20 202a 2049 6e70 m,E,l,h). * Inp │ │ │ │ +00012930: 7574 733a 0a20 2020 2020 202a 206d 2c20 uts:. * m, │ │ │ │ +00012940: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00012950: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00012960: 7472 6978 2c2c 2061 2070 7265 7365 6e74 trix,, a present │ │ │ │ +00012970: 6174 696f 6e20 6d61 7472 6978 2066 6f72 ation matrix for │ │ │ │ +00012980: 2061 0a20 2020 2020 2020 206d 6f64 756c a. modul │ │ │ │ +00012990: 650a 2020 2020 2020 2a20 452c 2061 202a e. * E, a * │ │ │ │ +000129a0: 6e6f 7465 2070 6f6c 796e 6f6d 6961 6c20 note polynomial │ │ │ │ +000129b0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +000129c0: 446f 6329 506f 6c79 6e6f 6d69 616c 5269 Doc)PolynomialRi │ │ │ │ +000129d0: 6e67 2c2c 2065 7874 6572 696f 720a 2020 ng,, exterior. │ │ │ │ +000129e0: 2020 2020 2020 616c 6765 6272 610a 2020 algebra. │ │ │ │ +000129f0: 2020 2020 2a20 6c2c 2061 6e20 2a6e 6f74 * l, an *not │ │ │ │ +00012a00: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00012a10: 756c 6179 3244 6f63 295a 5a2c 2c20 6c6f ulay2Doc)ZZ,, lo │ │ │ │ +00012a20: 7765 7220 636f 686f 6d6f 6c6f 6769 6361 wer cohomologica │ │ │ │ +00012a30: 6c20 6465 6772 6565 0a20 2020 2020 202a l degree. * │ │ │ │ +00012a40: 2068 2c20 616e 202a 6e6f 7465 2069 6e74 h, an *note int │ │ │ │ +00012a50: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ +00012a60: 446f 6329 5a5a 2c2c 2075 7070 6572 2062 Doc)ZZ,, upper b │ │ │ │ +00012a70: 6f75 6e64 206f 6e20 7468 650a 2020 2020 ound on the. │ │ │ │ +00012a80: 2020 2020 636f 686f 6d6f 6c6f 6769 6361 cohomologica │ │ │ │ +00012a90: 6c20 6465 6772 6565 0a20 202a 204f 7574 l degree. * Out │ │ │ │ +00012aa0: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00012ab0: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028 *note complex: ( │ │ │ │ +00012ac0: 436f 6d70 6c65 7865 7329 436f 6d70 6c65 Complexes)Comple │ │ │ │ +00012ad0: 782c 2c20 6120 6669 6e69 7465 2070 6965 x,, a finite pie │ │ │ │ +00012ae0: 6365 206f 6620 7468 6520 5461 7465 0a20 ce of the Tate. │ │ │ │ +00012af0: 2020 2020 2020 2072 6573 6f6c 7574 696f resolutio │ │ │ │ +00012b00: 6e0a 0a44 6573 6372 6970 7469 6f6e 0a3d n..Description.= │ │ │ │ +00012b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 ==========..This │ │ │ │ +00012b20: 2066 756e 6374 696f 6e20 7461 6b65 7320 function takes │ │ │ │ +00012b30: 6173 2069 6e70 7574 2061 2070 7265 7365 as input a prese │ │ │ │ +00012b40: 6e74 6174 696f 6e20 6d61 7472 6978 206d ntation matrix m │ │ │ │ +00012b50: 206f 6620 6120 6669 6e69 7465 6c79 2067 of a finitely g │ │ │ │ +00012b60: 656e 6572 6174 6564 0a67 7261 6465 6420 enerated.graded │ │ │ │ +00012b70: 532d 6d6f 6475 6c65 204d 2061 6e20 6578 S-module M an ex │ │ │ │ +00012b80: 7465 7269 6f72 2061 6c67 6562 7261 2045 terior algebra E │ │ │ │ +00012b90: 2061 6e64 2074 776f 2069 6e74 6567 6572 and two integer │ │ │ │ +00012ba0: 7320 6c20 616e 6420 682e 2049 6620 7220 s l and h. If r │ │ │ │ +00012bb0: 6973 2074 6865 0a72 6567 756c 6172 6974 is the.regularit │ │ │ │ +00012bc0: 7920 6f66 204d 2c20 7468 656e 2074 6869 y of M, then thi │ │ │ │ +00012bd0: 7320 6675 6e63 7469 6f6e 2063 6f6d 7075 s function compu │ │ │ │ +00012be0: 7465 7320 7468 6520 7069 6563 6520 6f66 tes the piece of │ │ │ │ +00012bf0: 2074 6865 2054 6174 6520 7265 736f 6c75 the Tate resolu │ │ │ │ +00012c00: 7469 6f6e 0a66 726f 6d20 636f 686f 6d6f tion.from cohomo │ │ │ │ +00012c10: 6c6f 6769 6361 6c20 6465 6772 6565 206c logical degree l │ │ │ │ +00012c20: 2074 6f20 636f 686f 6d6f 6c6f 6769 6361 to cohomologica │ │ │ │ +00012c30: 6c20 6465 6772 6565 206d 6178 2872 2b32 l degree max(r+2 │ │ │ │ +00012c40: 2c68 292e 2046 6f72 2069 6e73 7461 6e63 ,h). For instanc │ │ │ │ +00012c50: 652c 0a66 6f72 2074 6865 2068 6f6d 6f67 e,.for the homog │ │ │ │ +00012c60: 656e 656f 7573 2063 6f6f 7264 696e 6174 eneous coordinat │ │ │ │ +00012c70: 6520 7269 6e67 206f 6620 6120 706f 696e e ring of a poin │ │ │ │ +00012c80: 7420 696e 2074 6865 2070 726f 6a65 6374 t in the project │ │ │ │ +00012c90: 6976 6520 706c 616e 653a 0a2b 2d2d 2d2d ive plane:.+---- │ │ │ │ +00012ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ce0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ -00012cf0: 5a2f 3332 3030 335b 785f 302e 2e78 5f32 Z/32003[x_0..x_2 │ │ │ │ -00012d00: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ -00012d10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012cd0: 0a7c 6931 203a 2053 203d 205a 5a2f 3332 .|i1 : S = ZZ/32 │ │ │ │ +00012ce0: 3030 335b 785f 302e 2e78 5f32 5d3b 2020 003[x_0..x_2]; │ │ │ │ +00012cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012d00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00012d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 2d2d 2d2b ---------------+ │ │ │ │ -00012d50: 0a7c 6932 203a 2045 203d 205a 5a2f 3332 .|i2 : E = ZZ/32 │ │ │ │ -00012d60: 3030 335b 655f 302e 2e65 5f32 2c20 536b 003[e_0..e_2, Sk │ │ │ │ -00012d70: 6577 436f 6d6d 7574 6174 6976 653d 3e74 ewCommutative=>t │ │ │ │ -00012d80: 7275 655d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d rue];|.+-------- │ │ │ │ +00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00012d40: 203a 2045 203d 205a 5a2f 3332 3030 335b : E = ZZ/32003[ │ │ │ │ +00012d50: 655f 302e 2e65 5f32 2c20 536b 6577 436f e_0..e_2, SkewCo │ │ │ │ +00012d60: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ +00012d70: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +00012d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -00012dc0: 203a 206d 203d 206d 6174 7269 787b 7b78 : m = matrix{{x │ │ │ │ -00012dd0: 5f30 2c78 5f31 7d7d 3b20 2020 2020 2020 _0,x_1}}; │ │ │ │ +00012da0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206d -------+.|i3 : m │ │ │ │ +00012db0: 203d 206d 6174 7269 787b 7b78 5f30 2c78 = matrix{{x_0,x │ │ │ │ +00012dc0: 5f31 7d7d 3b20 2020 2020 2020 2020 2020 _1}}; │ │ │ │ +00012dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00012de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012df0: 207c 0a7c 2020 2020 2020 2020 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 │ │ │ │ -00012e20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00012e30: 2020 2020 2020 2031 2020 2020 2020 3220 1 2 │ │ │ │ -00012e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00012e60: 6f33 203a 204d 6174 7269 7820 5320 203c o3 : Matrix S < │ │ │ │ -00012e70: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ -00012e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00012e10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012e20: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00012e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e40: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +00012e50: 204d 6174 7269 7820 5320 203c 2d2d 2053 Matrix S <-- S │ │ │ │ +00012e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00012e80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00012e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00012ed0: 2072 6567 756c 6172 6974 7920 636f 6b65 regularity coke │ │ │ │ -00012ee0: 7220 6d20 2020 2020 2020 2020 2020 2020 r m │ │ │ │ -00012ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012f00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00012eb0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2072 6567 -----+.|i4 : reg │ │ │ │ +00012ec0: 756c 6172 6974 7920 636f 6b65 7220 6d20 ularity coker m │ │ │ │ +00012ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012ee0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 │ │ │ │ -00012f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f30: 2020 2020 207c 0a7c 6f34 203d 2030 2020 |.|o4 = 0 │ │ │ │ +00012f20: 207c 0a7c 6f34 203d 2030 2020 2020 2020 |.|o4 = 0 │ │ │ │ +00012f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012f50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012fa0: 2d2b 0a7c 6935 203a 2054 203d 2074 6174 -+.|i5 : T = tat │ │ │ │ -00012fb0: 6552 6573 6f6c 7574 696f 6e28 6d2c 452c eResolution(m,E, │ │ │ │ -00012fc0: 2d32 2c34 2920 2020 2020 2020 2020 2020 -2,4) │ │ │ │ -00012fd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00012f90: 6935 203a 2054 203d 2074 6174 6552 6573 i5 : T = tateRes │ │ │ │ +00012fa0: 6f6c 7574 696f 6e28 6d2c 452c 2d32 2c34 olution(m,E,-2,4 │ │ │ │ +00012fb0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00012fc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00013010: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00013020: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00013030: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00013040: 3120 207c 0a7c 6f35 203d 2045 2020 3c2d 1 |.|o5 = E <- │ │ │ │ -00013050: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20 - E <-- E <-- │ │ │ │ -00013060: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520 E <-- E <-- E │ │ │ │ -00013070: 203c 2d2d 2045 2020 207c 0a7c 2020 2020 <-- E |.| │ │ │ │ +00012ff0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00013000: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00013010: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00013020: 2020 2020 2031 2020 2020 2020 3120 207c 1 1 | │ │ │ │ +00013030: 0a7c 6f35 203d 2045 2020 3c2d 2d20 4520 .|o5 = E <-- E │ │ │ │ +00013040: 203c 2d2d 2045 2020 3c2d 2d20 4520 203c <-- E <-- E < │ │ │ │ +00013050: 2d2d 2045 2020 3c2d 2d20 4520 203c 2d2d -- E <-- E <-- │ │ │ │ +00013060: 2045 2020 207c 0a7c 2020 2020 2020 2020 E |.| │ │ │ │ +00013070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000130a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000130b0: 0a7c 2020 2020 2030 2020 2020 2020 3120 .| 0 1 │ │ │ │ -000130c0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -000130d0: 2020 2034 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ -000130e0: 2036 2020 207c 0a7c 2020 2020 2020 2020 6 |.| │ │ │ │ +00013090: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000130a0: 2020 2030 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ +000130b0: 2032 2020 2020 2020 3320 2020 2020 2034 2 3 4 │ │ │ │ +000130c0: 2020 2020 2020 3520 2020 2020 2036 2020 5 6 │ │ │ │ +000130d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000130e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000130f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013110: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00013120: 203a 2043 6f6d 706c 6578 2020 2020 2020 : Complex │ │ │ │ -00013130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00013100: 2020 2020 2020 207c 0a7c 6f35 203a 2043 |.|o5 : C │ │ │ │ +00013110: 6f6d 706c 6578 2020 2020 2020 2020 2020 omplex │ │ │ │ +00013120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013130: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013180: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2062 -------+.|i6 : b │ │ │ │ -00013190: 6574 7469 2054 2020 2020 2020 2020 2020 etti T │ │ │ │ -000131a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013170: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469 ---+.|i6 : betti │ │ │ │ +00013180: 2054 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ +00013190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000131a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000131b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00013200: 2020 3020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ -00013210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013220: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -00013230: 2074 6f74 616c 3a20 3120 3120 3120 3120 total: 1 1 1 1 │ │ │ │ -00013240: 3120 3120 3120 2020 2020 2020 2020 2020 1 1 1 │ │ │ │ -00013250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00013260: 0a7c 2020 2020 2020 2020 2d34 3a20 3120 .| -4: 1 │ │ │ │ -00013270: 3120 3120 3120 3120 3120 3120 2020 2020 1 1 1 1 1 1 │ │ │ │ -00013280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000131d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000131e0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ +000131f0: 3120 3220 3320 3420 3520 3620 2020 2020 1 2 3 4 5 6 │ │ │ │ +00013200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013210: 2020 2020 207c 0a7c 6f36 203d 2074 6f74 |.|o6 = tot │ │ │ │ +00013220: 616c 3a20 3120 3120 3120 3120 3120 3120 al: 1 1 1 1 1 1 │ │ │ │ +00013230: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013240: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00013250: 2020 2020 2020 2d34 3a20 3120 3120 3120 -4: 1 1 1 │ │ │ │ +00013260: 3120 3120 3120 3120 2020 2020 2020 2020 1 1 1 1 │ │ │ │ +00013270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013280: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -000132d0: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ -000132e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013300: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000132b0: 2020 2020 2020 207c 0a7c 6f36 203a 2042 |.|o6 : B │ │ │ │ +000132c0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +000132d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000132e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000132f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013330: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2054 -------+.|i7 : T │ │ │ │ -00013340: 2e64 645f 3120 2020 2020 2020 2020 2020 .dd_1 │ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013320: 2d2d 2d2b 0a7c 6937 203a 2054 2e64 645f ---+.|i7 : T.dd_ │ │ │ │ +00013330: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013350: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00013360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013380: 2020 2020 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2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ -00013430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013440: 2020 2020 207c 0a7c 6f37 203a 204d 6174 |.|o7 : Mat │ │ │ │ -00013450: 7269 7820 4520 203c 2d2d 2045 2020 2020 rix E <-- E │ │ │ │ -00013460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013470: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000133f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00013400: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00013410: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013430: 207c 0a7c 6f37 203a 204d 6174 7269 7820 |.|o7 : Matrix │ │ │ │ +00013440: 4520 203c 2d2d 2045 2020 2020 2020 2020 E <-- E │ │ │ │ +00013450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013460: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00013470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134b0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -000134c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000134d0: 2073 796d 4578 743a 2073 796d 4578 742c symExt: symExt, │ │ │ │ -000134e0: 202d 2d20 7468 6520 6669 7273 7420 6469 -- the first di │ │ │ │ -000134f0: 6666 6572 656e 7469 616c 206f 6620 7468 fferential of th │ │ │ │ -00013500: 6520 636f 6d70 6c65 7820 5228 4d29 0a0a e complex R(M).. │ │ │ │ -00013510: 5761 7973 2074 6f20 7573 6520 7461 7465 Ways to use tate │ │ │ │ -00013520: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ -00013530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00013540: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7461 =======.. * "ta │ │ │ │ -00013550: 7465 5265 736f 6c75 7469 6f6e 284d 6174 teResolution(Mat │ │ │ │ -00013560: 7269 782c 506f 6c79 6e6f 6d69 616c 5269 rix,PolynomialRi │ │ │ │ -00013570: 6e67 2c5a 5a2c 5a5a 2922 0a0a 466f 7220 ng,ZZ,ZZ)"..For │ │ │ │ -00013580: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00013590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000135a0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -000135b0: 6f74 6520 7461 7465 5265 736f 6c75 7469 ote tateResoluti │ │ │ │ -000135c0: 6f6e 3a20 7461 7465 5265 736f 6c75 7469 on: tateResoluti │ │ │ │ -000135d0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d on, is a *note m │ │ │ │ -000135e0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -000135f0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00013600: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000134a0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000134b0: 3d0a 0a20 202a 202a 6e6f 7465 2073 796d =.. * *note sym │ │ │ │ +000134c0: 4578 743a 2073 796d 4578 742c 202d 2d20 Ext: symExt, -- │ │ │ │ +000134d0: 7468 6520 6669 7273 7420 6469 6666 6572 the first differ │ │ │ │ +000134e0: 656e 7469 616c 206f 6620 7468 6520 636f ential of the co │ │ │ │ +000134f0: 6d70 6c65 7820 5228 4d29 0a0a 5761 7973 mplex R(M)..Ways │ │ │ │ +00013500: 2074 6f20 7573 6520 7461 7465 5265 736f to use tateReso │ │ │ │ +00013510: 6c75 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d lution:.======== │ │ │ │ +00013520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00013530: 3d3d 3d0a 0a20 202a 2022 7461 7465 5265 ===.. * "tateRe │ │ │ │ +00013540: 736f 6c75 7469 6f6e 284d 6174 7269 782c solution(Matrix, │ │ │ │ +00013550: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c5a PolynomialRing,Z │ │ │ │ +00013560: 5a2c 5a5a 2922 0a0a 466f 7220 7468 6520 Z,ZZ)"..For the │ │ │ │ +00013570: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00013580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00013590: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +000135a0: 7461 7465 5265 736f 6c75 7469 6f6e 3a20 tateResolution: │ │ │ │ +000135b0: 7461 7465 5265 736f 6c75 7469 6f6e 2c20 tateResolution, │ │ │ │ +000135c0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +000135d0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ +000135e0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +000135f0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00013660: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00013670: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00013680: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00013690: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -000136a0: 756c 6179 322d 312e 3235 2e30 362b 6473 ulay2-1.25.06+ds │ │ │ │ -000136b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -000136c0: 636b 6167 6573 2f42 4747 2e6d 323a 3735 ckages/BGG.m2:75 │ │ │ │ -000136d0: 333a 0a30 2e0a 1f0a 4669 6c65 3a20 4247 3:.0....File: BG │ │ │ │ -000136e0: 472e 696e 666f 2c20 4e6f 6465 3a20 756e G.info, Node: un │ │ │ │ -000136f0: 6976 6572 7361 6c45 7874 656e 7369 6f6e iversalExtension │ │ │ │ -00013700: 2c20 5072 6576 3a20 7461 7465 5265 736f , Prev: tateReso │ │ │ │ -00013710: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a lution, Up: Top. │ │ │ │ -00013720: 0a75 6e69 7665 7273 616c 4578 7465 6e73 .universalExtens │ │ │ │ -00013730: 696f 6e20 2d2d 2055 6e69 7665 7273 616c ion -- Universal │ │ │ │ -00013740: 2065 7874 656e 7369 6f6e 206f 6620 7665 extension of ve │ │ │ │ -00013750: 6374 6f72 2062 756e 646c 6573 206f 6e20 ctor bundles on │ │ │ │ -00013760: 505e 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a P^1.************ │ │ │ │ +00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00013650: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00013660: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00013670: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00013680: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00013690: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ +000136a0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +000136b0: 6573 2f42 4747 2e6d 323a 3735 333a 0a30 es/BGG.m2:753:.0 │ │ │ │ +000136c0: 2e0a 1f0a 4669 6c65 3a20 4247 472e 696e ....File: BGG.in │ │ │ │ +000136d0: 666f 2c20 4e6f 6465 3a20 756e 6976 6572 fo, Node: univer │ │ │ │ +000136e0: 7361 6c45 7874 656e 7369 6f6e 2c20 5072 salExtension, Pr │ │ │ │ +000136f0: 6576 3a20 7461 7465 5265 736f 6c75 7469 ev: tateResoluti │ │ │ │ +00013700: 6f6e 2c20 5570 3a20 546f 700a 0a75 6e69 on, Up: Top..uni │ │ │ │ +00013710: 7665 7273 616c 4578 7465 6e73 696f 6e20 versalExtension │ │ │ │ +00013720: 2d2d 2055 6e69 7665 7273 616c 2065 7874 -- Universal ext │ │ │ │ +00013730: 656e 7369 6f6e 206f 6620 7665 6374 6f72 ension of vector │ │ │ │ +00013740: 2062 756e 646c 6573 206f 6e20 505e 310a bundles on P^1. │ │ │ │ +00013750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00013760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000137a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -000137b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -000137c0: 756e 6976 6572 7361 6c45 7874 656e 7369 universalExtensi │ │ │ │ -000137d0: 6f6e 284c 612c 204c 6229 0a20 202a 2049 on(La, Lb). * I │ │ │ │ -000137e0: 6e70 7574 733a 0a20 2020 2020 202a 204c nputs:. * L │ │ │ │ -000137f0: 612c 2061 202a 6e6f 7465 206c 6973 743a a, a *note list: │ │ │ │ -00013800: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -00013810: 6973 742c 2c20 6f66 2069 6e74 6567 6572 ist,, of integer │ │ │ │ -00013820: 730a 2020 2020 2020 2a20 4c62 2c20 6120 s. * Lb, a │ │ │ │ -00013830: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -00013840: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -00013850: 206f 6620 696e 7465 6765 7273 0a20 202a of integers. * │ │ │ │ -00013860: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00013870: 2a20 452c 2061 202a 6e6f 7465 206d 6f64 * E, a *note mod │ │ │ │ -00013880: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -00013890: 6f63 294d 6f64 756c 652c 2c20 7265 7072 oc)Module,, repr │ │ │ │ -000138a0: 6573 656e 7469 6e67 2074 6865 2065 7874 esenting the ext │ │ │ │ -000138b0: 656e 7369 6f6e 0a0a 4465 7363 7269 7074 ension..Descript │ │ │ │ -000138c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -000138d0: 0a45 7665 7279 2076 6563 746f 7220 6275 .Every vector bu │ │ │ │ -000138e0: 6e64 6c65 2045 206f 6e20 245c 5050 5e31 ndle E on $\PP^1 │ │ │ │ -000138f0: 2420 7370 6c69 7473 2061 7320 6120 7375 $ splits as a su │ │ │ │ -00013900: 6d20 6f66 206c 696e 6520 6275 6e64 6c65 m of line bundle │ │ │ │ -00013910: 7320 4f4f 2861 5f69 292e 2049 6620 4c61 s OO(a_i). If La │ │ │ │ -00013920: 0a69 7320 6120 6c69 7374 206f 6620 696e .is a list of in │ │ │ │ -00013930: 7465 6765 7273 2c20 7765 2077 7269 7465 tegers, we write │ │ │ │ -00013940: 2045 284c 6129 2066 6f72 2074 6865 2064 E(La) for the d │ │ │ │ -00013950: 6972 6563 7420 7375 6d20 6f66 2074 6865 irect sum of the │ │ │ │ -00013960: 206c 696e 6520 6275 6e64 6c65 0a4f 4f28 line bundle.OO( │ │ │ │ -00013970: 4c61 5f69 292e 2020 4769 7665 6e20 7477 La_i). Given tw │ │ │ │ -00013980: 6f20 7375 6368 2062 756e 646c 6573 2073 o such bundles s │ │ │ │ -00013990: 7065 6369 6669 6564 2062 7920 7468 6520 pecified by the │ │ │ │ -000139a0: 6c69 7374 7320 4c61 2061 6e64 204c 6220 lists La and Lb │ │ │ │ -000139b0: 7468 6973 2073 6372 6970 740a 636f 6e73 this script.cons │ │ │ │ -000139c0: 7472 7563 7473 2061 206d 6f64 756c 6520 tructs a module │ │ │ │ -000139d0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ -000139e0: 2075 6e69 7665 7273 616c 2065 7874 656e universal exten │ │ │ │ -000139f0: 7369 6f6e 206f 6620 4528 4c62 2920 6279 sion of E(Lb) by │ │ │ │ -00013a00: 2045 284c 6129 2e20 4974 0a69 7320 6465 E(La). It.is de │ │ │ │ -00013a10: 6669 6e65 6420 6f6e 2074 6865 2070 726f fined on the pro │ │ │ │ -00013a20: 6475 6374 2076 6172 6965 7479 2045 7874 duct variety Ext │ │ │ │ -00013a30: 5e31 2845 284c 6129 2c20 4528 4c62 2929 ^1(E(La), E(Lb)) │ │ │ │ -00013a40: 2078 2024 5c50 505e 3124 2c20 616e 640a x $\PP^1$, and. │ │ │ │ -00013a50: 7265 7072 6573 656e 7465 6420 6865 7265 represented here │ │ │ │ -00013a60: 2062 7920 6120 6772 6164 6564 206d 6f64 by a graded mod │ │ │ │ -00013a70: 756c 6520 6f76 6572 2074 6865 2063 6f6f ule over the coo │ │ │ │ -00013a80: 7264 696e 6174 6520 7269 6e67 2053 203d rdinate ring S = │ │ │ │ -00013a90: 2041 5b79 5f30 2c79 5f31 5d20 6f66 0a74 A[y_0,y_1] of.t │ │ │ │ -00013aa0: 6869 7320 7661 7269 6574 793b 2068 6572 his variety; her │ │ │ │ -00013ab0: 6520 4120 6973 2074 6865 2063 6f6f 7264 e A is the coord │ │ │ │ -00013ac0: 696e 6174 6520 7269 6e67 206f 6620 4578 inate ring of Ex │ │ │ │ -00013ad0: 745e 3128 4528 4c61 292c 2045 284c 6229 t^1(E(La), E(Lb) │ │ │ │ -00013ae0: 292c 2077 6869 6368 2069 7320 610a 706f ), which is a.po │ │ │ │ -00013af0: 6c79 6e6f 6d69 616c 2072 696e 672e 0a0a lynomial ring... │ │ │ │ -00013b00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013790: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +000137a0: 2020 2020 2020 2020 4520 3d20 756e 6976 E = univ │ │ │ │ +000137b0: 6572 7361 6c45 7874 656e 7369 6f6e 284c ersalExtension(L │ │ │ │ +000137c0: 612c 204c 6229 0a20 202a 2049 6e70 7574 a, Lb). * Input │ │ │ │ +000137d0: 733a 0a20 2020 2020 202a 204c 612c 2061 s:. * La, a │ │ │ │ +000137e0: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ +000137f0: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ +00013800: 2c20 6f66 2069 6e74 6567 6572 730a 2020 , of integers. │ │ │ │ +00013810: 2020 2020 2a20 4c62 2c20 6120 2a6e 6f74 * Lb, a *not │ │ │ │ +00013820: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00013830: 7932 446f 6329 4c69 7374 2c2c 206f 6620 y2Doc)List,, of │ │ │ │ +00013840: 696e 7465 6765 7273 0a20 202a 204f 7574 integers. * Out │ │ │ │ +00013850: 7075 7473 3a0a 2020 2020 2020 2a20 452c puts:. * E, │ │ │ │ +00013860: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ +00013870: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00013880: 6f64 756c 652c 2c20 7265 7072 6573 656e odule,, represen │ │ │ │ +00013890: 7469 6e67 2074 6865 2065 7874 656e 7369 ting the extensi │ │ │ │ +000138a0: 6f6e 0a0a 4465 7363 7269 7074 696f 6e0a on..Description. │ │ │ │ +000138b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a45 7665 ===========..Eve │ │ │ │ +000138c0: 7279 2076 6563 746f 7220 6275 6e64 6c65 ry vector bundle │ │ │ │ +000138d0: 2045 206f 6e20 245c 5050 5e31 2420 7370 E on $\PP^1$ sp │ │ │ │ +000138e0: 6c69 7473 2061 7320 6120 7375 6d20 6f66 lits as a sum of │ │ │ │ +000138f0: 206c 696e 6520 6275 6e64 6c65 7320 4f4f line bundles OO │ │ │ │ +00013900: 2861 5f69 292e 2049 6620 4c61 0a69 7320 (a_i). If La.is │ │ │ │ +00013910: 6120 6c69 7374 206f 6620 696e 7465 6765 a list of intege │ │ │ │ +00013920: 7273 2c20 7765 2077 7269 7465 2045 284c rs, we write E(L │ │ │ │ +00013930: 6129 2066 6f72 2074 6865 2064 6972 6563 a) for the direc │ │ │ │ +00013940: 7420 7375 6d20 6f66 2074 6865 206c 696e t sum of the lin │ │ │ │ +00013950: 6520 6275 6e64 6c65 0a4f 4f28 4c61 5f69 e bundle.OO(La_i │ │ │ │ +00013960: 292e 2020 4769 7665 6e20 7477 6f20 7375 ). Given two su │ │ │ │ +00013970: 6368 2062 756e 646c 6573 2073 7065 6369 ch bundles speci │ │ │ │ +00013980: 6669 6564 2062 7920 7468 6520 6c69 7374 fied by the list │ │ │ │ +00013990: 7320 4c61 2061 6e64 204c 6220 7468 6973 s La and Lb this │ │ │ │ +000139a0: 2073 6372 6970 740a 636f 6e73 7472 7563 script.construc │ │ │ │ +000139b0: 7473 2061 206d 6f64 756c 6520 7265 7072 ts a module repr │ │ │ │ +000139c0: 6573 656e 7469 6e67 2074 6865 2075 6e69 esenting the uni │ │ │ │ +000139d0: 7665 7273 616c 2065 7874 656e 7369 6f6e versal extension │ │ │ │ +000139e0: 206f 6620 4528 4c62 2920 6279 2045 284c of E(Lb) by E(L │ │ │ │ +000139f0: 6129 2e20 4974 0a69 7320 6465 6669 6e65 a). It.is define │ │ │ │ +00013a00: 6420 6f6e 2074 6865 2070 726f 6475 6374 d on the product │ │ │ │ +00013a10: 2076 6172 6965 7479 2045 7874 5e31 2845 variety Ext^1(E │ │ │ │ +00013a20: 284c 6129 2c20 4528 4c62 2929 2078 2024 (La), E(Lb)) x $ │ │ │ │ +00013a30: 5c50 505e 3124 2c20 616e 640a 7265 7072 \PP^1$, and.repr │ │ │ │ +00013a40: 6573 656e 7465 6420 6865 7265 2062 7920 esented here by │ │ │ │ +00013a50: 6120 6772 6164 6564 206d 6f64 756c 6520 a graded module │ │ │ │ +00013a60: 6f76 6572 2074 6865 2063 6f6f 7264 696e over the coordin │ │ │ │ +00013a70: 6174 6520 7269 6e67 2053 203d 2041 5b79 ate ring S = A[y │ │ │ │ +00013a80: 5f30 2c79 5f31 5d20 6f66 0a74 6869 7320 _0,y_1] of.this │ │ │ │ +00013a90: 7661 7269 6574 793b 2068 6572 6520 4120 variety; here A │ │ │ │ +00013aa0: 6973 2074 6865 2063 6f6f 7264 696e 6174 is the coordinat │ │ │ │ +00013ab0: 6520 7269 6e67 206f 6620 4578 745e 3128 e ring of Ext^1( │ │ │ │ +00013ac0: 4528 4c61 292c 2045 284c 6229 292c 2077 E(La), E(Lb)), w │ │ │ │ +00013ad0: 6869 6368 2069 7320 610a 706f 6c79 6e6f hich is a.polyno │ │ │ │ +00013ae0: 6d69 616c 2072 696e 672e 0a0a 2b2d 2d2d mial ring...+--- │ │ │ │ +00013af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013b50: 7c69 3120 3a20 4d20 3d20 756e 6976 6572 |i1 : M = univer │ │ │ │ -00013b60: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013b70: 7d2c 207b 327d 2920 2020 2020 2020 2020 }, {2}) │ │ │ │ -00013b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013ba0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00013b40: 3a20 4d20 3d20 756e 6976 6572 7361 6c45 : M = universalE │ │ │ │ +00013b50: 7874 656e 7369 6f6e 287b 2d32 7d2c 207b xtension({-2}, { │ │ │ │ +00013b60: 327d 2920 2020 2020 2020 2020 2020 2020 2}) │ │ │ │ +00013b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013bf0: 7c6f 3120 3d20 636f 6b65 726e 656c 207b |o1 = cokernel { │ │ │ │ -00013c00: 322c 2030 7d20 7c20 785f 3020 785f 3120 2, 0} | x_0 x_1 │ │ │ │ -00013c10: 785f 3220 7c20 2020 2020 2020 2020 2020 x_2 | │ │ │ │ -00013c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013c30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013c40: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013c50: 312c 2031 7d20 7c20 795f 3020 3020 2020 1, 1} | y_0 0 │ │ │ │ -00013c60: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00013c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013c80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013c90: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013ca0: 312c 2031 7d20 7c20 795f 3120 795f 3020 1, 1} | y_1 y_0 │ │ │ │ -00013cb0: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00013cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013cd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013ce0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013cf0: 312c 2031 7d20 7c20 3020 2020 795f 3120 1, 1} | 0 y_1 │ │ │ │ -00013d00: 795f 3020 7c20 2020 2020 2020 2020 2020 y_0 | │ │ │ │ -00013d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013d30: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013d40: 312c 2031 7d20 7c20 3020 2020 3020 2020 1, 1} | 0 0 │ │ │ │ -00013d50: 795f 3120 7c20 2020 2020 2020 2020 2020 y_1 | │ │ │ │ -00013d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013d70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013d80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013bd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00013be0: 3d20 636f 6b65 726e 656c 207b 322c 2030 = cokernel {2, 0 │ │ │ │ +00013bf0: 7d20 7c20 785f 3020 785f 3120 785f 3220 } | x_0 x_1 x_2 │ │ │ │ +00013c00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013c30: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013c40: 7d20 7c20 795f 3020 3020 2020 3020 2020 } | y_0 0 0 │ │ │ │ +00013c50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013c80: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013c90: 7d20 7c20 795f 3120 795f 3020 3020 2020 } | y_1 y_0 0 │ │ │ │ +00013ca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013cd0: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013ce0: 7d20 7c20 3020 2020 795f 3120 795f 3020 } | 0 y_1 y_0 │ │ │ │ +00013cf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013d20: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013d30: 7d20 7c20 3020 2020 3020 2020 795f 3120 } | 0 0 y_1 │ │ │ │ +00013d40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013dd0: 7c20 2020 2020 205a 5a20 2020 2020 2020 | ZZ │ │ │ │ -00013de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013e00: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ -00013e10: 2020 2035 2020 2020 2020 2020 2020 7c0a 5 |. │ │ │ │ -00013e20: 7c6f 3120 3a20 2d2d 2d5b 7820 2e2e 7820 |o1 : ---[x ..x │ │ │ │ -00013e30: 5d5b 7920 2e2e 7920 5d2d 6d6f 6475 6c65 ][y ..y ]-module │ │ │ │ -00013e40: 2c20 7175 6f74 6965 6e74 206f 6620 282d , quotient of (- │ │ │ │ -00013e50: 2d2d 5b78 202e 2e78 205d 5b79 202e 2e79 --[x ..x ][y ..y │ │ │ │ -00013e60: 205d 2920 2020 2020 2020 2020 2020 7c0a ]) |. │ │ │ │ -00013e70: 7c20 2020 2020 3130 3120 2030 2020 2032 | 101 0 2 │ │ │ │ -00013e80: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ -00013e90: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00013ea0: 3031 2020 3020 2020 3220 2020 3020 2020 01 0 2 0 │ │ │ │ -00013eb0: 3120 2020 2020 2020 2020 2020 2020 7c0a 1 |. │ │ │ │ -00013ec0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013db0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013dc0: 2020 205a 5a20 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00013dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013de0: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ +00013df0: 2020 2020 2020 2020 2020 2020 2020 2035 5 │ │ │ │ +00013e00: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00013e10: 3a20 2d2d 2d5b 7820 2e2e 7820 5d5b 7920 : ---[x ..x ][y │ │ │ │ +00013e20: 2e2e 7920 5d2d 6d6f 6475 6c65 2c20 7175 ..y ]-module, qu │ │ │ │ +00013e30: 6f74 6965 6e74 206f 6620 282d 2d2d 5b78 otient of (---[x │ │ │ │ +00013e40: 202e 2e78 205d 5b79 202e 2e79 205d 2920 ..x ][y ..y ]) │ │ │ │ +00013e50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013e60: 2020 3130 3120 2030 2020 2032 2020 2030 101 0 2 0 │ │ │ │ +00013e70: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013e80: 2020 2020 2020 2020 2020 2031 3031 2020 101 │ │ │ │ +00013e90: 3020 2020 3220 2020 3020 2020 3120 2020 0 2 0 1 │ │ │ │ +00013ea0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00013eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013f10: 7c69 3220 3a20 4d20 3d20 756e 6976 6572 |i2 : M = univer │ │ │ │ -00013f20: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013f30: 2c2d 337d 2c20 7b32 2c33 7d29 2020 2020 ,-3}, {2,3}) │ │ │ │ -00013f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00013f00: 3a20 4d20 3d20 756e 6976 6572 7361 6c45 : M = universalE │ │ │ │ +00013f10: 7874 656e 7369 6f6e 287b 2d32 2c2d 337d xtension({-2,-3} │ │ │ │ +00013f20: 2c20 7b32 2c33 7d29 2020 2020 2020 2020 , {2,3}) │ │ │ │ +00013f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013fb0: 7c6f 3220 3d20 636f 6b65 726e 656c 207b |o2 = cokernel { │ │ │ │ -00013fc0: 322c 2030 7d20 7c20 785f 3079 5f31 2078 2, 0} | x_0y_1 x │ │ │ │ -00013fd0: 5f31 795f 3120 785f 3279 5f31 2078 5f33 _1y_1 x_2y_1 x_3 │ │ │ │ -00013fe0: 795f 3120 785f 3479 5f31 2078 5f35 795f y_1 x_4y_1 x_5y_ │ │ │ │ -00013ff0: 3120 785f 3679 5f31 2020 2020 2020 7c0a 1 x_6y_1 |. │ │ │ │ -00014000: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014010: 332c 2030 7d20 7c20 785f 3920 2020 2078 3, 0} | x_9 x │ │ │ │ -00014020: 5f31 3020 2020 785f 3131 2020 2078 5f31 _10 x_11 x_1 │ │ │ │ -00014030: 3220 2020 785f 3133 2020 2078 5f31 3420 2 x_13 x_14 │ │ │ │ -00014040: 2020 785f 3135 2020 2020 2020 2020 7c0a x_15 |. │ │ │ │ -00014050: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014060: 322c 2031 7d20 7c20 795f 3020 2020 2030 2, 1} | y_0 0 │ │ │ │ -00014070: 2020 2020 2020 3020 2020 2020 2030 2020 0 0 │ │ │ │ -00014080: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014090: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140a0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000140b0: 322c 2031 7d20 7c20 795f 3120 2020 2079 2, 1} | y_1 y │ │ │ │ -000140c0: 5f30 2020 2020 3020 2020 2020 2030 2020 _0 0 0 │ │ │ │ -000140d0: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -000140e0: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140f0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014100: 322c 2031 7d20 7c20 3020 2020 2020 2079 2, 1} | 0 y │ │ │ │ -00014110: 5f31 2020 2020 795f 3020 2020 2030 2020 _1 y_0 0 │ │ │ │ -00014120: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014130: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014140: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014150: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -00014160: 2020 2020 2020 795f 3120 2020 2079 5f30 y_1 y_0 │ │ │ │ -00014170: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014180: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014190: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000141a0: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -000141b0: 2020 2020 2020 3020 2020 2020 2079 5f31 0 y_1 │ │ │ │ -000141c0: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -000141d0: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000141e0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000141f0: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -00014200: 2020 2020 2020 3020 2020 2020 2030 2020 0 0 │ │ │ │ -00014210: 2020 2020 795f 3020 2020 2030 2020 2020 y_0 0 │ │ │ │ -00014220: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014230: 7c20 2020 2020 2020 2020 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See the docu │ │ │ │ -00014a20: 6d65 6e74 6174 696f 6e20 666f 7220 6469 mentation for di │ │ │ │ -00014a30: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ -00014a40: 2066 6f72 2061 0a63 6f6e 6a65 6374 7572 for a.conjectur │ │ │ │ -00014a50: 6520 6162 6f75 7420 7468 6520 6571 7561 e about the equa │ │ │ │ -00014a60: 7469 6f6e 7320 6f66 2074 6865 7365 2076 tions of these v │ │ │ │ -00014a70: 6172 6965 7469 6573 2e0a 0a53 6565 2061 arieties...See a │ │ │ │ -00014a80: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00014a90: 2a20 2a6e 6f74 6520 6469 7265 6374 496d * *note directIm │ │ │ │ -00014aa0: 6167 6543 6f6d 706c 6578 3a20 6469 7265 ageComplex: dire │ │ │ │ -00014ab0: 6374 496d 6167 6543 6f6d 706c 6578 2c20 ctImageComplex, │ │ │ │ -00014ac0: 2d2d 2064 6972 6563 7420 696d 6167 6520 -- direct image │ │ │ │ -00014ad0: 636f 6d70 6c65 780a 0a57 6179 7320 746f complex..Ways to │ │ │ │ -00014ae0: 2075 7365 2075 6e69 7665 7273 616c 4578 use universalEx │ │ │ │ -00014af0: 7465 6e73 696f 6e3a 0a3d 3d3d 3d3d 3d3d tension:.======= │ │ │ │ -00014b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00014b10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2275 ========.. * "u │ │ │ │ -00014b20: 6e69 7665 7273 616c 4578 7465 6e73 696f niversalExtensio │ │ │ │ -00014b30: 6e28 4c69 7374 2c4c 6973 7429 220a 0a46 n(List,List)"..F │ │ │ │ -00014b40: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00014b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00014b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00014b70: 202a 6e6f 7465 2075 6e69 7665 7273 616c *note universal │ │ │ │ -00014b80: 4578 7465 6e73 696f 6e3a 2075 6e69 7665 Extension: unive │ │ │ │ -00014b90: 7273 616c 4578 7465 6e73 696f 6e2c 2069 rsalExtension, i │ │ │ │ -00014ba0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -00014bb0: 0a66 756e 6374 696f 6e3a 2028 4d61 6361 .function: (Maca │ │ │ │ -00014bc0: 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2d2d 2d2d ---------------- │ │ │ │ -00000f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f40: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070 ----+..For the p │ │ │ │ -00000f50: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00000f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -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 │ │ │ │ -00001020: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 │ │ │ │ -000010a0: 653a 0a4e 6f64 653a 2054 6f70 7f32 3334 e:.Node: Top.234 │ │ │ │ -000010b0: 0a4e 6f64 653a 2072 756e 4265 6e63 686d .Node: runBenchm │ │ │ │ -000010c0: 6172 6b73 7f32 3033 350a 1f0a 456e 6420 arks.2035...End │ │ │ │ -000010d0: 5461 6720 5461 626c 650a Tag Table. │ │ │ │ +00000ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00000f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00000f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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: │ │ │ │ +00001000: 0a28 4d61 6361 756c 6179 3244 6f63 2943 .(Macaulay2Doc)C │ │ │ │ +00001010: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d ommand,...------ │ │ │ │ +00001020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001060: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00001070: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 │ │ │ │ +000010f0: 653a 0a4e 6f64 653a 2054 6f70 7f32 3334 e:.Node: Top.234 │ │ │ │ +00001100: 0a4e 6f64 653a 2072 756e 4265 6e63 686d .Node: runBenchm │ │ │ │ +00001110: 6172 6b73 7f32 3033 350a 1f0a 456e 6420 arks.2035...End │ │ │ │ +00001120: 5461 6720 5461 626c 650a Tag Table. │ │ ├── ./usr/share/info/Bertini.info.gz │ │ │ ├── Bertini.info │ │ │ │ @@ -2253,8157 +2253,8157 @@ │ │ │ │ 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 3733 3336 "/tmp/M2-7336 │ │ │ │ -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 │ │ │ │ -00008dc0: 645f 7270 0a20 2020 2020 2020 202c 203d d_rp. , = │ │ │ │ -00008dd0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00008de0: 616c 7565 2066 616c 7365 2c20 4f70 7469 alue false, Opti │ │ │ │ -00008df0: 6f6e 2074 6f20 7369 6c65 6e63 6520 6164 on to silence ad │ │ │ │ -00008e00: 6469 7469 6f6e 616c 206f 7574 7075 740a ditional output. │ │ │ │ -00008e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00008e20: 2020 202a 2053 2c20 6120 2a6e 6f74 6520 * S, a *note │ │ │ │ -00008e30: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ -00008e40: 446f 6329 4c69 7374 2c2c 2061 206c 6973 Doc)List,, a lis │ │ │ │ -00008e50: 7420 7768 6f73 6520 656e 7472 6965 7320 t whose entries │ │ │ │ -00008e60: 6172 6520 6c69 7374 7320 6f66 0a20 2020 are lists of. │ │ │ │ -00008e70: 2020 2020 2073 6f6c 7574 696f 6e73 2066 solutions f │ │ │ │ -00008e80: 6f72 2065 6163 6820 7461 7267 6574 2073 or each target s │ │ │ │ -00008e90: 7973 7465 6d0a 0a44 6573 6372 6970 7469 ystem..Descripti │ │ │ │ -00008ea0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00008eb0: 5468 6973 206d 6574 686f 6420 6e75 6d65 This method nume │ │ │ │ -00008ec0: 7269 6361 6c6c 7920 736f 6c76 6573 2073 rically solves s │ │ │ │ -00008ed0: 6576 6572 616c 2070 6f6c 796e 6f6d 6961 everal polynomia │ │ │ │ -00008ee0: 6c20 7379 7374 656d 7320 6672 6f6d 2061 l systems from a │ │ │ │ -00008ef0: 2070 6172 616d 6574 6572 697a 6564 0a66 parameterized.f │ │ │ │ -00008f00: 616d 696c 7920 6174 206f 6e63 652e 2020 amily at once. │ │ │ │ -00008f10: 5468 6520 6c69 7374 2046 2069 7320 6120 The list F is a │ │ │ │ -00008f20: 7379 7374 656d 206f 6620 706f 6c79 6e6f system of polyno │ │ │ │ -00008f30: 6d69 616c 7320 696e 2072 696e 6720 7661 mials in ring va │ │ │ │ -00008f40: 7269 6162 6c65 7320 616e 640a 7468 6520 riables and.the │ │ │ │ -00008f50: 7061 7261 6d65 7465 7273 206c 6973 7465 parameters liste │ │ │ │ -00008f60: 6420 696e 2050 2e20 2054 6865 206c 6973 d in P. The lis │ │ │ │ -00008f70: 7420 5420 6973 2074 6865 2073 6574 206f t T is the set o │ │ │ │ -00008f80: 6620 7061 7261 6d65 7465 7220 7661 6c75 f parameter valu │ │ │ │ -00008f90: 6573 2066 6f72 0a77 6869 6368 2073 6f6c es for.which sol │ │ │ │ -00008fa0: 7574 696f 6e73 2074 6f20 4620 6172 6520 utions to F are │ │ │ │ -00008fb0: 6465 7369 7265 642e 2020 426f 7468 2073 desired. Both s │ │ │ │ -00008fc0: 7461 6765 7320 6f66 2042 6572 7469 6e69 tages of Bertini │ │ │ │ -00008fd0: 2773 2070 6172 616d 6574 6572 2068 6f6d 's parameter hom │ │ │ │ -00008fe0: 6f74 6f70 790a 6d65 7468 6f64 2061 7265 otopy.method are │ │ │ │ -00008ff0: 2063 616c 6c65 6420 7769 7468 2062 6572 called with ber │ │ │ │ -00009000: 7469 6e69 5061 7261 6d65 7465 7248 6f6d tiniParameterHom │ │ │ │ -00009010: 6f74 6f70 792e 2046 6972 7374 2c20 4265 otopy. First, Be │ │ │ │ -00009020: 7274 696e 6920 6173 7369 676e 7320 610a rtini assigns a. │ │ │ │ -00009030: 7261 6e64 6f6d 2063 6f6d 706c 6578 206e random complex n │ │ │ │ -00009040: 756d 6265 7220 746f 2065 6163 6820 7061 umber to each pa │ │ │ │ -00009050: 7261 6d65 7465 7220 616e 6420 736f 6c76 rameter and solv │ │ │ │ -00009060: 6573 2074 6865 2072 6573 756c 7469 6e67 es the resulting │ │ │ │ -00009070: 2073 7973 7465 6d2c 2074 6865 6e2c 0a61 system, then,.a │ │ │ │ -00009080: 6674 6572 2074 6869 7320 696e 6974 6961 fter this initia │ │ │ │ -00009090: 6c20 7068 6173 652c 2042 6572 7469 6e69 l phase, Bertini │ │ │ │ -000090a0: 2063 6f6d 7075 7465 7320 736f 6c75 7469 computes soluti │ │ │ │ -000090b0: 6f6e 7320 666f 7220 6576 6572 7920 6769 ons for every gi │ │ │ │ -000090c0: 7665 6e20 6368 6f69 6365 206f 660a 7061 ven choice of.pa │ │ │ │ -000090d0: 7261 6d65 7465 7273 2075 7369 6e67 2061 rameters using a │ │ │ │ -000090e0: 206e 756d 6265 7220 6f66 2070 6174 6873 number of paths │ │ │ │ -000090f0: 2065 7175 616c 2074 6f20 7468 6520 6578 equal to the ex │ │ │ │ -00009100: 6163 7420 726f 6f74 2063 6f75 6e74 2069 act root count i │ │ │ │ -00009110: 6e20 7468 6520 6669 7273 740a 7374 6167 n the first.stag │ │ │ │ -00009120: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3132 3633 "/tmp/M2-1263 │ │ │ │ +00008d40: 3834 2d30 2f30 222c 204f 7074 696f 6e20 84-0/0", Option │ │ │ │ +00008d50: 746f 2063 6861 6e67 6520 6469 7265 6374 to change direct │ │ │ │ +00008d60: 6f72 7920 666f 7220 6669 6c65 2073 746f ory for file sto │ │ │ │ +00008d70: 7261 6765 2e0a 2020 2020 2020 2a20 2a6e rage.. * *n │ │ │ │ +00008d80: 6f74 6520 5665 7262 6f73 653a 2062 6572 ote Verbose: ber │ │ │ │ +00008d90: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70 tiniTrackHomotop │ │ │ │ +00008da0: 795f 6c70 5f70 645f 7064 5f70 645f 636d y_lp_pd_pd_pd_cm │ │ │ │ +00008db0: 5665 7262 6f73 653d 3e5f 7064 5f70 645f Verbose=>_pd_pd_ │ │ │ │ +00008dc0: 7064 5f72 700a 2020 2020 2020 2020 2c20 pd_rp. , │ │ │ │ +00008dd0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00008de0: 7661 6c75 6520 6661 6c73 652c 204f 7074 value false, Opt │ │ │ │ +00008df0: 696f 6e20 746f 2073 696c 656e 6365 2061 ion to silence a │ │ │ │ +00008e00: 6464 6974 696f 6e61 6c20 6f75 7470 7574 dditional output │ │ │ │ +00008e10: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00008e20: 2020 2020 2a20 532c 2061 202a 6e6f 7465 * S, a *note │ │ │ │ +00008e30: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ +00008e40: 3244 6f63 294c 6973 742c 2c20 6120 6c69 2Doc)List,, a li │ │ │ │ +00008e50: 7374 2077 686f 7365 2065 6e74 7269 6573 st whose entries │ │ │ │ +00008e60: 2061 7265 206c 6973 7473 206f 660a 2020 are lists of. │ │ │ │ +00008e70: 2020 2020 2020 736f 6c75 7469 6f6e 7320 solutions │ │ │ │ +00008e80: 666f 7220 6561 6368 2074 6172 6765 7420 for each target │ │ │ │ +00008e90: 7379 7374 656d 0a0a 4465 7363 7269 7074 system..Descript │ │ │ │ +00008ea0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00008eb0: 0a54 6869 7320 6d65 7468 6f64 206e 756d .This method num │ │ │ │ +00008ec0: 6572 6963 616c 6c79 2073 6f6c 7665 7320 erically solves │ │ │ │ +00008ed0: 7365 7665 7261 6c20 706f 6c79 6e6f 6d69 several polynomi │ │ │ │ +00008ee0: 616c 2073 7973 7465 6d73 2066 726f 6d20 al systems from │ │ │ │ +00008ef0: 6120 7061 7261 6d65 7465 7269 7a65 640a a parameterized. │ │ │ │ +00008f00: 6661 6d69 6c79 2061 7420 6f6e 6365 2e20 family at once. │ │ │ │ +00008f10: 2054 6865 206c 6973 7420 4620 6973 2061 The list F is a │ │ │ │ +00008f20: 2073 7973 7465 6d20 6f66 2070 6f6c 796e system of polyn │ │ │ │ +00008f30: 6f6d 6961 6c73 2069 6e20 7269 6e67 2076 omials in ring v │ │ │ │ +00008f40: 6172 6961 626c 6573 2061 6e64 0a74 6865 ariables and.the │ │ │ │ +00008f50: 2070 6172 616d 6574 6572 7320 6c69 7374 parameters list │ │ │ │ +00008f60: 6564 2069 6e20 502e 2020 5468 6520 6c69 ed in P. The li │ │ │ │ +00008f70: 7374 2054 2069 7320 7468 6520 7365 7420 st T is the set │ │ │ │ +00008f80: 6f66 2070 6172 616d 6574 6572 2076 616c of parameter val │ │ │ │ +00008f90: 7565 7320 666f 720a 7768 6963 6820 736f ues for.which so │ │ │ │ +00008fa0: 6c75 7469 6f6e 7320 746f 2046 2061 7265 lutions to F are │ │ │ │ +00008fb0: 2064 6573 6972 6564 2e20 2042 6f74 6820 desired. Both │ │ │ │ +00008fc0: 7374 6167 6573 206f 6620 4265 7274 696e stages of Bertin │ │ │ │ +00008fd0: 6927 7320 7061 7261 6d65 7465 7220 686f i's parameter ho │ │ │ │ +00008fe0: 6d6f 746f 7079 0a6d 6574 686f 6420 6172 motopy.method ar │ │ │ │ +00008ff0: 6520 6361 6c6c 6564 2077 6974 6820 6265 e called with be │ │ │ │ +00009000: 7274 696e 6950 6172 616d 6574 6572 486f rtiniParameterHo │ │ │ │ +00009010: 6d6f 746f 7079 2e20 4669 7273 742c 2042 motopy. First, B │ │ │ │ +00009020: 6572 7469 6e69 2061 7373 6967 6e73 2061 ertini assigns a │ │ │ │ +00009030: 0a72 616e 646f 6d20 636f 6d70 6c65 7820 .random complex │ │ │ │ +00009040: 6e75 6d62 6572 2074 6f20 6561 6368 2070 number to each p │ │ │ │ +00009050: 6172 616d 6574 6572 2061 6e64 2073 6f6c arameter and sol │ │ │ │ +00009060: 7665 7320 7468 6520 7265 7375 6c74 696e ves the resultin │ │ │ │ +00009070: 6720 7379 7374 656d 2c20 7468 656e 2c0a g system, then,. │ │ │ │ +00009080: 6166 7465 7220 7468 6973 2069 6e69 7469 after this initi │ │ │ │ +00009090: 616c 2070 6861 7365 2c20 4265 7274 696e al phase, Bertin │ │ │ │ +000090a0: 6920 636f 6d70 7574 6573 2073 6f6c 7574 i computes solut │ │ │ │ +000090b0: 696f 6e73 2066 6f72 2065 7665 7279 2067 ions for every g │ │ │ │ +000090c0: 6976 656e 2063 686f 6963 6520 6f66 0a70 iven choice of.p │ │ │ │ +000090d0: 6172 616d 6574 6572 7320 7573 696e 6720 arameters using │ │ │ │ +000090e0: 6120 6e75 6d62 6572 206f 6620 7061 7468 a number of path │ │ │ │ +000090f0: 7320 6571 7561 6c20 746f 2074 6865 2065 s equal to the e │ │ │ │ +00009100: 7861 6374 2072 6f6f 7420 636f 756e 7420 xact root count │ │ │ │ +00009110: 696e 2074 6865 2066 6972 7374 0a73 7461 in the first.sta │ │ │ │ +00009120: 6765 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ge...+---------- │ │ │ │ 00009130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009170: 2d2d 2b0a 7c69 3120 3a20 523d 4343 5b75 --+.|i1 : R=CC[u │ │ │ │ -00009180: 312c 7532 2c75 332c 782c 795d 3b20 2020 1,u2,u3,x,y]; │ │ │ │ +00009170: 2d2d 2d2b 0a7c 6931 203a 2052 3d43 435b ---+.|i1 : R=CC[ │ │ │ │ +00009180: 7531 2c75 322c 7533 2c78 2c79 5d3b 2020 u1,u2,u3,x,y]; │ │ │ │ 00009190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000091a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000091b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000091c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000091c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000091d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000091e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000091f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009210: 2d2d 2b0a 7c69 3220 3a20 6631 3d75 312a --+.|i2 : f1=u1* │ │ │ │ -00009220: 2879 2d31 292b 7532 2a28 792d 3229 2b75 (y-1)+u2*(y-2)+u │ │ │ │ -00009230: 332a 2879 2d33 293b 202d 2d70 6172 616d 3*(y-3); --param │ │ │ │ -00009240: 6574 6572 7320 6172 6520 7531 2c20 7532 eters are u1, u2 │ │ │ │ -00009250: 2c20 616e 6420 7533 2020 2020 2020 2020 , and u3 │ │ │ │ -00009260: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00009210: 2d2d 2d2b 0a7c 6932 203a 2066 313d 7531 ---+.|i2 : f1=u1 │ │ │ │ +00009220: 2a28 792d 3129 2b75 322a 2879 2d32 292b *(y-1)+u2*(y-2)+ │ │ │ │ +00009230: 7533 2a28 792d 3329 3b20 2d2d 7061 7261 u3*(y-3); --para │ │ │ │ +00009240: 6d65 7465 7273 2061 7265 2075 312c 2075 meters are u1, u │ │ │ │ +00009250: 322c 2061 6e64 2075 3320 2020 2020 2020 2, and u3 │ │ │ │ +00009260: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00009270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000092a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000092b0: 2d2d 2b0a 7c69 3320 3a20 6632 3d28 782d --+.|i3 : f2=(x- │ │ │ │ -000092c0: 3131 292a 2878 2d31 3229 2a28 782d 3133 11)*(x-12)*(x-13 │ │ │ │ -000092d0: 292d 7531 3b20 2020 2020 2020 2020 2020 )-u1; │ │ │ │ +000092b0: 2d2d 2d2b 0a7c 6933 203a 2066 323d 2878 ---+.|i3 : f2=(x │ │ │ │ +000092c0: 2d31 3129 2a28 782d 3132 292a 2878 2d31 -11)*(x-12)*(x-1 │ │ │ │ +000092d0: 3329 2d75 313b 2020 2020 2020 2020 2020 3)-u1; │ │ │ │ 000092e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000092f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009300: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00009300: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00009310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009350: 2d2d 2b0a 7c69 3420 3a20 7061 7261 6d56 --+.|i4 : paramV │ │ │ │ -00009360: 616c 7565 7330 3d7b 312c 302c 307d 3b20 alues0={1,0,0}; │ │ │ │ +00009350: 2d2d 2d2b 0a7c 6934 203a 2070 6172 616d ---+.|i4 : param │ │ │ │ +00009360: 5661 6c75 6573 303d 7b31 2c30 2c30 7d3b Values0={1,0,0}; │ │ │ │ 00009370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000093a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000093a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000093b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000093c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000093d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000093e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000093f0: 2d2d 2b0a 7c69 3520 3a20 7061 7261 6d56 --+.|i5 : paramV │ │ │ │ -00009400: 616c 7565 7331 3d7b 302c 312b 322a 6969 alues1={0,1+2*ii │ │ │ │ -00009410: 2c30 7d3b 2020 2020 2020 2020 2020 2020 ,0}; │ │ │ │ +000093f0: 2d2d 2d2b 0a7c 6935 203a 2070 6172 616d ---+.|i5 : param │ │ │ │ +00009400: 5661 6c75 6573 313d 7b30 2c31 2b32 2a69 Values1={0,1+2*i │ │ │ │ +00009410: 692c 307d 3b20 2020 2020 2020 2020 2020 i,0}; │ │ │ │ 00009420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009440: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00009440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00009450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009490: 2d2d 2b0a 7c69 3620 3a20 6250 483d 6265 --+.|i6 : bPH=be │ │ │ │ -000094a0: 7274 696e 6950 6172 616d 6574 6572 486f rtiniParameterHo │ │ │ │ -000094b0: 6d6f 746f 7079 2820 7b66 312c 6632 7d2c motopy( {f1,f2}, │ │ │ │ -000094c0: 207b 7531 2c75 322c 7533 7d2c 7b70 6172 {u1,u2,u3},{par │ │ │ │ -000094d0: 616d 5661 6c75 6573 3020 2c70 6172 616d amValues0 ,param │ │ │ │ -000094e0: 5661 7c0a 7c20 2020 2020 2020 2020 2020 Va|.| │ │ │ │ +00009490: 2d2d 2d2b 0a7c 6936 203a 2062 5048 3d62 ---+.|i6 : bPH=b │ │ │ │ +000094a0: 6572 7469 6e69 5061 7261 6d65 7465 7248 ertiniParameterH │ │ │ │ +000094b0: 6f6d 6f74 6f70 7928 207b 6631 2c66 327d omotopy( {f1,f2} │ │ │ │ +000094c0: 2c20 7b75 312c 7532 2c75 337d 2c7b 7061 , {u1,u2,u3},{pa │ │ │ │ +000094d0: 7261 6d56 616c 7565 7330 202c 7061 7261 ramValues0 ,para │ │ │ │ +000094e0: 6d56 617c 0a7c 2020 2020 2020 2020 2020 mVa|.| │ │ │ │ 000094f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009530: 2020 7c0a 7c6f 3620 3d20 7b7b 7b31 312e |.|o6 = {{{11. │ │ │ │ -00009540: 3333 3736 2d2e 3536 3232 382a 6969 2c20 3376-.56228*ii, │ │ │ │ -00009550: 317d 2c20 7b31 312e 3333 3736 2b2e 3536 1}, {11.3376+.56 │ │ │ │ -00009560: 3232 382a 6969 2c20 317d 2c20 7b31 332e 228*ii, 1}, {13. │ │ │ │ -00009570: 3332 3437 2c20 317d 7d2c 207b 7b31 312c 3247, 1}}, {{11, │ │ │ │ -00009580: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00009530: 2020 207c 0a7c 6f36 203d 207b 7b7b 3131 |.|o6 = {{{11 │ │ │ │ +00009540: 2e33 3337 362d 2e35 3632 3238 2a69 692c .3376-.56228*ii, │ │ │ │ +00009550: 2031 7d2c 207b 3131 2e33 3337 362b 2e35 1}, {11.3376+.5 │ │ │ │ +00009560: 3632 3238 2a69 692c 2031 7d2c 207b 3133 6228*ii, 1}, {13 │ │ │ │ +00009570: 2e33 3234 372c 2031 7d7d 2c20 7b7b 3131 .3247, 1}}, {{11 │ │ │ │ +00009580: 2c20 207c 0a7c 2020 2020 202d 2d2d 2d2d , |.| ----- │ │ │ │ 00009590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000095a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000095b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000095c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000095d0: 2d2d 7c0a 7c20 2020 2020 327d 2c20 7b31 --|.| 2}, {1 │ │ │ │ -000095e0: 322c 2032 7d2c 207b 3133 2c20 327d 7d7d 2, 2}, {13, 2}}} │ │ │ │ -000095f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000095d0: 2d2d 2d7c 0a7c 2020 2020 2032 7d2c 207b ---|.| 2}, { │ │ │ │ +000095e0: 3132 2c20 327d 2c20 7b31 332c 2032 7d7d 12, 2}, {13, 2}} │ │ │ │ +000095f0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ 00009600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009620: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009620: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009670: 2020 7c0a 7c6f 3620 3a20 4c69 7374 2020 |.|o6 : List │ │ │ │ +00009670: 2020 207c 0a7c 6f36 203a 204c 6973 7420 |.|o6 : List │ │ │ │ 00009680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000096a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000096b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000096c0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +000096c0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 000096d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000096e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000096f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009710: 2d2d 7c0a 7c6c 7565 7331 207d 2920 2020 --|.|lues1 }) │ │ │ │ +00009710: 2d2d 2d7c 0a7c 6c75 6573 3120 7d29 2020 ---|.|lues1 }) │ │ │ │ 00009720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009760: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00009760: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00009770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000097b0: 2d2d 2b0a 7c69 3720 3a20 6250 485f 302d --+.|i7 : bPH_0- │ │ │ │ -000097c0: 2d74 6865 2073 6f6c 7574 696f 6e73 2074 -the solutions t │ │ │ │ -000097d0: 6f20 7468 6520 7379 7374 656d 2077 6974 o the system wit │ │ │ │ -000097e0: 6820 7061 7261 6d65 7465 7273 2073 6574 h parameters set │ │ │ │ -000097f0: 2065 7175 616c 2074 6f20 2020 2020 2020 equal to │ │ │ │ -00009800: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000097b0: 2d2d 2d2b 0a7c 6937 203a 2062 5048 5f30 ---+.|i7 : bPH_0 │ │ │ │ +000097c0: 2d2d 7468 6520 736f 6c75 7469 6f6e 7320 --the solutions │ │ │ │ +000097d0: 746f 2074 6865 2073 7973 7465 6d20 7769 to the system wi │ │ │ │ +000097e0: 7468 2070 6172 616d 6574 6572 7320 7365 th parameters se │ │ │ │ +000097f0: 7420 6571 7561 6c20 746f 2020 2020 2020 t equal to │ │ │ │ +00009800: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009850: 2020 7c0a 7c6f 3720 3d20 7b7b 3131 2e33 |.|o7 = {{11.3 │ │ │ │ -00009860: 3337 362d 2e35 3632 3238 2a69 692c 2031 376-.56228*ii, 1 │ │ │ │ -00009870: 7d2c 207b 3131 2e33 3337 362b 2e35 3632 }, {11.3376+.562 │ │ │ │ -00009880: 3238 2a69 692c 2031 7d2c 207b 3133 2e33 28*ii, 1}, {13.3 │ │ │ │ -00009890: 3234 372c 2031 7d7d 2020 2020 2020 2020 247, 1}} │ │ │ │ -000098a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009850: 2020 207c 0a7c 6f37 203d 207b 7b31 312e |.|o7 = {{11. │ │ │ │ +00009860: 3333 3736 2d2e 3536 3232 382a 6969 2c20 3376-.56228*ii, │ │ │ │ +00009870: 317d 2c20 7b31 312e 3333 3736 2b2e 3536 1}, {11.3376+.56 │ │ │ │ +00009880: 3232 382a 6969 2c20 317d 2c20 7b31 332e 228*ii, 1}, {13. │ │ │ │ +00009890: 3332 3437 2c20 317d 7d20 2020 2020 2020 3247, 1}} │ │ │ │ +000098a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000098b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000098c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000098d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000098e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000098f0: 2020 7c0a 7c6f 3720 3a20 4c69 7374 2020 |.|o7 : List │ │ │ │ +000098f0: 2020 207c 0a7c 6f37 203a 204c 6973 7420 |.|o7 : List │ │ │ │ 00009900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009940: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +00009940: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 00009950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009990: 2d2d 7c0a 7c70 6172 616d 5661 6c75 6573 --|.|paramValues │ │ │ │ -000099a0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00009990: 2d2d 2d7c 0a7c 7061 7261 6d56 616c 7565 ---|.|paramValue │ │ │ │ +000099a0: 7330 2020 2020 2020 2020 2020 2020 2020 s0 │ │ │ │ 000099b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000099c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000099d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000099e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000099e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000099f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009a30: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ +00009a30: 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ---+.+---------- │ │ │ │ 00009a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009a80: 2d2d 2b0a 7c69 3820 3a20 523d 4343 5b78 --+.|i8 : R=CC[x │ │ │ │ -00009a90: 2c79 2c7a 2c75 312c 7532 5d20 2020 2020 ,y,z,u1,u2] │ │ │ │ +00009a80: 2d2d 2d2b 0a7c 6938 203a 2052 3d43 435b ---+.|i8 : R=CC[ │ │ │ │ +00009a90: 782c 792c 7a2c 7531 2c75 325d 2020 2020 x,y,z,u1,u2] │ │ │ │ 00009aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009ad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009b20: 2020 7c0a 7c6f 3820 3d20 5220 2020 2020 |.|o8 = R │ │ │ │ +00009b20: 2020 207c 0a7c 6f38 203d 2052 2020 2020 |.|o8 = R │ │ │ │ 00009b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009b70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009b70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009bc0: 2020 7c0a 7c6f 3820 3a20 506f 6c79 6e6f |.|o8 : Polyno │ │ │ │ -00009bd0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00009bc0: 2020 207c 0a7c 6f38 203a 2050 6f6c 796e |.|o8 : Polyn │ │ │ │ +00009bd0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ 00009be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009c10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00009c10: 2020 207c 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│ │ │ │ 00009e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009e90: 2d2d 2b0a 7c69 3130 203a 2066 323d 7531 --+.|i10 : f2=u1 │ │ │ │ -00009ea0: 2a78 2b75 322a 7920 2020 2020 2020 2020 *x+u2*y │ │ │ │ +00009e90: 2d2d 2d2b 0a7c 6931 3020 3a20 6632 3d75 ---+.|i10 : f2=u │ │ │ │ +00009ea0: 312a 782b 7532 2a79 2020 2020 2020 2020 1*x+u2*y │ │ │ │ 00009eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009f30: 2020 7c0a 7c6f 3130 203d 2078 2a75 3120 |.|o10 = x*u1 │ │ │ │ -00009f40: 2b20 792a 7532 2020 2020 2020 2020 2020 + y*u2 │ │ │ │ +00009f30: 2020 207c 0a7c 6f31 3020 3d20 782a 7531 |.|o10 = x*u1 │ │ │ │ +00009f40: 202b 2079 2a75 3220 2020 2020 2020 2020 + y*u2 │ │ │ │ 00009f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009f80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00009f80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009fd0: 2020 7c0a 7c6f 3130 203a 2052 2020 2020 |.|o10 : R │ │ │ │ +00009fd0: 2020 207c 0a7c 6f31 3020 3a20 5220 2020 |.|o10 : R │ │ │ │ 00009fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a020: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000a020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a070: 2d2d 2b0a 7c69 3131 203a 2066 696e 616c --+.|i11 : final │ │ │ │ -0000a080: 5061 7261 6d65 7465 7273 303d 7b30 2c31 Parameters0={0,1 │ │ │ │ -0000a090: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000a070: 2d2d 2d2b 0a7c 6931 3120 3a20 6669 6e61 ---+.|i11 : fina │ │ │ │ +0000a080: 6c50 6172 616d 6574 6572 7330 3d7b 302c lParameters0={0, │ │ │ │ +0000a090: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ 0000a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a0c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a0c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a110: 2020 7c0a 7c6f 3131 203d 207b 302c 2031 |.|o11 = {0, 1 │ │ │ │ -0000a120: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000a110: 2020 207c 0a7c 6f31 3120 3d20 7b30 2c20 |.|o11 = {0, │ │ │ │ +0000a120: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ 0000a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a160: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a160: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a1b0: 2020 7c0a 7c6f 3131 203a 204c 6973 7420 |.|o11 : List │ │ │ │ +0000a1b0: 2020 207c 0a7c 6f31 3120 3a20 4c69 7374 |.|o11 : List │ │ │ │ 0000a1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a200: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000a200: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a250: 2d2d 2b0a 7c69 3132 203a 2066 696e 616c --+.|i12 : final │ │ │ │ -0000a260: 5061 7261 6d65 7465 7273 313d 7b31 2c30 Parameters1={1,0 │ │ │ │ -0000a270: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000a250: 2d2d 2d2b 0a7c 6931 3220 3a20 6669 6e61 ---+.|i12 : fina │ │ │ │ +0000a260: 6c50 6172 616d 6574 6572 7331 3d7b 312c lParameters1={1, │ │ │ │ +0000a270: 307d 2020 2020 2020 2020 2020 2020 2020 0} │ │ │ │ 0000a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a2a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a2a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a2f0: 2020 7c0a 7c6f 3132 203d 207b 312c 2030 |.|o12 = {1, 0 │ │ │ │ -0000a300: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000a2f0: 2020 207c 0a7c 6f31 3220 3d20 7b31 2c20 |.|o12 = {1, │ │ │ │ +0000a300: 307d 2020 2020 2020 2020 2020 2020 2020 0} │ │ │ │ 0000a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a340: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a340: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a390: 2020 7c0a 7c6f 3132 203a 204c 6973 7420 |.|o12 : List │ │ │ │ +0000a390: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374 |.|o12 : List │ │ │ │ 0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a3e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000a3e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000a3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a430: 2d2d 2b0a 7c69 3133 203a 2062 5048 3d62 --+.|i13 : bPH=b │ │ │ │ -0000a440: 6572 7469 6e69 5061 7261 6d65 7465 7248 ertiniParameterH │ │ │ │ -0000a450: 6f6d 6f74 6f70 7928 207b 6631 2c66 327d omotopy( {f1,f2} │ │ │ │ -0000a460: 2c20 7b75 312c 7532 7d2c 7b66 696e 616c , {u1,u2},{final │ │ │ │ -0000a470: 5061 7261 6d65 7465 7273 3020 2c66 696e Parameters0 ,fin │ │ │ │ -0000a480: 616c 7c0a 7c20 2020 2020 2020 2020 2020 al|.| │ │ │ │ +0000a430: 2d2d 2d2b 0a7c 6931 3320 3a20 6250 483d ---+.|i13 : bPH= │ │ │ │ +0000a440: 6265 7274 696e 6950 6172 616d 6574 6572 bertiniParameter │ │ │ │ +0000a450: 486f 6d6f 746f 7079 2820 7b66 312c 6632 Homotopy( {f1,f2 │ │ │ │ +0000a460: 7d2c 207b 7531 2c75 327d 2c7b 6669 6e61 }, {u1,u2},{fina │ │ │ │ +0000a470: 6c50 6172 616d 6574 6572 7330 202c 6669 lParameters0 ,fi │ │ │ │ +0000a480: 6e61 6c7c 0a7c 2020 2020 2020 2020 2020 nal|.| │ │ │ │ 0000a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a4d0: 2020 7c0a 7c6f 3133 203d 207b 7b7b 312c |.|o13 = {{{1, │ │ │ │ -0000a4e0: 2031 2e30 3436 3334 652d 3137 2d31 2e30 1.04634e-17-1.0 │ │ │ │ -0000a4f0: 3134 3438 652d 3137 2a69 692c 202d 317d 1448e-17*ii, -1} │ │ │ │ -0000a500: 2c20 7b31 2c20 312e 3332 3131 3165 2d31 , {1, 1.32111e-1 │ │ │ │ -0000a510: 372b 362e 3431 3865 2d32 302a 6969 2c20 7+6.418e-20*ii, │ │ │ │ -0000a520: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ +0000a4d0: 2020 207c 0a7c 6f31 3320 3d20 7b7b 7b31 |.|o13 = {{{1 │ │ │ │ +0000a4e0: 2c20 312e 3034 3633 3465 2d31 372d 312e , 1.04634e-17-1. │ │ │ │ +0000a4f0: 3031 3434 3865 2d31 372a 6969 2c20 2d31 01448e-17*ii, -1 │ │ │ │ +0000a500: 7d2c 207b 312c 2031 2e33 3231 3131 652d }, {1, 1.32111e- │ │ │ │ +0000a510: 3137 2b36 2e34 3138 652d 3230 2a69 692c 17+6.418e-20*ii, │ │ │ │ +0000a520: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ 0000a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a570: 2d2d 7c0a 7c20 2020 2020 2031 7d7d 2c20 --|.| 1}}, │ │ │ │ -0000a580: 7b7b 392e 3937 3833 3365 2d31 392b 312e {{9.97833e-19+1. │ │ │ │ -0000a590: 3039 3138 3565 2d31 382a 6969 2c20 312c 09185e-18*ii, 1, │ │ │ │ -0000a5a0: 2031 7d2c 207b 2d35 2e34 3138 3834 652d 1}, {-5.41884e- │ │ │ │ -0000a5b0: 3136 2b31 2e34 3132 3031 652d 3136 2a69 16+1.41201e-16*i │ │ │ │ -0000a5c0: 692c 7c0a 7c20 2020 2020 202d 2d2d 2d2d i,|.| ----- │ │ │ │ +0000a570: 2d2d 2d7c 0a7c 2020 2020 2020 317d 7d2c ---|.| 1}}, │ │ │ │ +0000a580: 207b 7b39 2e39 3738 3333 652d 3139 2b31 {{9.97833e-19+1 │ │ │ │ +0000a590: 2e30 3931 3835 652d 3138 2a69 692c 2031 .09185e-18*ii, 1 │ │ │ │ +0000a5a0: 2c20 317d 2c20 7b2d 352e 3431 3838 3465 , 1}, {-5.41884e │ │ │ │ +0000a5b0: 2d31 362b 312e 3431 3230 3165 2d31 362a -16+1.41201e-16* │ │ │ │ +0000a5c0: 6969 2c7c 0a7c 2020 2020 2020 2d2d 2d2d ii,|.| ---- │ │ │ │ 0000a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a610: 2d2d 7c0a 7c20 2020 2020 2031 2c20 2d31 --|.| 1, -1 │ │ │ │ -0000a620: 7d7d 7d20 2020 2020 2020 2020 2020 2020 }}} │ │ │ │ +0000a610: 2d2d 2d7c 0a7c 2020 2020 2020 312c 202d ---|.| 1, - │ │ │ │ +0000a620: 317d 7d7d 2020 2020 2020 2020 2020 2020 1}}} │ │ │ │ 0000a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a660: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a660: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a6b0: 2020 7c0a 7c6f 3133 203a 204c 6973 7420 |.|o13 : List │ │ │ │ +0000a6b0: 2020 207c 0a7c 6f31 3320 3a20 4c69 7374 |.|o13 : List │ │ │ │ 0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a700: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0000a700: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a750: 2d2d 7c0a 7c50 6172 616d 6574 6572 7331 --|.|Parameters1 │ │ │ │ -0000a760: 207d 2c48 6f6d 5661 7269 6162 6c65 4772 },HomVariableGr │ │ │ │ -0000a770: 6f75 703d 3e7b 782c 792c 7a7d 2920 2020 oup=>{x,y,z}) │ │ │ │ +0000a750: 2d2d 2d7c 0a7c 5061 7261 6d65 7465 7273 ---|.|Parameters │ │ │ │ +0000a760: 3120 7d2c 486f 6d56 6172 6961 626c 6547 1 },HomVariableG │ │ │ │ +0000a770: 726f 7570 3d3e 7b78 2c79 2c7a 7d29 2020 roup=>{x,y,z}) │ │ │ │ 0000a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a7a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000a7a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a7f0: 2d2d 2b0a 7c69 3134 203a 2062 5048 5f30 --+.|i14 : bPH_0 │ │ │ │ -0000a800: 2d2d 5468 6520 7477 6f20 736f 6c75 7469 --The two soluti │ │ │ │ -0000a810: 6f6e 7320 666f 7220 6669 6e61 6c50 6172 ons for finalPar │ │ │ │ -0000a820: 616d 6574 6572 7330 2020 2020 2020 2020 ameters0 │ │ │ │ +0000a7f0: 2d2d 2d2b 0a7c 6931 3420 3a20 6250 485f ---+.|i14 : bPH_ │ │ │ │ +0000a800: 302d 2d54 6865 2074 776f 2073 6f6c 7574 0--The two solut │ │ │ │ +0000a810: 696f 6e73 2066 6f72 2066 696e 616c 5061 ions for finalPa │ │ │ │ +0000a820: 7261 6d65 7465 7273 3020 2020 2020 2020 rameters0 │ │ │ │ 0000a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a840: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000a840: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a890: 2020 7c0a 7c6f 3134 203d 207b 7b31 2c20 |.|o14 = {{1, │ │ │ │ -0000a8a0: 312e 3034 3633 3465 2d31 372d 312e 3031 1.04634e-17-1.01 │ │ │ │ -0000a8b0: 3434 3865 2d31 372a 6969 2c20 2d31 7d2c 448e-17*ii, -1}, │ │ │ │ -0000a8c0: 207b 312c 2031 2e33 3231 3131 652d 3137 {1, 1.32111e-17 │ │ │ │ -0000a8d0: 2b36 2e34 3138 652d 3230 2a69 692c 2031 +6.418e-20*ii, 1 │ │ │ │ -0000a8e0: 7d7d 7c0a 7c20 2020 2020 2020 2020 2020 }}|.| │ │ │ │ +0000a890: 2020 207c 0a7c 6f31 3420 3d20 7b7b 312c |.|o14 = {{1, │ │ │ │ +0000a8a0: 2031 2e30 3436 3334 652d 3137 2d31 2e30 1.04634e-17-1.0 │ │ │ │ +0000a8b0: 3134 3438 652d 3137 2a69 692c 202d 317d 1448e-17*ii, -1} │ │ │ │ +0000a8c0: 2c20 7b31 2c20 312e 3332 3131 3165 2d31 , {1, 1.32111e-1 │ │ │ │ +0000a8d0: 372b 362e 3431 3865 2d32 302a 6969 2c20 7+6.418e-20*ii, │ │ │ │ +0000a8e0: 317d 7d7c 0a7c 2020 2020 2020 2020 2020 1}}|.| │ │ │ │ 0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a930: 2020 7c0a 7c6f 3134 203a 204c 6973 7420 |.|o14 : List │ │ │ │ +0000a930: 2020 207c 0a7c 6f31 3420 3a20 4c69 7374 |.|o14 : List │ │ │ │ 0000a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a980: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000a980: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000a990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000a9d0: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ +0000a9d0: 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ---+.+---------- │ │ │ │ 0000a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000aa20: 2d2d 2b0a 7c69 3135 203a 2066 696e 5061 --+.|i15 : finPa │ │ │ │ -0000aa30: 7261 6d56 616c 7565 733d 7b7b 317d 2c7b ramValues={{1},{ │ │ │ │ -0000aa40: 327d 7d20 2020 2020 2020 2020 2020 2020 2}} │ │ │ │ +0000aa20: 2d2d 2d2b 0a7c 6931 3520 3a20 6669 6e50 ---+.|i15 : finP │ │ │ │ +0000aa30: 6172 616d 5661 6c75 6573 3d7b 7b31 7d2c aramValues={{1}, │ │ │ │ +0000aa40: 7b32 7d7d 2020 2020 2020 2020 2020 2020 {2}} │ │ │ │ 0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000aa70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000aa70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000aac0: 2020 7c0a 7c6f 3135 203d 207b 7b31 7d2c |.|o15 = {{1}, │ │ │ │ -0000aad0: 207b 327d 7d20 2020 2020 2020 2020 2020 {2}} │ │ │ │ +0000aac0: 2020 207c 0a7c 6f31 3520 3d20 7b7b 317d |.|o15 = {{1} │ │ │ │ +0000aad0: 2c20 7b32 7d7d 2020 2020 2020 2020 2020 , {2}} │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ab10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000ab10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ab60: 2020 7c0a 7c6f 3135 203a 204c 6973 7420 |.|o15 : List │ │ │ │ +0000ab60: 2020 207c 0a7c 6f31 3520 3a20 4c69 7374 |.|o15 : List │ │ │ │ 0000ab70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000abb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000abb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ac00: 2d2d 2b0a 7c69 3136 203a 2062 5048 313d --+.|i16 : bPH1= │ │ │ │ -0000ac10: 6265 7274 696e 6950 6172 616d 6574 6572 bertiniParameter │ │ │ │ -0000ac20: 486f 6d6f 746f 7079 2820 7b22 785e 322d Homotopy( {"x^2- │ │ │ │ -0000ac30: 7531 227d 2c20 2020 2020 2020 2020 2020 u1"}, │ │ │ │ +0000ac00: 2d2d 2d2b 0a7c 6931 3620 3a20 6250 4831 ---+.|i16 : bPH1 │ │ │ │ +0000ac10: 3d62 6572 7469 6e69 5061 7261 6d65 7465 =bertiniParamete │ │ │ │ +0000ac20: 7248 6f6d 6f74 6f70 7928 207b 2278 5e32 rHomotopy( {"x^2 │ │ │ │ +0000ac30: 2d75 3122 7d2c 2020 2020 2020 2020 2020 -u1"}, │ │ │ │ 0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ac50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000ac50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000aca0: 2020 7c0a 7c6f 3136 203d 207b 7b7b 2d31 |.|o16 = {{{-1 │ │ │ │ -0000acb0: 7d2c 207b 317d 7d2c 207b 7b2d 312e 3431 }, {1}}, {{-1.41 │ │ │ │ -0000acc0: 3432 317d 2c20 7b31 2e34 3134 3231 7d7d 421}, {1.41421}} │ │ │ │ -0000acd0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000aca0: 2020 207c 0a7c 6f31 3620 3d20 7b7b 7b2d |.|o16 = {{{- │ │ │ │ +0000acb0: 317d 2c20 7b31 7d7d 2c20 7b7b 2d31 2e34 1}, {1}}, {{-1.4 │ │ │ │ +0000acc0: 3134 3231 7d2c 207b 312e 3431 3432 317d 1421}, {1.41421} │ │ │ │ +0000acd0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 0000ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000acf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000acf0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ad40: 2020 7c0a 7c6f 3136 203a 204c 6973 7420 |.|o16 : List │ │ │ │ +0000ad40: 2020 207c 0a7c 6f31 3620 3a20 4c69 7374 |.|o16 : List │ │ │ │ 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 2020 2020 2020 2020 2020 │ │ │ │ -0000ad90: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0000ad90: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ade0: 2d2d 7c0a 7c7b 7531 7d2c 6669 6e50 6172 --|.|{u1},finPar │ │ │ │ -0000adf0: 616d 5661 6c75 6573 2c41 6666 5661 7269 amValues,AffVari │ │ │ │ -0000ae00: 6162 6c65 4772 6f75 703d 3e7b 787d 2920 ableGroup=>{x}) │ │ │ │ +0000ade0: 2d2d 2d7c 0a7c 7b75 317d 2c66 696e 5061 ---|.|{u1},finPa │ │ │ │ +0000adf0: 7261 6d56 616c 7565 732c 4166 6656 6172 ramValues,AffVar │ │ │ │ +0000ae00: 6961 626c 6547 726f 7570 3d3e 7b78 7d29 iableGroup=>{x}) │ │ │ │ 0000ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ae30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000ae30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ae80: 2d2d 2b0a 7c69 3137 203a 2062 5048 323d --+.|i17 : bPH2= │ │ │ │ -0000ae90: 6265 7274 696e 6950 6172 616d 6574 6572 bertiniParameter │ │ │ │ -0000aea0: 486f 6d6f 746f 7079 2820 7b22 785e 322d Homotopy( {"x^2- │ │ │ │ -0000aeb0: 7531 227d 2c20 2020 2020 2020 2020 2020 u1"}, │ │ │ │ +0000ae80: 2d2d 2d2b 0a7c 6931 3720 3a20 6250 4832 ---+.|i17 : bPH2 │ │ │ │ +0000ae90: 3d62 6572 7469 6e69 5061 7261 6d65 7465 =bertiniParamete │ │ │ │ +0000aea0: 7248 6f6d 6f74 6f70 7928 207b 2278 5e32 rHomotopy( {"x^2 │ │ │ │ +0000aeb0: 2d75 3122 7d2c 2020 2020 2020 2020 2020 -u1"}, │ │ │ │ 0000aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000aed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000aed0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000af20: 2020 7c0a 7c6f 3137 203d 207b 7b7b 2d31 |.|o17 = {{{-1 │ │ │ │ -0000af30: 7d2c 207b 317d 7d2c 207b 7b2d 312e 3431 }, {1}}, {{-1.41 │ │ │ │ -0000af40: 3432 317d 2c20 7b31 2e34 3134 3231 7d7d 421}, {1.41421}} │ │ │ │ -0000af50: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000af20: 2020 207c 0a7c 6f31 3720 3d20 7b7b 7b2d |.|o17 = {{{- │ │ │ │ +0000af30: 317d 2c20 7b31 7d7d 2c20 7b7b 2d31 2e34 1}, {1}}, {{-1.4 │ │ │ │ +0000af40: 3134 3231 7d2c 207b 312e 3431 3432 317d 1421}, {1.41421} │ │ │ │ +0000af50: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 0000af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000af70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000af70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000afc0: 2020 7c0a 7c6f 3137 203a 204c 6973 7420 |.|o17 : List │ │ │ │ +0000afc0: 2020 207c 0a7c 6f31 3720 3a20 4c69 7374 |.|o17 : List │ │ │ │ 0000afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b010: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0000b010: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b060: 2d2d 7c0a 7c7b 7531 7d2c 6669 6e50 6172 --|.|{u1},finPar │ │ │ │ -0000b070: 616d 5661 6c75 6573 2c41 6666 5661 7269 amValues,AffVari │ │ │ │ -0000b080: 6162 6c65 4772 6f75 703d 3e7b 787d 2c4f ableGroup=>{x},O │ │ │ │ -0000b090: 7574 7075 7453 7479 6c65 3d3e 224f 7574 utputStyle=>"Out │ │ │ │ -0000b0a0: 536f 6c75 7469 6f6e 7322 2920 2020 2020 Solutions") │ │ │ │ -0000b0b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000b060: 2d2d 2d7c 0a7c 7b75 317d 2c66 696e 5061 ---|.|{u1},finPa │ │ │ │ +0000b070: 7261 6d56 616c 7565 732c 4166 6656 6172 ramValues,AffVar │ │ │ │ +0000b080: 6961 626c 6547 726f 7570 3d3e 7b78 7d2c iableGroup=>{x}, │ │ │ │ +0000b090: 4f75 7470 7574 5374 796c 653d 3e22 4f75 OutputStyle=>"Ou │ │ │ │ +0000b0a0: 7453 6f6c 7574 696f 6e73 2229 2020 2020 tSolutions") │ │ │ │ +0000b0b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b100: 2d2d 2b0a 7c69 3138 203a 2063 6c61 7373 --+.|i18 : class │ │ │ │ -0000b110: 2062 5048 315f 305f 3020 2020 2020 2020 bPH1_0_0 │ │ │ │ +0000b100: 2d2d 2d2b 0a7c 6931 3820 3a20 636c 6173 ---+.|i18 : clas │ │ │ │ +0000b110: 7320 6250 4831 5f30 5f30 2020 2020 2020 s bPH1_0_0 │ │ │ │ 0000b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b150: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b150: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b1a0: 2020 7c0a 7c6f 3138 203d 2050 6f69 6e74 |.|o18 = Point │ │ │ │ -0000b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b1a0: 2020 207c 0a7c 6f31 3820 3d20 506f 696e |.|o18 = Poin │ │ │ │ +0000b1b0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0000b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b1f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b1f0: 2020 207c 0a7c 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 2020 2020 │ │ │ │ -0000b240: 2020 7c0a 7c6f 3138 203a 2054 7970 6520 |.|o18 : Type │ │ │ │ +0000b240: 2020 207c 0a7c 6f31 3820 3a20 5479 7065 |.|o18 : Type │ │ │ │ 0000b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b260: 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000b290: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000b2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b2e0: 2d2d 2b0a 7c69 3139 203a 2063 6c61 7373 --+.|i19 : class │ │ │ │ -0000b2f0: 2062 5048 325f 305f 3020 2020 2020 2020 bPH2_0_0 │ │ │ │ +0000b2e0: 2d2d 2d2b 0a7c 6931 3920 3a20 636c 6173 ---+.|i19 : clas │ │ │ │ +0000b2f0: 7320 6250 4832 5f30 5f30 2020 2020 2020 s bPH2_0_0 │ │ │ │ 0000b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b330: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b330: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b380: 2020 7c0a 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2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b470: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000b470: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000b480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b4c0: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ +0000b4c0: 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ---+.+---------- │ │ │ │ 0000b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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---------------- │ │ │ │ -0000b650: 2d2d 2b0a 7c69 3231 203a 206d 616b 6544 --+.|i21 : makeD │ │ │ │ -0000b660: 6972 6563 746f 7279 2064 6972 313b 2020 irectory dir1; │ │ │ │ +0000b650: 2d2d 2d2b 0a7c 6932 3120 3a20 6d61 6b65 ---+.|i21 : make │ │ │ │ +0000b660: 4469 7265 6374 6f72 7920 6469 7231 3b20 Directory dir1; │ │ │ │ 0000b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b6a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000b6a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b6f0: 2d2d 2b0a 7c69 3232 203a 2062 5048 353d --+.|i22 : bPH5= │ │ │ │ -0000b700: 6265 7274 696e 6950 6172 616d 6574 6572 bertiniParameter │ │ │ │ -0000b710: 486f 6d6f 746f 7079 2820 7b22 785e 322d Homotopy( {"x^2- │ │ │ │ -0000b720: 7531 227d 2c20 2020 2020 2020 2020 2020 u1"}, │ │ │ │ +0000b6f0: 2d2d 2d2b 0a7c 6932 3220 3a20 6250 4835 ---+.|i22 : bPH5 │ │ │ │ +0000b700: 3d62 6572 7469 6e69 5061 7261 6d65 7465 =bertiniParamete │ │ │ │ +0000b710: 7248 6f6d 6f74 6f70 7928 207b 2278 5e32 rHomotopy( {"x^2 │ │ │ │ +0000b720: 2d75 3122 7d2c 2020 2020 2020 2020 2020 -u1"}, │ │ │ │ 0000b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b740: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0000b740: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b790: 2d2d 7c0a 7c7b 7531 7d2c 7b7b 317d 2c7b --|.|{u1},{{1},{ │ │ │ │ -0000b7a0: 327d 7d2c 4166 6656 6172 6961 626c 6547 2}},AffVariableG │ │ │ │ -0000b7b0: 726f 7570 3d3e 7b78 7d2c 4f75 7470 7574 roup=>{x},Output │ │ │ │ -0000b7c0: 5374 796c 653d 3e22 4f75 744e 6f6e 6522 Style=>"OutNone" │ │ │ │ -0000b7d0: 2c54 6f70 4469 7265 6374 6f72 793d 3e20 ,TopDirectory=> │ │ │ │ -0000b7e0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0000b790: 2d2d 2d7c 0a7c 7b75 317d 2c7b 7b31 7d2c ---|.|{u1},{{1}, │ │ │ │ +0000b7a0: 7b32 7d7d 2c41 6666 5661 7269 6162 6c65 {2}},AffVariable │ │ │ │ +0000b7b0: 4772 6f75 703d 3e7b 787d 2c4f 7574 7075 Group=>{x},Outpu │ │ │ │ +0000b7c0: 7453 7479 6c65 3d3e 224f 7574 4e6f 6e65 tStyle=>"OutNone │ │ │ │ +0000b7d0: 222c 546f 7044 6972 6563 746f 7279 3d3e ",TopDirectory=> │ │ │ │ +0000b7e0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b830: 2d2d 7c0a 7c64 6972 3129 2020 2020 2020 --|.|dir1) │ │ │ │ +0000b830: 2d2d 2d7c 0a7c 6469 7231 2920 2020 2020 ---|.|dir1) │ │ │ │ 0000b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b880: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000b880: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b8d0: 2d2d 2b0a 7c69 3233 203a 2042 303d 696d --+.|i23 : B0=im │ │ │ │ -0000b8e0: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ -0000b8f0: 6528 6469 7231 2c4e 616d 6553 6f6c 7574 e(dir1,NameSolut │ │ │ │ -0000b900: 696f 6e73 4669 6c65 3d3e 2270 685f 6a61 ionsFile=>"ph_ja │ │ │ │ -0000b910: 6465 5f30 2229 2020 2020 2020 2020 2020 de_0") │ │ │ │ -0000b920: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b8d0: 2d2d 2d2b 0a7c 6932 3320 3a20 4230 3d69 ---+.|i23 : B0=i │ │ │ │ +0000b8e0: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669 mportSolutionsFi │ │ │ │ +0000b8f0: 6c65 2864 6972 312c 4e61 6d65 536f 6c75 le(dir1,NameSolu │ │ │ │ +0000b900: 7469 6f6e 7346 696c 653d 3e22 7068 5f6a tionsFile=>"ph_j │ │ │ │ +0000b910: 6164 655f 3022 2920 2020 2020 2020 2020 ade_0") │ │ │ │ +0000b920: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b970: 2020 7c0a 7c6f 3233 203d 207b 7b2d 317d |.|o23 = {{-1} │ │ │ │ -0000b980: 2c20 7b31 7d7d 2020 2020 2020 2020 2020 , {1}} │ │ │ │ +0000b970: 2020 207c 0a7c 6f32 3320 3d20 7b7b 2d31 |.|o23 = {{-1 │ │ │ │ +0000b980: 7d2c 207b 317d 7d20 2020 2020 2020 2020 }, {1}} │ │ │ │ 0000b990: 2020 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 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b9c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ba10: 2020 7c0a 7c6f 3233 203a 204c 6973 7420 |.|o23 : List │ │ │ │ +0000ba10: 2020 207c 0a7c 6f32 3320 3a20 4c69 7374 |.|o23 : List │ │ │ │ 0000ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ba60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000ba60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000bab0: 2d2d 2b0a 7c69 3234 203a 2042 313d 696d --+.|i24 : B1=im │ │ │ │ -0000bac0: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ -0000bad0: 6528 6469 7231 2c4e 616d 6553 6f6c 7574 e(dir1,NameSolut │ │ │ │ -0000bae0: 696f 6e73 4669 6c65 3d3e 2270 685f 6a61 ionsFile=>"ph_ja │ │ │ │ -0000baf0: 6465 5f31 2229 2020 2020 2020 2020 2020 de_1") │ │ │ │ -0000bb00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000bab0: 2d2d 2d2b 0a7c 6932 3420 3a20 4231 3d69 ---+.|i24 : B1=i │ │ │ │ +0000bac0: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669 mportSolutionsFi │ │ │ │ +0000bad0: 6c65 2864 6972 312c 4e61 6d65 536f 6c75 le(dir1,NameSolu │ │ │ │ +0000bae0: 7469 6f6e 7346 696c 653d 3e22 7068 5f6a tionsFile=>"ph_j │ │ │ │ +0000baf0: 6164 655f 3122 2920 2020 2020 2020 2020 ade_1") │ │ │ │ +0000bb00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000bb50: 2020 7c0a 7c6f 3234 203d 207b 7b2d 312e |.|o24 = {{-1. │ │ │ │ -0000bb60: 3431 3432 317d 2c20 7b31 2e34 3134 3231 41421}, {1.41421 │ │ │ │ -0000bb70: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ +0000bb50: 2020 207c 0a7c 6f32 3420 3d20 7b7b 2d31 |.|o24 = {{-1 │ │ │ │ +0000bb60: 2e34 3134 3231 7d2c 207b 312e 3431 3432 .41421}, {1.4142 │ │ │ │ +0000bb70: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ 0000bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000bba0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000bba0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbc0: 2020 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 7c0a 7c6f 3234 203a 204c 6973 7420 |.|o24 : List │ │ │ │ +0000bbf0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374 |.|o24 : List │ │ │ │ 0000bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000bc40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000bc40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000bc90: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -0000bca0: 2062 6572 7469 6e69 5061 7261 6d65 7465 bertiniParamete │ │ │ │ -0000bcb0: 7248 6f6d 6f74 6f70 793a 0a3d 3d3d 3d3d rHomotopy:.===== │ │ │ │ +0000bc90: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573 ---+..Ways to us │ │ │ │ +0000bca0: 6520 6265 7274 696e 6950 6172 616d 6574 e bertiniParamet │ │ │ │ +0000bcb0: 6572 486f 6d6f 746f 7079 3a0a 3d3d 3d3d erHomotopy:.==== │ │ │ │ 0000bcc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 0000bcd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000bce0: 0a0a 2020 2a20 2262 6572 7469 6e69 5061 .. * "bertiniPa │ │ │ │ -0000bcf0: 7261 6d65 7465 7248 6f6d 6f74 6f70 7928 rameterHomotopy( │ │ │ │ -0000bd00: 4c69 7374 2c4c 6973 742c 4c69 7374 2922 List,List,List)" │ │ │ │ -0000bd10: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0000bd20: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0000bd30: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0000bd40: 6563 7420 2a6e 6f74 6520 6265 7274 696e ect *note bertin │ │ │ │ -0000bd50: 6950 6172 616d 6574 6572 486f 6d6f 746f iParameterHomoto │ │ │ │ -0000bd60: 7079 3a20 6265 7274 696e 6950 6172 616d py: bertiniParam │ │ │ │ -0000bd70: 6574 6572 486f 6d6f 746f 7079 2c20 6973 eterHomotopy, is │ │ │ │ -0000bd80: 2061 202a 6e6f 7465 0a6d 6574 686f 6420 a *note.method │ │ │ │ -0000bd90: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ -0000bda0: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ -0000bdb0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0000bdc0: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ -0000bdd0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0000bce0: 3d0a 0a20 202a 2022 6265 7274 696e 6950 =.. * "bertiniP │ │ │ │ +0000bcf0: 6172 616d 6574 6572 486f 6d6f 746f 7079 arameterHomotopy │ │ │ │ +0000bd00: 284c 6973 742c 4c69 7374 2c4c 6973 7429 (List,List,List) │ │ │ │ +0000bd10: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0000bd20: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0000bd30: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0000bd40: 6a65 6374 202a 6e6f 7465 2062 6572 7469 ject *note berti │ │ │ │ +0000bd50: 6e69 5061 7261 6d65 7465 7248 6f6d 6f74 niParameterHomot │ │ │ │ +0000bd60: 6f70 793a 2062 6572 7469 6e69 5061 7261 opy: bertiniPara │ │ │ │ +0000bd70: 6d65 7465 7248 6f6d 6f74 6f70 792c 2069 meterHomotopy, i │ │ │ │ +0000bd80: 7320 6120 2a6e 6f74 650a 6d65 7468 6f64 s a *note.method │ │ │ │ +0000bd90: 2066 756e 6374 696f 6e20 7769 7468 206f function with o │ │ │ │ +0000bda0: 7074 696f 6e73 3a20 284d 6163 6175 6c61 ptions: (Macaula │ │ │ │ +0000bdb0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0000bdc0: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +0000bdd0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0000bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000be20: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0000be30: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0000be40: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0000be50: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0000be60: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ -0000be70: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0000be80: 7061 636b 6167 6573 2f42 6572 7469 6e69 packages/Bertini │ │ │ │ -0000be90: 2e6d 323a 0a33 3032 343a 302e 0a1f 0a46 .m2:.3024:0....F │ │ │ │ -0000bea0: 696c 653a 2042 6572 7469 6e69 2e69 6e66 ile: Bertini.inf │ │ │ │ -0000beb0: 6f2c 204e 6f64 653a 2062 6572 7469 6e69 o, Node: bertini │ │ │ │ -0000bec0: 506f 7344 696d 536f 6c76 652c 204e 6578 PosDimSolve, Nex │ │ │ │ -0000bed0: 743a 2062 6572 7469 6e69 5265 6669 6e65 t: bertiniRefine │ │ │ │ -0000bee0: 536f 6c73 2c20 5072 6576 3a20 6265 7274 Sols, Prev: bert │ │ │ │ -0000bef0: 696e 6950 6172 616d 6574 6572 486f 6d6f iniParameterHomo │ │ │ │ -0000bf00: 746f 7079 2c20 5570 3a20 546f 700a 0a62 topy, Up: Top..b │ │ │ │ -0000bf10: 6572 7469 6e69 506f 7344 696d 536f 6c76 ertiniPosDimSolv │ │ │ │ -0000bf20: 6520 2d2d 2061 206d 6169 6e20 6d65 7468 e -- a main meth │ │ │ │ -0000bf30: 6f64 2074 6861 7420 6973 2075 7365 6420 od that is used │ │ │ │ -0000bf40: 746f 2070 726f 6475 6365 2077 6974 6e65 to produce witne │ │ │ │ -0000bf50: 7373 2073 6574 730a 2a2a 2a2a 2a2a 2a2a ss sets.******** │ │ │ │ +0000be20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0000be30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0000be40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0000be50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0000be60: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +0000be70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0000be80: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ +0000be90: 692e 6d32 3a0a 3330 3234 3a30 2e0a 1f0a i.m2:.3024:0.... │ │ │ │ +0000bea0: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ +0000beb0: 666f 2c20 4e6f 6465 3a20 6265 7274 696e fo, Node: bertin │ │ │ │ +0000bec0: 6950 6f73 4469 6d53 6f6c 7665 2c20 4e65 iPosDimSolve, Ne │ │ │ │ +0000bed0: 7874 3a20 6265 7274 696e 6952 6566 696e xt: bertiniRefin │ │ │ │ +0000bee0: 6553 6f6c 732c 2050 7265 763a 2062 6572 eSols, Prev: ber │ │ │ │ +0000bef0: 7469 6e69 5061 7261 6d65 7465 7248 6f6d tiniParameterHom │ │ │ │ +0000bf00: 6f74 6f70 792c 2055 703a 2054 6f70 0a0a otopy, Up: Top.. │ │ │ │ +0000bf10: 6265 7274 696e 6950 6f73 4469 6d53 6f6c bertiniPosDimSol │ │ │ │ +0000bf20: 7665 202d 2d20 6120 6d61 696e 206d 6574 ve -- a main met │ │ │ │ +0000bf30: 686f 6420 7468 6174 2069 7320 7573 6564 hod that is used │ │ │ │ +0000bf40: 2074 6f20 7072 6f64 7563 6520 7769 746e to produce witn │ │ │ │ +0000bf50: 6573 7320 7365 7473 0a2a 2a2a 2a2a 2a2a ess sets.******* │ │ │ │ 0000bf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000bf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000bf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000bf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000bfa0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -0000bfb0: 2020 2020 2020 5620 3d20 6265 7274 696e V = bertin │ │ │ │ -0000bfc0: 6950 6f73 4469 6d53 6f6c 7665 2049 0a20 iPosDimSolve I. │ │ │ │ -0000bfd0: 2020 2020 2020 2056 203d 2062 6572 7469 V = berti │ │ │ │ -0000bfe0: 6e69 506f 7344 696d 536f 6c76 6520 460a niPosDimSolve F. │ │ │ │ -0000bff0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0000c000: 2020 2a20 462c 2061 202a 6e6f 7465 206c * F, a *note l │ │ │ │ -0000c010: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -0000c020: 6f63 294c 6973 742c 2c20 6120 6c69 7374 oc)List,, a list │ │ │ │ -0000c030: 206f 6620 7269 6e67 2065 6c65 6d65 6e74 of ring element │ │ │ │ -0000c040: 7320 6465 6669 6e69 6e67 0a20 2020 2020 s defining. │ │ │ │ -0000c050: 2020 2061 2076 6172 6965 7479 0a20 202a a variety. * │ │ │ │ -0000c060: 202a 6e6f 7465 204f 7074 696f 6e61 6c20 *note Optional │ │ │ │ -0000c070: 696e 7075 7473 3a20 284d 6163 6175 6c61 inputs: (Macaula │ │ │ │ -0000c080: 7932 446f 6329 7573 696e 6720 6675 6e63 y2Doc)using func │ │ │ │ -0000c090: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -0000c0a0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ -0000c0b0: 2020 202a 2042 6572 7469 6e69 496e 7075 * BertiniInpu │ │ │ │ -0000c0c0: 7443 6f6e 6669 6775 7261 7469 6f6e 2028 tConfiguration ( │ │ │ │ -0000c0d0: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -0000c0e0: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -0000c0f0: 6566 6175 6c74 2076 616c 7565 0a20 2020 efault value. │ │ │ │ -0000c100: 2020 2020 207b 7d2c 0a20 2020 2020 202a {},. * │ │ │ │ -0000c110: 202a 6e6f 7465 2049 7350 726f 6a65 6374 *note IsProject │ │ │ │ -0000c120: 6976 653a 2049 7350 726f 6a65 6374 6976 ive: IsProjectiv │ │ │ │ -0000c130: 652c 203d 3e20 2e2e 2e2c 2064 6566 6175 e, => ..., defau │ │ │ │ -0000c140: 6c74 2076 616c 7565 202d 312c 206f 7074 lt value -1, opt │ │ │ │ -0000c150: 696f 6e61 6c0a 2020 2020 2020 2020 6172 ional. ar │ │ │ │ -0000c160: 6775 6d65 6e74 2074 6f20 7370 6563 6966 gument to specif │ │ │ │ -0000c170: 7920 7768 6574 6865 7220 746f 2075 7365 y whether to use │ │ │ │ -0000c180: 2068 6f6d 6f67 656e 656f 7573 2063 6f6f homogeneous coo │ │ │ │ -0000c190: 7264 696e 6174 6573 0a20 2020 2020 202a rdinates. * │ │ │ │ -0000c1a0: 202a 6e6f 7465 2056 6572 626f 7365 3a20 *note Verbose: │ │ │ │ -0000c1b0: 6265 7274 696e 6954 7261 636b 486f 6d6f bertiniTrackHomo │ │ │ │ -0000c1c0: 746f 7079 5f6c 705f 7064 5f70 645f 7064 topy_lp_pd_pd_pd │ │ │ │ -0000c1d0: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f _cmVerbose=>_pd_ │ │ │ │ -0000c1e0: 7064 5f70 645f 7270 0a20 2020 2020 2020 pd_pd_rp. │ │ │ │ -0000c1f0: 202c 203d 3e20 2e2e 2e2c 2064 6566 6175 , => ..., defau │ │ │ │ -0000c200: 6c74 2076 616c 7565 2066 616c 7365 2c20 lt value false, │ │ │ │ -0000c210: 4f70 7469 6f6e 2074 6f20 7369 6c65 6e63 Option to silenc │ │ │ │ -0000c220: 6520 6164 6469 7469 6f6e 616c 206f 7574 e additional out │ │ │ │ -0000c230: 7075 740a 2020 2a20 4f75 7470 7574 733a put. * Outputs: │ │ │ │ -0000c240: 0a20 2020 2020 202a 2056 2c20 6120 2a6e . * V, a *n │ │ │ │ -0000c250: 6f74 6520 6e75 6d65 7269 6361 6c20 7661 ote numerical va │ │ │ │ -0000c260: 7269 6574 793a 2028 4e41 4774 7970 6573 riety: (NAGtypes │ │ │ │ -0000c270: 294e 756d 6572 6963 616c 5661 7269 6574 )NumericalVariet │ │ │ │ -0000c280: 792c 2c20 6120 6e75 6d65 7269 6361 6c0a y,, a numerical. │ │ │ │ -0000c290: 2020 2020 2020 2020 6972 7265 6475 6369 irreduci │ │ │ │ -0000c2a0: 626c 6520 6465 636f 6d70 6f73 6974 696f ble decompositio │ │ │ │ -0000c2b0: 6e20 6f66 2074 6865 2076 6172 6965 7479 n of the variety │ │ │ │ -0000c2c0: 2064 6566 696e 6564 2062 7920 460a 0a44 defined by F..D │ │ │ │ -0000c2d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0000c2e0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6d65 7468 ======..The meth │ │ │ │ -0000c2f0: 6f64 2062 6572 7469 6e69 506f 7344 696d od bertiniPosDim │ │ │ │ -0000c300: 536f 6c76 6520 6361 6c6c 7320 2042 6572 Solve calls Ber │ │ │ │ -0000c310: 7469 6e69 2074 6f20 6669 6e64 2061 206e tini to find a n │ │ │ │ -0000c320: 756d 6572 6963 616c 2069 7272 6564 7563 umerical irreduc │ │ │ │ -0000c330: 6962 6c65 0a64 6563 6f6d 706f 7369 7469 ible.decompositi │ │ │ │ -0000c340: 6f6e 206f 6620 7468 6520 7a65 726f 2d73 on of the zero-s │ │ │ │ -0000c350: 6574 206f 6620 462e 2020 5468 6520 6465 et of F. The de │ │ │ │ -0000c360: 636f 6d70 6f73 6974 696f 6e20 6973 2072 composition is r │ │ │ │ -0000c370: 6574 7572 6e65 6420 6173 2074 6865 202a eturned as the * │ │ │ │ -0000c380: 6e6f 7465 0a4e 756d 6572 6963 616c 5661 note.NumericalVa │ │ │ │ -0000c390: 7269 6574 793a 2028 4e41 4774 7970 6573 riety: (NAGtypes │ │ │ │ -0000c3a0: 294e 756d 6572 6963 616c 5661 7269 6574 )NumericalVariet │ │ │ │ -0000c3b0: 792c 204e 562e 2020 5769 746e 6573 7320 y, NV. Witness │ │ │ │ -0000c3c0: 7365 7473 206f 6620 4e56 2063 6f6e 7461 sets of NV conta │ │ │ │ -0000c3d0: 696e 0a61 7070 726f 7869 6d61 7469 6f6e in.approximation │ │ │ │ -0000c3e0: 7320 746f 2073 6f6c 7574 696f 6e73 206f s to solutions o │ │ │ │ -0000c3f0: 6620 7468 6520 7379 7374 656d 2046 3d30 f the system F=0 │ │ │ │ -0000c400: 2e20 4265 7274 696e 6920 2831 2920 7772 . Bertini (1) wr │ │ │ │ -0000c410: 6974 6573 2074 6865 2073 7973 7465 6d20 ites the system │ │ │ │ -0000c420: 746f 0a74 656d 706f 7261 7279 2066 696c to.temporary fil │ │ │ │ -0000c430: 6573 2c20 2832 2920 696e 766f 6b65 7320 es, (2) invokes │ │ │ │ -0000c440: 4265 7274 696e 6927 7320 736f 6c76 6572 Bertini's solver │ │ │ │ -0000c450: 2077 6974 6820 5472 6163 6b54 7970 6520 with TrackType │ │ │ │ -0000c460: 3d3e 2031 2c20 2833 2920 4265 7274 696e => 1, (3) Bertin │ │ │ │ -0000c470: 690a 7573 6573 2061 2063 6173 6361 6465 i.uses a cascade │ │ │ │ -0000c480: 2068 6f6d 6f74 6f70 7920 746f 2066 696e homotopy to fin │ │ │ │ -0000c490: 6420 7769 746e 6573 7320 7375 7065 7273 d witness supers │ │ │ │ -0000c4a0: 6574 7320 696e 2065 6163 6820 6469 6d65 ets in each dime │ │ │ │ -0000c4b0: 6e73 696f 6e2c 2028 3429 0a72 656d 6f76 nsion, (4).remov │ │ │ │ -0000c4c0: 6573 2065 7874 7261 2070 6f69 6e74 7320 es extra points │ │ │ │ -0000c4d0: 7573 696e 6720 6120 6d65 6d62 6572 7368 using a membersh │ │ │ │ -0000c4e0: 6970 2074 6573 7420 6f72 206c 6f63 616c ip test or local │ │ │ │ -0000c4f0: 2064 696d 656e 7369 6f6e 2074 6573 742c dimension test, │ │ │ │ -0000c500: 2028 3529 0a64 6566 6c61 7465 7320 7369 (5).deflates si │ │ │ │ -0000c510: 6e67 756c 6172 2077 6974 6e65 7373 2070 ngular witness p │ │ │ │ -0000c520: 6f69 6e74 732c 2061 6e64 2066 696e 616c oints, and final │ │ │ │ -0000c530: 6c79 2028 3629 2064 6563 6f6d 706f 7365 ly (6) decompose │ │ │ │ -0000c540: 7320 7573 696e 6720 610a 636f 6d62 696e s using a.combin │ │ │ │ -0000c550: 6174 696f 6e20 6f66 206d 6f6e 6f64 726f ation of monodro │ │ │ │ -0000c560: 6d79 2061 6e64 2061 206c 696e 6561 7220 my and a linear │ │ │ │ -0000c570: 7472 6163 6520 7465 7374 0a0a 2b2d 2d2d trace test..+--- │ │ │ │ +0000bfa0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0000bfb0: 2020 2020 2020 2056 203d 2062 6572 7469 V = berti │ │ │ │ +0000bfc0: 6e69 506f 7344 696d 536f 6c76 6520 490a niPosDimSolve I. │ │ │ │ +0000bfd0: 2020 2020 2020 2020 5620 3d20 6265 7274 V = bert │ │ │ │ +0000bfe0: 696e 6950 6f73 4469 6d53 6f6c 7665 2046 iniPosDimSolve F │ │ │ │ +0000bff0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0000c000: 2020 202a 2046 2c20 6120 2a6e 6f74 6520 * F, a *note │ │ │ │ +0000c010: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0000c020: 446f 6329 4c69 7374 2c2c 2061 206c 6973 Doc)List,, a lis │ │ │ │ +0000c030: 7420 6f66 2072 696e 6720 656c 656d 656e t of ring elemen │ │ │ │ +0000c040: 7473 2064 6566 696e 696e 670a 2020 2020 ts defining. │ │ │ │ +0000c050: 2020 2020 6120 7661 7269 6574 790a 2020 a variety. │ │ │ │ +0000c060: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c * *note Optional │ │ │ │ +0000c070: 2069 6e70 7574 733a 2028 4d61 6361 756c inputs: (Macaul │ │ │ │ +0000c080: 6179 3244 6f63 2975 7369 6e67 2066 756e ay2Doc)using fun │ │ │ │ +0000c090: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ +0000c0a0: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020 onal inputs,:. │ │ │ │ +0000c0b0: 2020 2020 2a20 4265 7274 696e 6949 6e70 * BertiniInp │ │ │ │ +0000c0c0: 7574 436f 6e66 6967 7572 6174 696f 6e20 utConfiguration │ │ │ │ +0000c0d0: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +0000c0e0: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ +0000c0f0: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ +0000c100: 2020 2020 2020 7b7d 2c0a 2020 2020 2020 {},. │ │ │ │ +0000c110: 2a20 2a6e 6f74 6520 4973 5072 6f6a 6563 * *note IsProjec │ │ │ │ +0000c120: 7469 7665 3a20 4973 5072 6f6a 6563 7469 tive: IsProjecti │ │ │ │ +0000c130: 7665 2c20 3d3e 202e 2e2e 2c20 6465 6661 ve, => ..., defa │ │ │ │ +0000c140: 756c 7420 7661 6c75 6520 2d31 2c20 6f70 ult value -1, op │ │ │ │ +0000c150: 7469 6f6e 616c 0a20 2020 2020 2020 2061 tional. a │ │ │ │ +0000c160: 7267 756d 656e 7420 746f 2073 7065 6369 rgument to speci │ │ │ │ +0000c170: 6679 2077 6865 7468 6572 2074 6f20 7573 fy whether to us │ │ │ │ +0000c180: 6520 686f 6d6f 6765 6e65 6f75 7320 636f e homogeneous co │ │ │ │ +0000c190: 6f72 6469 6e61 7465 730a 2020 2020 2020 ordinates. │ │ │ │ +0000c1a0: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a * *note Verbose: │ │ │ │ +0000c1b0: 2062 6572 7469 6e69 5472 6163 6b48 6f6d bertiniTrackHom │ │ │ │ +0000c1c0: 6f74 6f70 795f 6c70 5f70 645f 7064 5f70 otopy_lp_pd_pd_p │ │ │ │ +0000c1d0: 645f 636d 5665 7262 6f73 653d 3e5f 7064 d_cmVerbose=>_pd │ │ │ │ +0000c1e0: 5f70 645f 7064 5f72 700a 2020 2020 2020 _pd_pd_rp. │ │ │ │ +0000c1f0: 2020 2c20 3d3e 202e 2e2e 2c20 6465 6661 , => ..., defa │ │ │ │ +0000c200: 756c 7420 7661 6c75 6520 6661 6c73 652c ult value false, │ │ │ │ +0000c210: 204f 7074 696f 6e20 746f 2073 696c 656e Option to silen │ │ │ │ +0000c220: 6365 2061 6464 6974 696f 6e61 6c20 6f75 ce additional ou │ │ │ │ +0000c230: 7470 7574 0a20 202a 204f 7574 7075 7473 tput. * Outputs │ │ │ │ +0000c240: 3a0a 2020 2020 2020 2a20 562c 2061 202a :. * V, a * │ │ │ │ +0000c250: 6e6f 7465 206e 756d 6572 6963 616c 2076 note numerical v │ │ │ │ +0000c260: 6172 6965 7479 3a20 284e 4147 7479 7065 ariety: (NAGtype │ │ │ │ +0000c270: 7329 4e75 6d65 7269 6361 6c56 6172 6965 s)NumericalVarie │ │ │ │ +0000c280: 7479 2c2c 2061 206e 756d 6572 6963 616c ty,, a numerical │ │ │ │ +0000c290: 0a20 2020 2020 2020 2069 7272 6564 7563 . irreduc │ │ │ │ +0000c2a0: 6962 6c65 2064 6563 6f6d 706f 7369 7469 ible decompositi │ │ │ │ +0000c2b0: 6f6e 206f 6620 7468 6520 7661 7269 6574 on of the variet │ │ │ │ +0000c2c0: 7920 6465 6669 6e65 6420 6279 2046 0a0a y defined by F.. │ │ │ │ +0000c2d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0000c2e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206d 6574 =======..The met │ │ │ │ +0000c2f0: 686f 6420 6265 7274 696e 6950 6f73 4469 hod bertiniPosDi │ │ │ │ +0000c300: 6d53 6f6c 7665 2063 616c 6c73 2020 4265 mSolve calls Be │ │ │ │ +0000c310: 7274 696e 6920 746f 2066 696e 6420 6120 rtini to find a │ │ │ │ +0000c320: 6e75 6d65 7269 6361 6c20 6972 7265 6475 numerical irredu │ │ │ │ +0000c330: 6369 626c 650a 6465 636f 6d70 6f73 6974 cible.decomposit │ │ │ │ +0000c340: 696f 6e20 6f66 2074 6865 207a 6572 6f2d ion of the zero- │ │ │ │ +0000c350: 7365 7420 6f66 2046 2e20 2054 6865 2064 set of F. The d │ │ │ │ +0000c360: 6563 6f6d 706f 7369 7469 6f6e 2069 7320 ecomposition is │ │ │ │ +0000c370: 7265 7475 726e 6564 2061 7320 7468 6520 returned as the │ │ │ │ +0000c380: 2a6e 6f74 650a 4e75 6d65 7269 6361 6c56 *note.NumericalV │ │ │ │ +0000c390: 6172 6965 7479 3a20 284e 4147 7479 7065 ariety: (NAGtype │ │ │ │ +0000c3a0: 7329 4e75 6d65 7269 6361 6c56 6172 6965 s)NumericalVarie │ │ │ │ +0000c3b0: 7479 2c20 4e56 2e20 2057 6974 6e65 7373 ty, NV. Witness │ │ │ │ +0000c3c0: 2073 6574 7320 6f66 204e 5620 636f 6e74 sets of NV cont │ │ │ │ +0000c3d0: 6169 6e0a 6170 7072 6f78 696d 6174 696f ain.approximatio │ │ │ │ +0000c3e0: 6e73 2074 6f20 736f 6c75 7469 6f6e 7320 ns to solutions │ │ │ │ +0000c3f0: 6f66 2074 6865 2073 7973 7465 6d20 463d of the system F= │ │ │ │ +0000c400: 302e 2042 6572 7469 6e69 2028 3129 2077 0. Bertini (1) w │ │ │ │ +0000c410: 7269 7465 7320 7468 6520 7379 7374 656d rites the system │ │ │ │ +0000c420: 2074 6f0a 7465 6d70 6f72 6172 7920 6669 to.temporary fi │ │ │ │ +0000c430: 6c65 732c 2028 3229 2069 6e76 6f6b 6573 les, (2) invokes │ │ │ │ +0000c440: 2042 6572 7469 6e69 2773 2073 6f6c 7665 Bertini's solve │ │ │ │ +0000c450: 7220 7769 7468 2054 7261 636b 5479 7065 r with TrackType │ │ │ │ +0000c460: 203d 3e20 312c 2028 3329 2042 6572 7469 => 1, (3) Berti │ │ │ │ +0000c470: 6e69 0a75 7365 7320 6120 6361 7363 6164 ni.uses a cascad │ │ │ │ +0000c480: 6520 686f 6d6f 746f 7079 2074 6f20 6669 e homotopy to fi │ │ │ │ +0000c490: 6e64 2077 6974 6e65 7373 2073 7570 6572 nd witness super │ │ │ │ +0000c4a0: 7365 7473 2069 6e20 6561 6368 2064 696d sets in each dim │ │ │ │ +0000c4b0: 656e 7369 6f6e 2c20 2834 290a 7265 6d6f ension, (4).remo │ │ │ │ +0000c4c0: 7665 7320 6578 7472 6120 706f 696e 7473 ves extra points │ │ │ │ +0000c4d0: 2075 7369 6e67 2061 206d 656d 6265 7273 using a members │ │ │ │ +0000c4e0: 6869 7020 7465 7374 206f 7220 6c6f 6361 hip test or loca │ │ │ │ +0000c4f0: 6c20 6469 6d65 6e73 696f 6e20 7465 7374 l dimension test │ │ │ │ +0000c500: 2c20 2835 290a 6465 666c 6174 6573 2073 , (5).deflates s │ │ │ │ +0000c510: 696e 6775 6c61 7220 7769 746e 6573 7320 ingular witness │ │ │ │ +0000c520: 706f 696e 7473 2c20 616e 6420 6669 6e61 points, and fina │ │ │ │ +0000c530: 6c6c 7920 2836 2920 6465 636f 6d70 6f73 lly (6) decompos │ │ │ │ +0000c540: 6573 2075 7369 6e67 2061 0a63 6f6d 6269 es using a.combi │ │ │ │ +0000c550: 6e61 7469 6f6e 206f 6620 6d6f 6e6f 6472 nation of monodr │ │ │ │ +0000c560: 6f6d 7920 616e 6420 6120 6c69 6e65 6172 omy and a linear │ │ │ │ +0000c570: 2074 7261 6365 2074 6573 740a 0a2b 2d2d trace test..+-- │ │ │ │ 0000c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -0000c5d0: 3a20 5220 3d20 5151 5b78 2c79 2c7a 5d20 : R = QQ[x,y,z] │ │ │ │ +0000c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0000c5d0: 203a 2052 203d 2051 515b 782c 792c 7a5d : R = QQ[x,y,z] │ │ │ │ 0000c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c610: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000c610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c660: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0000c670: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +0000c660: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0000c670: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0000c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000c6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c700: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0000c710: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ -0000c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000c700: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0000c710: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0000c720: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0000c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000c750: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -0000c7b0: 3a20 4620 3d20 7b28 795e 322b 785e 322b : F = {(y^2+x^2+ │ │ │ │ -0000c7c0: 7a5e 322d 3129 2a78 2c28 795e 322b 785e z^2-1)*x,(y^2+x^ │ │ │ │ -0000c7d0: 322b 7a5e 322d 3129 2a79 7d20 2020 2020 2+z^2-1)*y} │ │ │ │ +0000c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0000c7b0: 203a 2046 203d 207b 2879 5e32 2b78 5e32 : F = {(y^2+x^2 │ │ │ │ +0000c7c0: 2b7a 5e32 2d31 292a 782c 2879 5e32 2b78 +z^2-1)*x,(y^2+x │ │ │ │ +0000c7d0: 5e32 2b7a 5e32 2d31 292a 797d 2020 2020 ^2+z^2-1)*y} │ │ │ │ 0000c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c7f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000c7f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c840: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0000c850: 2020 2020 3320 2020 2020 2032 2020 2020 3 2 │ │ │ │ -0000c860: 2020 3220 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -0000c870: 3320 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ +0000c840: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000c850: 2020 2020 2033 2020 2020 2020 3220 2020 3 2 │ │ │ │ +0000c860: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0000c870: 2033 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ 0000c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c890: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0000c8a0: 3d20 7b78 2020 2b20 782a 7920 202b 2078 = {x + x*y + x │ │ │ │ -0000c8b0: 2a7a 2020 2d20 782c 2078 2079 202b 2079 *z - x, x y + y │ │ │ │ -0000c8c0: 2020 2b20 792a 7a20 202d 2079 7d20 2020 + y*z - y} │ │ │ │ +0000c890: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0000c8a0: 203d 207b 7820 202b 2078 2a79 2020 2b20 = {x + x*y + │ │ │ │ +0000c8b0: 782a 7a20 202d 2078 2c20 7820 7920 2b20 x*z - x, x y + │ │ │ │ +0000c8c0: 7920 202b 2079 2a7a 2020 2d20 797d 2020 y + y*z - y} │ │ │ │ 0000c8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000c8e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c930: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0000c940: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0000c930: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0000c940: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 0000c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000c980: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000c980: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -0000c9e0: 3a20 5320 3d20 6265 7274 696e 6950 6f73 : S = bertiniPos │ │ │ │ -0000c9f0: 4469 6d53 6f6c 7665 2046 2020 2020 2020 DimSolve F │ │ │ │ +0000c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +0000c9e0: 203a 2053 203d 2062 6572 7469 6e69 506f : S = bertiniPo │ │ │ │ +0000c9f0: 7344 696d 536f 6c76 6520 4620 2020 2020 sDimSolve F │ │ │ │ 0000ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ca20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000ca20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ca70: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -0000ca80: 3d20 5320 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0000ca70: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0000ca80: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ 0000ca90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000caa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cac0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000cac0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cb10: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -0000cb20: 3a20 4e75 6d65 7269 6361 6c56 6172 6965 : NumericalVarie │ │ │ │ -0000cb30: 7479 2020 2020 2020 2020 2020 2020 2020 ty │ │ │ │ +0000cb10: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0000cb20: 203a 204e 756d 6572 6963 616c 5661 7269 : NumericalVari │ │ │ │ +0000cb30: 6574 7920 2020 2020 2020 2020 2020 2020 ety │ │ │ │ 0000cb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cb60: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000cb60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -0000cbc0: 3a20 5323 315f 3023 506f 696e 7473 202d : S#1_0#Points - │ │ │ │ -0000cbd0: 2d20 315f 3020 6368 6f6f 7365 7320 7468 - 1_0 chooses th │ │ │ │ -0000cbe0: 6520 6669 7273 7420 7769 746e 6573 7320 e first witness │ │ │ │ -0000cbf0: 7365 7420 696e 2064 696d 656e 7369 6f6e set in dimension │ │ │ │ -0000cc00: 2031 2020 2020 2020 2020 7c0a 7c20 2020 1 |.| │ │ │ │ +0000cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +0000cbc0: 203a 2053 2331 5f30 2350 6f69 6e74 7320 : S#1_0#Points │ │ │ │ +0000cbd0: 2d2d 2031 5f30 2063 686f 6f73 6573 2074 -- 1_0 chooses t │ │ │ │ +0000cbe0: 6865 2066 6972 7374 2077 6974 6e65 7373 he first witness │ │ │ │ +0000cbf0: 2073 6574 2069 6e20 6469 6d65 6e73 696f set in dimensio │ │ │ │ +0000cc00: 6e20 3120 2020 2020 2020 207c 0a7c 2020 n 1 |.| │ │ │ │ 0000cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cc50: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0000cc60: 3d20 7b7b 322e 3634 3436 3865 2d35 392b = {{2.64468e-59+ │ │ │ │ -0000cc70: 312e 3833 3934 3965 2d35 392a 6969 2c20 1.83949e-59*ii, │ │ │ │ -0000cc80: 2d31 2e30 3837 3765 2d36 302b 332e 3337 -1.0877e-60+3.37 │ │ │ │ -0000cc90: 3538 3365 2d35 392a 6969 2c20 2e32 3631 583e-59*ii, .261 │ │ │ │ -0000cca0: 3234 3620 2020 2020 2020 7c0a 7c20 2020 246 |.| │ │ │ │ +0000cc50: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0000cc60: 203d 207b 7b32 2e36 3434 3638 652d 3539 = {{2.64468e-59 │ │ │ │ +0000cc70: 2b31 2e38 3339 3439 652d 3539 2a69 692c +1.83949e-59*ii, │ │ │ │ +0000cc80: 202d 312e 3038 3737 652d 3630 2b33 2e33 -1.0877e-60+3.3 │ │ │ │ +0000cc90: 3735 3833 652d 3539 2a69 692c 202e 3236 7583e-59*ii, .26 │ │ │ │ +0000cca0: 3132 3436 2020 2020 2020 207c 0a7c 2020 1246 |.| │ │ │ │ 0000ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ccc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ccf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0000cd00: 3a20 5665 7274 6963 616c 4c69 7374 2020 : VerticalList │ │ │ │ +0000ccf0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0000cd00: 203a 2056 6572 7469 6361 6c4c 6973 7420 : VerticalList │ │ │ │ 0000cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cd40: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d |.|--- │ │ │ │ +0000cd40: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ 0000cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cd90: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2b 2e31 ----------|.|+.1 │ │ │ │ -0000cda0: 3436 3031 382a 6969 7d7d 2020 2020 2020 46018*ii}} │ │ │ │ +0000cd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2b2e -----------|.|+. │ │ │ │ +0000cda0: 3134 3630 3138 2a69 697d 7d20 2020 2020 146018*ii}} │ │ │ │ 0000cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cde0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000cde0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a45 6163 ----------+..Eac │ │ │ │ -0000ce40: 6820 2a6e 6f74 6520 5769 746e 6573 7353 h *note WitnessS │ │ │ │ -0000ce50: 6574 3a20 284e 4147 7479 7065 7329 5769 et: (NAGtypes)Wi │ │ │ │ -0000ce60: 746e 6573 7353 6574 2c20 6973 2061 6363 tnessSet, is acc │ │ │ │ -0000ce70: 6573 7365 6420 6279 2064 696d 656e 7369 essed by dimensi │ │ │ │ -0000ce80: 6f6e 2061 6e64 2074 6865 6e0a 6c69 7374 on and then.list │ │ │ │ -0000ce90: 2070 6f73 6974 696f 6e2e 0a0a 2b2d 2d2d position...+--- │ │ │ │ +0000ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4561 -----------+..Ea │ │ │ │ +0000ce40: 6368 202a 6e6f 7465 2057 6974 6e65 7373 ch *note Witness │ │ │ │ +0000ce50: 5365 743a 2028 4e41 4774 7970 6573 2957 Set: (NAGtypes)W │ │ │ │ +0000ce60: 6974 6e65 7373 5365 742c 2069 7320 6163 itnessSet, is ac │ │ │ │ +0000ce70: 6365 7373 6564 2062 7920 6469 6d65 6e73 cessed by dimens │ │ │ │ +0000ce80: 696f 6e20 616e 6420 7468 656e 0a6c 6973 ion and then.lis │ │ │ │ +0000ce90: 7420 706f 7369 7469 6f6e 2e0a 0a2b 2d2d t position...+-- │ │ │ │ 0000cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000cec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ced0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0000cef0: 3a20 5323 3120 2d2d 6669 7273 7420 7370 : S#1 --first sp │ │ │ │ -0000cf00: 6563 6966 7920 6469 6d65 6e73 696f 6e20 ecify dimension │ │ │ │ +0000cee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +0000cef0: 203a 2053 2331 202d 2d66 6972 7374 2073 : S#1 --first s │ │ │ │ +0000cf00: 7065 6369 6679 2064 696d 656e 7369 6f6e pecify dimension │ │ │ │ 0000cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cf30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000cf30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cf80: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -0000cf90: 3d20 7b28 6469 6d3d 312c 6465 673d 3129 = {(dim=1,deg=1) │ │ │ │ -0000cfa0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000cf80: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +0000cf90: 203d 207b 2864 696d 3d31 2c64 6567 3d31 = {(dim=1,deg=1 │ │ │ │ +0000cfa0: 297d 2020 2020 2020 2020 2020 2020 2020 )} │ │ │ │ 0000cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000cfd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000cfd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000cfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d020: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -0000d030: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0000d020: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +0000d030: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 0000d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d070: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000d070: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000d080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0000d0d0: 3a20 7065 656b 206f 6f5f 3020 2d2d 7468 : peek oo_0 --th │ │ │ │ -0000d0e0: 656e 206c 6973 7420 706f 7369 7469 6f6e en list position │ │ │ │ -0000d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +0000d0d0: 203a 2070 6565 6b20 6f6f 5f30 202d 2d74 : peek oo_0 --t │ │ │ │ +0000d0e0: 6865 6e20 6c69 7374 2070 6f73 6974 696f hen list positio │ │ │ │ +0000d0f0: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ 0000d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d110: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000d110: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000d120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d160: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -0000d170: 3d20 5769 746e 6573 7353 6574 7b63 6163 = WitnessSet{cac │ │ │ │ -0000d180: 6865 203d 3e20 4361 6368 6554 6162 6c65 he => CacheTable │ │ │ │ -0000d190: 7b2e 2e2e 332e 2e2e 7d20 2020 2020 2020 {...3...} │ │ │ │ +0000d160: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0000d170: 203d 2057 6974 6e65 7373 5365 747b 6361 = WitnessSet{ca │ │ │ │ +0000d180: 6368 6520 3d3e 2043 6163 6865 5461 626c che => CacheTabl │ │ │ │ +0000d190: 657b 2e2e 2e33 2e2e 2e7d 2020 2020 2020 e{...3...} │ │ │ │ 0000d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d1b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0000d1c0: 2020 2020 2020 2020 2020 2020 2045 7175 Equ │ │ │ │ -0000d1d0: 6174 696f 6e73 203d 3e20 7b2d 337d 207c ations => {-3} | │ │ │ │ -0000d1e0: 2078 332b 7879 322b 787a 322d 7820 7c20 x3+xy2+xz2-x | │ │ │ │ +0000d1b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000d1c0: 2020 2020 2020 2020 2020 2020 2020 4571 Eq │ │ │ │ +0000d1d0: 7561 7469 6f6e 7320 3d3e 207b 2d33 7d20 uations => {-3} │ │ │ │ +0000d1e0: 7c20 7833 2b78 7932 2b78 7a32 2d78 207c | x3+xy2+xz2-x | │ │ │ │ 0000d1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d200: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000d200: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d220: 2020 2020 2020 2020 2020 7b2d 337d 207c {-3} | │ │ │ │ -0000d230: 2078 3279 2b79 332b 797a 322d 7920 7c20 x2y+y3+yz2-y | │ │ │ │ +0000d220: 2020 2020 2020 2020 2020 207b 2d33 7d20 {-3} │ │ │ │ +0000d230: 7c20 7832 792b 7933 2b79 7a32 2d79 207c | x2y+y3+yz2-y | │ │ │ │ 0000d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d250: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0000d260: 2020 2020 2020 2020 2020 2020 2050 6f69 Poi │ │ │ │ -0000d270: 6e74 7320 3d3e 207b 7b32 2e36 3434 3638 nts => {{2.64468 │ │ │ │ -0000d280: 652d 3539 2b31 2e38 3339 3439 652d 3539 e-59+1.83949e-59 │ │ │ │ -0000d290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0000d2b0: 2020 2020 2020 2020 2020 2020 2053 6c69 Sli │ │ │ │ -0000d2c0: 6365 203d 3e20 7c20 2e30 3733 3838 332b ce => | .073883+ │ │ │ │ -0000d2d0: 312e 3531 3332 3969 6920 312e 3338 3336 1.51329ii 1.3836 │ │ │ │ -0000d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2f0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d |.|--- │ │ │ │ +0000d250: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000d260: 2020 2020 2020 2020 2020 2020 2020 506f Po │ │ │ │ +0000d270: 696e 7473 203d 3e20 7b7b 322e 3634 3436 ints => {{2.6446 │ │ │ │ +0000d280: 3865 2d35 392b 312e 3833 3934 3965 2d35 8e-59+1.83949e-5 │ │ │ │ +0000d290: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ +0000d2a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000d2b0: 2020 2020 2020 2020 2020 2020 2020 536c Sl │ │ │ │ +0000d2c0: 6963 6520 3d3e 207c 202e 3037 3338 3833 ice => | .073883 │ │ │ │ +0000d2d0: 2b31 2e35 3133 3239 6969 2031 2e33 3833 +1.51329ii 1.383 │ │ │ │ +0000d2e0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +0000d2f0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ 0000d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d340: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +0000d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ 0000d350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d380: 2020 7d20 2020 2020 2020 2020 2020 2020 } │ │ │ │ -0000d390: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000d380: 2020 207d 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000d390: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000d3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d3e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000d3e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000d3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d430: 2020 2020 2020 2020 2020 7c0a 7c2a 6969 |.|*ii │ │ │ │ -0000d440: 2c20 2d31 2e30 3837 3765 2d36 302b 332e , -1.0877e-60+3. │ │ │ │ -0000d450: 3337 3538 3365 2d35 392a 6969 2c20 2e32 37583e-59*ii, .2 │ │ │ │ -0000d460: 3631 3234 362b 2e31 3436 3031 382a 6969 61246+.146018*ii │ │ │ │ -0000d470: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -0000d480: 2020 2020 2020 2020 2020 7c0a 7c2b 2e31 |.|+.1 │ │ │ │ -0000d490: 3836 3538 3869 6920 2d32 2e30 3231 3933 86588ii -2.02193 │ │ │ │ -0000d4a0: 2b2e 3735 3736 3736 6969 202e 3633 3838 +.757676ii .6388 │ │ │ │ -0000d4b0: 3535 2b2e 3039 3732 3939 3169 6920 7c20 55+.0972991ii | │ │ │ │ +0000d430: 2020 2020 2020 2020 2020 207c 0a7c 2a69 |.|*i │ │ │ │ +0000d440: 692c 202d 312e 3038 3737 652d 3630 2b33 i, -1.0877e-60+3 │ │ │ │ +0000d450: 2e33 3735 3833 652d 3539 2a69 692c 202e .37583e-59*ii, . │ │ │ │ +0000d460: 3236 3132 3436 2b2e 3134 3630 3138 2a69 261246+.146018*i │ │ │ │ +0000d470: 697d 7d20 2020 2020 2020 2020 2020 2020 i}} │ │ │ │ +0000d480: 2020 2020 2020 2020 2020 207c 0a7c 2b2e |.|+. │ │ │ │ +0000d490: 3138 3635 3838 6969 202d 322e 3032 3139 186588ii -2.0219 │ │ │ │ +0000d4a0: 332b 2e37 3537 3637 3669 6920 2e36 3338 3+.757676ii .638 │ │ │ │ +0000d4b0: 3835 352b 2e30 3937 3239 3931 6969 207c 855+.0972991ii | │ │ │ │ 0000d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d4d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0000d4d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0000d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 ----------+..In │ │ │ │ -0000d530: 7468 6520 6578 616d 706c 652c 2077 6520 the example, we │ │ │ │ -0000d540: 6669 6e64 2074 776f 2063 6f6d 706f 6e65 find two compone │ │ │ │ -0000d550: 6e74 732c 206f 6e65 2063 6f6d 706f 6e65 nts, one compone │ │ │ │ -0000d560: 6e74 2068 6173 2064 696d 656e 7369 6f6e nt has dimension │ │ │ │ -0000d570: 2031 2061 6e64 0a64 6567 7265 6520 3120 1 and.degree 1 │ │ │ │ -0000d580: 616e 6420 7468 6520 6f74 6865 7220 6861 and the other ha │ │ │ │ -0000d590: 7320 6469 6d65 6e73 696f 6e20 3220 616e s dimension 2 an │ │ │ │ -0000d5a0: 6420 6465 6772 6565 2032 2e20 2057 6520 d degree 2. We │ │ │ │ -0000d5b0: 6765 7420 7468 6520 7361 6d65 2072 6573 get the same res │ │ │ │ -0000d5c0: 756c 7473 0a75 7369 6e67 2073 796d 626f ults.using symbo │ │ │ │ -0000d5d0: 6c69 6320 6d65 7468 6f64 732e 0a0a 2b2d lic methods...+- │ │ │ │ +0000d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ +0000d530: 2074 6865 2065 7861 6d70 6c65 2c20 7765 the example, we │ │ │ │ +0000d540: 2066 696e 6420 7477 6f20 636f 6d70 6f6e find two compon │ │ │ │ +0000d550: 656e 7473 2c20 6f6e 6520 636f 6d70 6f6e ents, one compon │ │ │ │ +0000d560: 656e 7420 6861 7320 6469 6d65 6e73 696f ent has dimensio │ │ │ │ +0000d570: 6e20 3120 616e 640a 6465 6772 6565 2031 n 1 and.degree 1 │ │ │ │ +0000d580: 2061 6e64 2074 6865 206f 7468 6572 2068 and the other h │ │ │ │ +0000d590: 6173 2064 696d 656e 7369 6f6e 2032 2061 as dimension 2 a │ │ │ │ +0000d5a0: 6e64 2064 6567 7265 6520 322e 2020 5765 nd degree 2. We │ │ │ │ +0000d5b0: 2067 6574 2074 6865 2073 616d 6520 7265 get the same re │ │ │ │ +0000d5c0: 7375 6c74 730a 7573 696e 6720 7379 6d62 sults.using symb │ │ │ │ +0000d5d0: 6f6c 6963 206d 6574 686f 6473 2e0a 0a2b olic methods...+ │ │ │ │ 0000d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0000d610: 203a 2050 443d 7072 696d 6172 7944 6563 : PD=primaryDec │ │ │ │ -0000d620: 6f6d 706f 7369 7469 6f6e 2820 6964 6561 omposition( idea │ │ │ │ -0000d630: 6c20 4629 2020 2020 2020 7c0a 7c20 2020 l F) |.| │ │ │ │ +0000d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000d610: 3720 3a20 5044 3d70 7269 6d61 7279 4465 7 : PD=primaryDe │ │ │ │ +0000d620: 636f 6d70 6f73 6974 696f 6e28 2069 6465 composition( ide │ │ │ │ +0000d630: 616c 2046 2920 2020 2020 207c 0a7c 2020 al F) |.| │ │ │ │ 0000d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d660: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0000d670: 2020 2020 2020 2020 2032 2020 2020 3220 2 2 │ │ │ │ -0000d680: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000d690: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20 |.|o7 = │ │ │ │ -0000d6a0: 7b69 6465 616c 2878 2020 2b20 7920 202b {ideal(x + y + │ │ │ │ -0000d6b0: 207a 2020 2d20 3129 2c20 6964 6561 6c20 z - 1), ideal │ │ │ │ -0000d6c0: 2879 2c20 7829 7d7c 0a7c 2020 2020 2020 (y, x)}|.| │ │ │ │ +0000d660: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000d670: 2020 2020 2020 2020 2020 3220 2020 2032 2 2 │ │ │ │ +0000d680: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0000d690: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +0000d6a0: 207b 6964 6561 6c28 7820 202b 2079 2020 {ideal(x + y │ │ │ │ +0000d6b0: 2b20 7a20 202d 2031 292c 2069 6465 616c + z - 1), ideal │ │ │ │ +0000d6c0: 2028 792c 2078 297d 7c0a 7c20 2020 2020 (y, x)}|.| │ │ │ │ 0000d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d6f0: 2020 2020 2020 7c0a 7c6f 3720 3a20 4c69 |.|o7 : Li │ │ │ │ -0000d700: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +0000d6f0: 2020 2020 2020 207c 0a7c 6f37 203a 204c |.|o7 : L │ │ │ │ +0000d700: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 0000d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d720: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0000d720: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0000d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d750: 2d2d 2d2d 2b0a 7c69 3820 3a20 6469 6d20 ----+.|i8 : dim │ │ │ │ -0000d760: 5044 5f30 2020 2020 2020 2020 2020 2020 PD_0 │ │ │ │ +0000d750: 2d2d 2d2d 2d2b 0a7c 6938 203a 2064 696d -----+.|i8 : dim │ │ │ │ +0000d760: 2050 445f 3020 2020 2020 2020 2020 2020 PD_0 │ │ │ │ 0000d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d780: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000d780: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0000d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d7b0: 2020 7c0a 7c6f 3820 3d20 3220 2020 2020 |.|o8 = 2 │ │ │ │ +0000d7b0: 2020 207c 0a7c 6f38 203d 2032 2020 2020 |.|o8 = 2 │ │ │ │ 0000d7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d7e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0000d7e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0000d7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d810: 2b0a 7c69 3920 3a20 6465 6772 6565 2050 +.|i9 : degree P │ │ │ │ -0000d820: 445f 3020 2020 2020 2020 2020 2020 2020 D_0 │ │ │ │ -0000d830: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000d840: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0000d810: 2d2b 0a7c 6939 203a 2064 6567 7265 6520 -+.|i9 : degree │ │ │ │ +0000d820: 5044 5f30 2020 2020 2020 2020 2020 2020 PD_0 │ │ │ │ +0000d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d840: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d860: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000d870: 7c6f 3920 3d20 3220 2020 2020 2020 2020 |o9 = 2 │ │ │ │ +0000d860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0000d870: 0a7c 6f39 203d 2032 2020 2020 2020 2020 .|o9 = 2 │ │ │ │ 0000d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d890: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0000d8a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0000d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0000d8d0: 3130 203a 2064 696d 2050 445f 3120 2020 10 : dim PD_1 │ │ │ │ +0000d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000d8d0: 6931 3020 3a20 6469 6d20 5044 5f31 2020 i10 : dim PD_1 │ │ │ │ 0000d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000d8f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d920: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ -0000d930: 203d 2031 2020 2020 2020 2020 2020 2020 = 1 │ │ │ │ +0000d920: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0000d930: 3020 3d20 3120 2020 2020 2020 2020 2020 0 = 1 │ │ │ │ 0000d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d950: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0000d950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0000d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d980: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ -0000d990: 2064 6567 7265 6520 5044 5f31 2020 2020 degree PD_1 │ │ │ │ +0000d980: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ +0000d990: 3a20 6465 6772 6565 2050 445f 3120 2020 : degree PD_1 │ │ │ │ 0000d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d9b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0000d9b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0000d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d9e0: 2020 2020 2020 7c0a 7c6f 3131 203d 2031 |.|o11 = 1 │ │ │ │ -0000d9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d9e0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ +0000d9f0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0000da00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000da10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0000da10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0000da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000da40: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -0000da50: 7365 2062 6572 7469 6e69 506f 7344 696d se bertiniPosDim │ │ │ │ -0000da60: 536f 6c76 653a 0a3d 3d3d 3d3d 3d3d 3d3d Solve:.========= │ │ │ │ +0000da40: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ +0000da50: 7573 6520 6265 7274 696e 6950 6f73 4469 use bertiniPosDi │ │ │ │ +0000da60: 6d53 6f6c 7665 3a0a 3d3d 3d3d 3d3d 3d3d mSolve:.======== │ │ │ │ 0000da70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000da80: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2262 6572 ======.. * "ber │ │ │ │ -0000da90: 7469 6e69 506f 7344 696d 536f 6c76 6528 tiniPosDimSolve( │ │ │ │ -0000daa0: 4964 6561 6c29 220a 2020 2a20 2262 6572 Ideal)". * "ber │ │ │ │ -0000dab0: 7469 6e69 506f 7344 696d 536f 6c76 6528 tiniPosDimSolve( │ │ │ │ -0000dac0: 4c69 7374 2922 0a0a 466f 7220 7468 6520 List)"..For the │ │ │ │ -0000dad0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0000dae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0000daf0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0000db00: 6265 7274 696e 6950 6f73 4469 6d53 6f6c bertiniPosDimSol │ │ │ │ -0000db10: 7665 3a20 6265 7274 696e 6950 6f73 4469 ve: bertiniPosDi │ │ │ │ -0000db20: 6d53 6f6c 7665 2c20 6973 2061 202a 6e6f mSolve, is a *no │ │ │ │ -0000db30: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ -0000db40: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ -0000db50: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0000db60: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ -0000db70: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +0000da80: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6265 =======.. * "be │ │ │ │ +0000da90: 7274 696e 6950 6f73 4469 6d53 6f6c 7665 rtiniPosDimSolve │ │ │ │ +0000daa0: 2849 6465 616c 2922 0a20 202a 2022 6265 (Ideal)". * "be │ │ │ │ +0000dab0: 7274 696e 6950 6f73 4469 6d53 6f6c 7665 rtiniPosDimSolve │ │ │ │ +0000dac0: 284c 6973 7429 220a 0a46 6f72 2074 6865 (List)"..For the │ │ │ │ +0000dad0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +0000dae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0000daf0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +0000db00: 2062 6572 7469 6e69 506f 7344 696d 536f bertiniPosDimSo │ │ │ │ +0000db10: 6c76 653a 2062 6572 7469 6e69 506f 7344 lve: bertiniPosD │ │ │ │ +0000db20: 696d 536f 6c76 652c 2069 7320 6120 2a6e imSolve, is a *n │ │ │ │ +0000db30: 6f74 6520 6d65 7468 6f64 0a66 756e 6374 ote method.funct │ │ │ │ +0000db40: 696f 6e20 7769 7468 206f 7074 696f 6e73 ion with options │ │ │ │ +0000db50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0000db60: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +0000db70: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ 0000db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0000dbd0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0000dbe0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0000dbf0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0000dc00: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0000dc10: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -0000dc20: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0000dc30: 6573 2f42 6572 7469 6e69 2e6d 323a 0a32 es/Bertini.m2:.2 │ │ │ │ -0000dc40: 3735 363a 302e 0a1f 0a46 696c 653a 2042 756:0....File: B │ │ │ │ -0000dc50: 6572 7469 6e69 2e69 6e66 6f2c 204e 6f64 ertini.info, Nod │ │ │ │ -0000dc60: 653a 2062 6572 7469 6e69 5265 6669 6e65 e: bertiniRefine │ │ │ │ -0000dc70: 536f 6c73 2c20 4e65 7874 3a20 6265 7274 Sols, Next: bert │ │ │ │ -0000dc80: 696e 6953 616d 706c 652c 2050 7265 763a iniSample, Prev: │ │ │ │ -0000dc90: 2062 6572 7469 6e69 506f 7344 696d 536f bertiniPosDimSo │ │ │ │ -0000dca0: 6c76 652c 2055 703a 2054 6f70 0a0a 6265 lve, Up: Top..be │ │ │ │ -0000dcb0: 7274 696e 6952 6566 696e 6553 6f6c 7320 rtiniRefineSols │ │ │ │ -0000dcc0: 2d2d 2073 6861 7270 656e 2073 6f6c 7574 -- sharpen solut │ │ │ │ -0000dcd0: 696f 6e73 2074 6f20 6120 7072 6573 6372 ions to a prescr │ │ │ │ -0000dce0: 6962 6564 206e 756d 6265 7220 6f66 2064 ibed number of d │ │ │ │ -0000dcf0: 6967 6974 730a 2a2a 2a2a 2a2a 2a2a 2a2a igits.********** │ │ │ │ +0000dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +0000dbd0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +0000dbe0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +0000dbf0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +0000dc00: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +0000dc10: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ +0000dc20: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +0000dc30: 6765 732f 4265 7274 696e 692e 6d32 3a0a ges/Bertini.m2:. │ │ │ │ +0000dc40: 3237 3536 3a30 2e0a 1f0a 4669 6c65 3a20 2756:0....File: │ │ │ │ +0000dc50: 4265 7274 696e 692e 696e 666f 2c20 4e6f Bertini.info, No │ │ │ │ +0000dc60: 6465 3a20 6265 7274 696e 6952 6566 696e de: bertiniRefin │ │ │ │ +0000dc70: 6553 6f6c 732c 204e 6578 743a 2062 6572 eSols, Next: ber │ │ │ │ +0000dc80: 7469 6e69 5361 6d70 6c65 2c20 5072 6576 tiniSample, Prev │ │ │ │ +0000dc90: 3a20 6265 7274 696e 6950 6f73 4469 6d53 : bertiniPosDimS │ │ │ │ +0000dca0: 6f6c 7665 2c20 5570 3a20 546f 700a 0a62 olve, Up: Top..b │ │ │ │ +0000dcb0: 6572 7469 6e69 5265 6669 6e65 536f 6c73 ertiniRefineSols │ │ │ │ +0000dcc0: 202d 2d20 7368 6172 7065 6e20 736f 6c75 -- sharpen solu │ │ │ │ +0000dcd0: 7469 6f6e 7320 746f 2061 2070 7265 7363 tions to a presc │ │ │ │ +0000dce0: 7269 6265 6420 6e75 6d62 6572 206f 6620 ribed number of │ │ │ │ +0000dcf0: 6469 6769 7473 0a2a 2a2a 2a2a 2a2a 2a2a digits.********* │ │ │ │ 0000dd00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000dd10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000dd20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000dd30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -0000dd40: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -0000dd50: 2020 2053 203d 2062 6572 7469 6e69 5265 S = bertiniRe │ │ │ │ -0000dd60: 6669 6e65 536f 6c73 2849 4644 2c20 642c fineSols(IFD, d, │ │ │ │ -0000dd70: 2057 2c20 4f46 4429 0a20 2020 2020 2020 W, OFD). │ │ │ │ -0000dd80: 2053 203d 2062 6572 7469 6e69 5265 6669 S = bertiniRefi │ │ │ │ -0000dd90: 6e65 536f 6c73 2864 2c20 572c 204f 4644 neSols(d, W, OFD │ │ │ │ -0000dda0: 290a 2020 2020 2020 2020 5320 3d20 6265 ). S = be │ │ │ │ -0000ddb0: 7274 696e 6952 6566 696e 6553 6f6c 7328 rtiniRefineSols( │ │ │ │ -0000ddc0: 642c 2057 290a 2020 2a20 496e 7075 7473 d, W). * Inputs │ │ │ │ -0000ddd0: 3a0a 2020 2020 2020 2a20 4946 442c 2061 :. * IFD, a │ │ │ │ -0000dde0: 202a 6e6f 7465 2073 7472 696e 673a 2028 *note string: ( │ │ │ │ -0000ddf0: 4d61 6361 756c 6179 3244 6f63 2953 7472 Macaulay2Doc)Str │ │ │ │ -0000de00: 696e 672c 2c20 6120 6469 7265 6374 6f72 ing,, a director │ │ │ │ -0000de10: 7920 7768 6572 6520 7468 6520 696e 7075 y where the inpu │ │ │ │ -0000de20: 740a 2020 2020 2020 2020 6669 6c65 206f t. file o │ │ │ │ -0000de30: 6620 6120 6265 7274 696e 695a 6572 6f44 f a bertiniZeroD │ │ │ │ -0000de40: 696d 2073 6f6c 7665 2069 7320 7374 6f72 im solve is stor │ │ │ │ -0000de50: 6564 2061 6c6f 6e67 2077 6974 6820 7468 ed along with th │ │ │ │ -0000de60: 6174 2072 756e 7320 6f75 7470 7574 0a20 at runs output. │ │ │ │ -0000de70: 2020 2020 2020 2066 696c 6573 0a20 2020 files. │ │ │ │ -0000de80: 2020 202a 2064 2c20 616e 202a 6e6f 7465 * d, an *note │ │ │ │ -0000de90: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -0000dea0: 6c61 7932 446f 6329 5a5a 2c2c 2061 6e20 lay2Doc)ZZ,, an │ │ │ │ -0000deb0: 696e 7465 6765 7220 7370 6563 6966 7969 integer specifyi │ │ │ │ -0000dec0: 6e67 2074 6865 0a20 2020 2020 2020 206e ng the. n │ │ │ │ -0000ded0: 756d 6265 7220 6f66 2064 6967 6974 7320 umber of digits │ │ │ │ -0000dee0: 6f66 2070 7265 6369 7369 6f6e 0a20 2020 of precision. │ │ │ │ -0000def0: 2020 202a 2057 2c20 6120 2a6e 6f74 6520 * W, a *note │ │ │ │ -0000df00: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ -0000df10: 446f 6329 4c69 7374 2c2c 2061 206c 6973 Doc)List,, a lis │ │ │ │ -0000df20: 7420 6f66 2070 6f69 6e74 7320 746f 2062 t of points to b │ │ │ │ -0000df30: 6520 7368 6172 7065 6e65 640a 2020 2020 e sharpened. │ │ │ │ -0000df40: 2020 2a20 4f46 442c 2061 202a 6e6f 7465 * OFD, a *note │ │ │ │ -0000df50: 2073 7472 696e 673a 2028 4d61 6361 756c string: (Macaul │ │ │ │ -0000df60: 6179 3244 6f63 2953 7472 696e 672c 2c20 ay2Doc)String,, │ │ │ │ -0000df70: 6120 6469 7265 6374 6f72 7920 7768 6572 a directory wher │ │ │ │ -0000df80: 6520 7468 650a 2020 2020 2020 2020 6f75 e the. ou │ │ │ │ -0000df90: 7470 7574 2066 696c 6573 206f 6620 7468 tput files of th │ │ │ │ -0000dfa0: 6520 7265 6669 6e65 6d65 6e74 2061 7265 e refinement are │ │ │ │ -0000dfb0: 2073 746f 7265 640a 2020 2a20 2a6e 6f74 stored. * *not │ │ │ │ -0000dfc0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ -0000dfd0: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ -0000dfe0: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ -0000dff0: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -0000e000: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -0000e010: 4164 6469 7469 6f6e 616c 4669 6c65 7320 AdditionalFiles │ │ │ │ -0000e020: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -0000e030: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -0000e040: 6465 6661 756c 7420 7661 6c75 6520 7b7d default value {} │ │ │ │ -0000e050: 2c20 0a20 2020 2020 202a 202a 6e6f 7465 , . * *note │ │ │ │ -0000e060: 2049 7350 726f 6a65 6374 6976 653a 2049 IsProjective: I │ │ │ │ -0000e070: 7350 726f 6a65 6374 6976 652c 203d 3e20 sProjective, => │ │ │ │ -0000e080: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0000e090: 7565 202d 312c 206f 7074 696f 6e61 6c0a ue -1, optional. │ │ │ │ -0000e0a0: 2020 2020 2020 2020 6172 6775 6d65 6e74 argument │ │ │ │ -0000e0b0: 2074 6f20 7370 6563 6966 7920 7768 6574 to specify whet │ │ │ │ -0000e0c0: 6865 7220 746f 2075 7365 2068 6f6d 6f67 her to use homog │ │ │ │ -0000e0d0: 656e 656f 7573 2063 6f6f 7264 696e 6174 eneous coordinat │ │ │ │ -0000e0e0: 6573 0a20 2020 2020 202a 204e 616d 6542 es. * NameB │ │ │ │ -0000e0f0: 2749 6e70 7574 4669 6c65 2028 6d69 7373 'InputFile (miss │ │ │ │ -0000e100: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -0000e110: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -0000e120: 6c74 2076 616c 7565 2022 696e 7075 7422 lt value "input" │ │ │ │ -0000e130: 2c20 0a20 2020 2020 202a 202a 6e6f 7465 , . * *note │ │ │ │ -0000e140: 2056 6572 626f 7365 3a20 6265 7274 696e Verbose: bertin │ │ │ │ -0000e150: 6954 7261 636b 486f 6d6f 746f 7079 5f6c iTrackHomotopy_l │ │ │ │ -0000e160: 705f 7064 5f70 645f 7064 5f63 6d56 6572 p_pd_pd_pd_cmVer │ │ │ │ -0000e170: 626f 7365 3d3e 5f70 645f 7064 5f70 645f bose=>_pd_pd_pd_ │ │ │ │ -0000e180: 7270 0a20 2020 2020 2020 202c 203d 3e20 rp. , => │ │ │ │ -0000e190: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0000e1a0: 7565 2066 616c 7365 2c20 4f70 7469 6f6e ue false, Option │ │ │ │ -0000e1b0: 2074 6f20 7369 6c65 6e63 6520 6164 6469 to silence addi │ │ │ │ -0000e1c0: 7469 6f6e 616c 206f 7574 7075 740a 2020 tional output. │ │ │ │ -0000e1d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0000e1e0: 202a 2053 2c20 6120 2a6e 6f74 6520 6c69 * S, a *note li │ │ │ │ -0000e1f0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -0000e200: 6329 4c69 7374 2c2c 2061 206c 6973 7420 c)List,, a list │ │ │ │ -0000e210: 6f66 2073 6f6c 7574 696f 6e73 206f 6620 of solutions of │ │ │ │ -0000e220: 7479 7065 2050 6f69 6e74 0a0a 4465 7363 type Point..Desc │ │ │ │ -0000e230: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0000e240: 3d3d 3d0a 0a54 6869 7320 6d65 7468 6f64 ===..This method │ │ │ │ -0000e250: 2074 616b 6573 2074 6865 206c 6973 7420 takes the list │ │ │ │ -0000e260: 5720 6f66 2073 6f6c 7574 696f 6e73 2066 W of solutions f │ │ │ │ -0000e270: 726f 6d20 6120 6265 7274 696e 695a 6572 rom a bertiniZer │ │ │ │ -0000e280: 6f44 696d 536f 6c76 6520 616e 640a 7368 oDimSolve and.sh │ │ │ │ -0000e290: 6172 7065 6e73 2074 6865 6d20 746f 2064 arpens them to d │ │ │ │ -0000e2a0: 2064 6967 6974 7320 7573 696e 6720 7468 digits using th │ │ │ │ -0000e2b0: 6520 7368 6172 7065 6e69 6e67 206d 6f64 e sharpening mod │ │ │ │ -0000e2c0: 756c 6520 6f66 2042 6572 7469 6e69 2e20 ule of Bertini. │ │ │ │ -0000e2d0: 5768 656e 2049 4644 2069 730a 6f6d 6974 When IFD is.omit │ │ │ │ -0000e2e0: 7465 6420 7468 6520 696e 666f 726d 6174 ted the informat │ │ │ │ -0000e2f0: 696f 6e20 6973 2070 756c 6c65 6420 6672 ion is pulled fr │ │ │ │ -0000e300: 6f6d 2074 6865 2063 6163 6865 206f 6620 om the cache of │ │ │ │ -0000e310: 7468 6520 6669 7273 7420 706f 696e 7420 the first point │ │ │ │ -0000e320: 696e 2057 2e20 5768 656e 0a4f 4644 2069 in W. When.OFD i │ │ │ │ -0000e330: 7320 6f6d 6974 7465 6420 6120 7465 6d70 s omitted a temp │ │ │ │ -0000e340: 6f72 6172 7920 6469 7265 6374 6f72 7920 orary directory │ │ │ │ -0000e350: 6973 2063 7265 6174 6564 2e0a 0a2b 2d2d is created...+-- │ │ │ │ +0000dd30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0000dd40: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0000dd50: 2020 2020 5320 3d20 6265 7274 696e 6952 S = bertiniR │ │ │ │ +0000dd60: 6566 696e 6553 6f6c 7328 4946 442c 2064 efineSols(IFD, d │ │ │ │ +0000dd70: 2c20 572c 204f 4644 290a 2020 2020 2020 , W, OFD). │ │ │ │ +0000dd80: 2020 5320 3d20 6265 7274 696e 6952 6566 S = bertiniRef │ │ │ │ +0000dd90: 696e 6553 6f6c 7328 642c 2057 2c20 4f46 ineSols(d, W, OF │ │ │ │ +0000dda0: 4429 0a20 2020 2020 2020 2053 203d 2062 D). S = b │ │ │ │ +0000ddb0: 6572 7469 6e69 5265 6669 6e65 536f 6c73 ertiniRefineSols │ │ │ │ +0000ddc0: 2864 2c20 5729 0a20 202a 2049 6e70 7574 (d, W). * Input │ │ │ │ +0000ddd0: 733a 0a20 2020 2020 202a 2049 4644 2c20 s:. * IFD, │ │ │ │ +0000dde0: 6120 2a6e 6f74 6520 7374 7269 6e67 3a20 a *note string: │ │ │ │ +0000ddf0: 284d 6163 6175 6c61 7932 446f 6329 5374 (Macaulay2Doc)St │ │ │ │ +0000de00: 7269 6e67 2c2c 2061 2064 6972 6563 746f ring,, a directo │ │ │ │ +0000de10: 7279 2077 6865 7265 2074 6865 2069 6e70 ry where the inp │ │ │ │ +0000de20: 7574 0a20 2020 2020 2020 2066 696c 6520 ut. file │ │ │ │ +0000de30: 6f66 2061 2062 6572 7469 6e69 5a65 726f of a bertiniZero │ │ │ │ +0000de40: 4469 6d20 736f 6c76 6520 6973 2073 746f Dim solve is sto │ │ │ │ +0000de50: 7265 6420 616c 6f6e 6720 7769 7468 2074 red along with t │ │ │ │ +0000de60: 6861 7420 7275 6e73 206f 7574 7075 740a hat runs output. │ │ │ │ +0000de70: 2020 2020 2020 2020 6669 6c65 730a 2020 files. │ │ │ │ +0000de80: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74 * d, an *not │ │ │ │ +0000de90: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +0000dea0: 756c 6179 3244 6f63 295a 5a2c 2c20 616e ulay2Doc)ZZ,, an │ │ │ │ +0000deb0: 2069 6e74 6567 6572 2073 7065 6369 6679 integer specify │ │ │ │ +0000dec0: 696e 6720 7468 650a 2020 2020 2020 2020 ing the. │ │ │ │ +0000ded0: 6e75 6d62 6572 206f 6620 6469 6769 7473 number of digits │ │ │ │ +0000dee0: 206f 6620 7072 6563 6973 696f 6e0a 2020 of precision. │ │ │ │ +0000def0: 2020 2020 2a20 572c 2061 202a 6e6f 7465 * W, a *note │ │ │ │ +0000df00: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ +0000df10: 3244 6f63 294c 6973 742c 2c20 6120 6c69 2Doc)List,, a li │ │ │ │ +0000df20: 7374 206f 6620 706f 696e 7473 2074 6f20 st of points to │ │ │ │ +0000df30: 6265 2073 6861 7270 656e 6564 0a20 2020 be sharpened. │ │ │ │ +0000df40: 2020 202a 204f 4644 2c20 6120 2a6e 6f74 * OFD, a *not │ │ │ │ +0000df50: 6520 7374 7269 6e67 3a20 284d 6163 6175 e string: (Macau │ │ │ │ +0000df60: 6c61 7932 446f 6329 5374 7269 6e67 2c2c lay2Doc)String,, │ │ │ │ +0000df70: 2061 2064 6972 6563 746f 7279 2077 6865 a directory whe │ │ │ │ +0000df80: 7265 2074 6865 0a20 2020 2020 2020 206f re the. o │ │ │ │ +0000df90: 7574 7075 7420 6669 6c65 7320 6f66 2074 utput files of t │ │ │ │ +0000dfa0: 6865 2072 6566 696e 656d 656e 7420 6172 he refinement ar │ │ │ │ +0000dfb0: 6520 7374 6f72 6564 0a20 202a 202a 6e6f e stored. * *no │ │ │ │ +0000dfc0: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +0000dfd0: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +0000dfe0: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +0000dff0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +0000e000: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +0000e010: 2041 6464 6974 696f 6e61 6c46 696c 6573 AdditionalFiles │ │ │ │ +0000e020: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +0000e030: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +0000e040: 2064 6566 6175 6c74 2076 616c 7565 207b default value { │ │ │ │ +0000e050: 7d2c 200a 2020 2020 2020 2a20 2a6e 6f74 }, . * *not │ │ │ │ +0000e060: 6520 4973 5072 6f6a 6563 7469 7665 3a20 e IsProjective: │ │ │ │ +0000e070: 4973 5072 6f6a 6563 7469 7665 2c20 3d3e IsProjective, => │ │ │ │ +0000e080: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +0000e090: 6c75 6520 2d31 2c20 6f70 7469 6f6e 616c lue -1, optional │ │ │ │ +0000e0a0: 0a20 2020 2020 2020 2061 7267 756d 656e . argumen │ │ │ │ +0000e0b0: 7420 746f 2073 7065 6369 6679 2077 6865 t to specify whe │ │ │ │ +0000e0c0: 7468 6572 2074 6f20 7573 6520 686f 6d6f ther to use homo │ │ │ │ +0000e0d0: 6765 6e65 6f75 7320 636f 6f72 6469 6e61 geneous coordina │ │ │ │ +0000e0e0: 7465 730a 2020 2020 2020 2a20 4e61 6d65 tes. * Name │ │ │ │ +0000e0f0: 4227 496e 7075 7446 696c 6520 286d 6973 B'InputFile (mis │ │ │ │ +0000e100: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ +0000e110: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ +0000e120: 756c 7420 7661 6c75 6520 2269 6e70 7574 ult value "input │ │ │ │ +0000e130: 222c 200a 2020 2020 2020 2a20 2a6e 6f74 ", . * *not │ │ │ │ +0000e140: 6520 5665 7262 6f73 653a 2062 6572 7469 e Verbose: berti │ │ │ │ +0000e150: 6e69 5472 6163 6b48 6f6d 6f74 6f70 795f niTrackHomotopy_ │ │ │ │ +0000e160: 6c70 5f70 645f 7064 5f70 645f 636d 5665 lp_pd_pd_pd_cmVe │ │ │ │ +0000e170: 7262 6f73 653d 3e5f 7064 5f70 645f 7064 rbose=>_pd_pd_pd │ │ │ │ +0000e180: 5f72 700a 2020 2020 2020 2020 2c20 3d3e _rp. , => │ │ │ │ +0000e190: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +0000e1a0: 6c75 6520 6661 6c73 652c 204f 7074 696f lue false, Optio │ │ │ │ +0000e1b0: 6e20 746f 2073 696c 656e 6365 2061 6464 n to silence add │ │ │ │ +0000e1c0: 6974 696f 6e61 6c20 6f75 7470 7574 0a20 itional output. │ │ │ │ +0000e1d0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +0000e1e0: 2020 2a20 532c 2061 202a 6e6f 7465 206c * S, a *note l │ │ │ │ +0000e1f0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +0000e200: 6f63 294c 6973 742c 2c20 6120 6c69 7374 oc)List,, a list │ │ │ │ +0000e210: 206f 6620 736f 6c75 7469 6f6e 7320 6f66 of solutions of │ │ │ │ +0000e220: 2074 7970 6520 506f 696e 740a 0a44 6573 type Point..Des │ │ │ │ +0000e230: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0000e240: 3d3d 3d3d 0a0a 5468 6973 206d 6574 686f ====..This metho │ │ │ │ +0000e250: 6420 7461 6b65 7320 7468 6520 6c69 7374 d takes the list │ │ │ │ +0000e260: 2057 206f 6620 736f 6c75 7469 6f6e 7320 W of solutions │ │ │ │ +0000e270: 6672 6f6d 2061 2062 6572 7469 6e69 5a65 from a bertiniZe │ │ │ │ +0000e280: 726f 4469 6d53 6f6c 7665 2061 6e64 0a73 roDimSolve and.s │ │ │ │ +0000e290: 6861 7270 656e 7320 7468 656d 2074 6f20 harpens them to │ │ │ │ +0000e2a0: 6420 6469 6769 7473 2075 7369 6e67 2074 d digits using t │ │ │ │ +0000e2b0: 6865 2073 6861 7270 656e 696e 6720 6d6f he sharpening mo │ │ │ │ +0000e2c0: 6475 6c65 206f 6620 4265 7274 696e 692e dule of Bertini. │ │ │ │ +0000e2d0: 2057 6865 6e20 4946 4420 6973 0a6f 6d69 When IFD is.omi │ │ │ │ +0000e2e0: 7474 6564 2074 6865 2069 6e66 6f72 6d61 tted the informa │ │ │ │ +0000e2f0: 7469 6f6e 2069 7320 7075 6c6c 6564 2066 tion is pulled f │ │ │ │ +0000e300: 726f 6d20 7468 6520 6361 6368 6520 6f66 rom the cache of │ │ │ │ +0000e310: 2074 6865 2066 6972 7374 2070 6f69 6e74 the first point │ │ │ │ +0000e320: 2069 6e20 572e 2057 6865 6e0a 4f46 4420 in W. When.OFD │ │ │ │ +0000e330: 6973 206f 6d69 7474 6564 2061 2074 656d is omitted a tem │ │ │ │ +0000e340: 706f 7261 7279 2064 6972 6563 746f 7279 porary directory │ │ │ │ +0000e350: 2069 7320 6372 6561 7465 642e 0a0a 2b2d is created...+- │ │ │ │ 0000e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0000e3b0: 203a 2052 203d 2043 435b 782c 795d 3b20 : R = CC[x,y]; │ │ │ │ +0000e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000e3b0: 3120 3a20 5220 3d20 4343 5b78 2c79 5d3b 1 : R = CC[x,y]; │ │ │ │ 0000e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e3f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000e3f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0000e450: 203a 2046 203d 207b 785e 322d 322c 795e : F = {x^2-2,y^ │ │ │ │ -0000e460: 322d 327d 3b20 2020 2020 2020 2020 2020 2-2}; │ │ │ │ +0000e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000e450: 3220 3a20 4620 3d20 7b78 5e32 2d32 2c79 2 : F = {x^2-2,y │ │ │ │ +0000e460: 5e32 2d32 7d3b 2020 2020 2020 2020 2020 ^2-2}; │ │ │ │ 0000e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e490: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000e490: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000e4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0000e4f0: 203a 2057 203d 2062 6572 7469 6e69 5a65 : W = bertiniZe │ │ │ │ -0000e500: 726f 4469 6d53 6f6c 7665 2028 4629 2020 roDimSolve (F) │ │ │ │ +0000e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000e4f0: 3320 3a20 5720 3d20 6265 7274 696e 695a 3 : W = bertiniZ │ │ │ │ +0000e500: 6572 6f44 696d 536f 6c76 6520 2846 2920 eroDimSolve (F) │ │ │ │ 0000e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e530: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000e530: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e580: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0000e590: 203d 207b 7b31 2e34 3134 3231 2c20 312e = {{1.41421, 1. │ │ │ │ -0000e5a0: 3431 3432 317d 2c20 7b31 2e34 3134 3231 41421}, {1.41421 │ │ │ │ -0000e5b0: 2c20 2d31 2e34 3134 3231 7d2c 207b 2d31 , -1.41421}, {-1 │ │ │ │ -0000e5c0: 2e34 3134 3231 2c20 312e 3431 3432 317d .41421, 1.41421} │ │ │ │ -0000e5d0: 2c20 2020 2020 2020 2020 207c 0a7c 2020 , |.| │ │ │ │ -0000e5e0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +0000e580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000e590: 3320 3d20 7b7b 312e 3431 3432 312c 2031 3 = {{1.41421, 1 │ │ │ │ +0000e5a0: 2e34 3134 3231 7d2c 207b 312e 3431 3432 .41421}, {1.4142 │ │ │ │ +0000e5b0: 312c 202d 312e 3431 3432 317d 2c20 7b2d 1, -1.41421}, {- │ │ │ │ +0000e5c0: 312e 3431 3432 312c 2031 2e34 3134 3231 1.41421, 1.41421 │ │ │ │ +0000e5d0: 7d2c 2020 2020 2020 2020 2020 7c0a 7c20 }, |.| │ │ │ │ +0000e5e0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 0000e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0000e630: 2020 207b 2d31 2e34 3134 3231 2c20 2d31 {-1.41421, -1 │ │ │ │ -0000e640: 2e34 3134 3231 7d7d 2020 2020 2020 2020 .41421}} │ │ │ │ +0000e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +0000e630: 2020 2020 7b2d 312e 3431 3432 312c 202d {-1.41421, - │ │ │ │ +0000e640: 312e 3431 3432 317d 7d20 2020 2020 2020 1.41421}} │ │ │ │ 0000e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e670: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000e670: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e6c0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0000e6d0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0000e6c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000e6d0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ 0000e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000e710: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0000e770: 203a 2053 203d 2062 6572 7469 6e69 5265 : S = bertiniRe │ │ │ │ -0000e780: 6669 6e65 536f 6c73 2028 3130 302c 5729 fineSols (100,W) │ │ │ │ -0000e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000e770: 3420 3a20 5320 3d20 6265 7274 696e 6952 4 : S = bertiniR │ │ │ │ +0000e780: 6566 696e 6553 6f6c 7320 2831 3030 2c57 efineSols (100,W │ │ │ │ +0000e790: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0000e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000e7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e800: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0000e810: 203d 207b 7b31 2e34 3134 3231 2c20 312e = {{1.41421, 1. │ │ │ │ -0000e820: 3431 3432 317d 2c20 7b31 2e34 3134 3231 41421}, {1.41421 │ │ │ │ -0000e830: 2c20 2d31 2e34 3134 3231 7d2c 207b 2d31 , -1.41421}, {-1 │ │ │ │ -0000e840: 2e34 3134 3231 2c20 312e 3431 3432 317d .41421, 1.41421} │ │ │ │ -0000e850: 2c20 2020 2020 2020 2020 207c 0a7c 2020 , |.| │ │ │ │ -0000e860: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +0000e800: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000e810: 3420 3d20 7b7b 312e 3431 3432 312c 2031 4 = {{1.41421, 1 │ │ │ │ +0000e820: 2e34 3134 3231 7d2c 207b 312e 3431 3432 .41421}, {1.4142 │ │ │ │ +0000e830: 312c 202d 312e 3431 3432 317d 2c20 7b2d 1, -1.41421}, {- │ │ │ │ +0000e840: 312e 3431 3432 312c 2031 2e34 3134 3231 1.41421, 1.41421 │ │ │ │ +0000e850: 7d2c 2020 2020 2020 2020 2020 7c0a 7c20 }, |.| │ │ │ │ +0000e860: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 0000e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0000e8b0: 2020 207b 2d31 2e34 3134 3231 2c20 2d31 {-1.41421, -1 │ │ │ │ -0000e8c0: 2e34 3134 3231 7d7d 2020 2020 2020 2020 .41421}} │ │ │ │ +0000e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +0000e8b0: 2020 2020 7b2d 312e 3431 3432 312c 202d {-1.41421, - │ │ │ │ +0000e8c0: 312e 3431 3432 317d 7d20 2020 2020 2020 1.41421}} │ │ │ │ 0000e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000e8f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e940: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0000e950: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0000e940: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000e950: 3420 3a20 4c69 7374 2020 2020 2020 2020 4 : List │ │ │ │ 0000e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000e990: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ -0000e9f0: 203a 2063 6f6f 7264 7320 3d20 636f 6f72 : coords = coor │ │ │ │ -0000ea00: 6469 6e61 7465 7320 535f 3020 2020 2020 dinates S_0 │ │ │ │ +0000e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000e9f0: 3520 3a20 636f 6f72 6473 203d 2063 6f6f 5 : coords = coo │ │ │ │ +0000ea00: 7264 696e 6174 6573 2053 5f30 2020 2020 rdinates S_0 │ │ │ │ 0000ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ea30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000ea30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ea80: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -0000ea90: 203d 207b 312e 3431 3432 312c 2031 2e34 = {1.41421, 1.4 │ │ │ │ -0000eaa0: 3134 3231 7d20 2020 2020 2020 2020 2020 1421} │ │ │ │ +0000ea80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000ea90: 3520 3d20 7b31 2e34 3134 3231 2c20 312e 5 = {1.41421, 1. │ │ │ │ +0000eaa0: 3431 3432 317d 2020 2020 2020 2020 2020 41421} │ │ │ │ 0000eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ead0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000ead0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eb20: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -0000eb30: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0000eb20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000eb30: 3520 3a20 4c69 7374 2020 2020 2020 2020 5 : List │ │ │ │ 0000eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eb70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000eb70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000eb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -0000ebd0: 203a 2063 6f6f 7264 735f 3020 2020 2020 : coords_0 │ │ │ │ +0000ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0000ebd0: 3620 3a20 636f 6f72 6473 5f30 2020 2020 6 : coords_0 │ │ │ │ 0000ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ec10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000ec10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ec60: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0000ec70: 203d 2031 2e34 3134 3231 3335 3632 3337 = 1.41421356237 │ │ │ │ -0000ec80: 3330 3935 3034 3838 3031 3638 3837 3234 3095048801688724 │ │ │ │ -0000ec90: 3230 3936 3938 3037 3835 3639 3637 3138 2096980785696718 │ │ │ │ -0000eca0: 3735 3337 3639 3438 3037 3331 3736 3637 7537694807317667 │ │ │ │ -0000ecb0: 3937 3337 3939 3037 3332 347c 0a7c 2020 97379907324|.| │ │ │ │ -0000ecc0: 2020 2037 3834 3632 3130 3730 3338 3835 7846210703885 │ │ │ │ -0000ecd0: 3033 3837 3533 3433 3237 3634 3135 3733 0387534327641573 │ │ │ │ -0000ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ec60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000ec70: 3620 3d20 312e 3431 3432 3133 3536 3233 6 = 1.4142135623 │ │ │ │ +0000ec80: 3733 3039 3530 3438 3830 3136 3838 3732 7309504880168872 │ │ │ │ +0000ec90: 3432 3039 3639 3830 3738 3536 3936 3731 4209698078569671 │ │ │ │ +0000eca0: 3837 3533 3736 3934 3830 3733 3137 3636 8753769480731766 │ │ │ │ +0000ecb0: 3739 3733 3739 3930 3733 3234 7c0a 7c20 797379907324|.| │ │ │ │ +0000ecc0: 2020 2020 3738 3436 3231 3037 3033 3838 784621070388 │ │ │ │ +0000ecd0: 3530 3338 3735 3334 3332 3736 3431 3537 5038753432764157 │ │ │ │ +0000ece0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0000ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ed00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0000ed00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ed50: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0000ed60: 203a 2043 4320 286f 6620 7072 6563 6973 : CC (of precis │ │ │ │ -0000ed70: 696f 6e20 3333 3329 2020 2020 2020 2020 ion 333) │ │ │ │ +0000ed50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0000ed60: 3620 3a20 4343 2028 6f66 2070 7265 6369 6 : CC (of preci │ │ │ │ +0000ed70: 7369 6f6e 2033 3333 2920 2020 2020 2020 sion 333) │ │ │ │ 0000ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eda0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000eda0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 2a6e -----------+..*n │ │ │ │ -0000ee00: 6f74 6520 6265 7274 696e 6952 6566 696e ote bertiniRefin │ │ │ │ -0000ee10: 6553 6f6c 733a 2062 6572 7469 6e69 5265 eSols: bertiniRe │ │ │ │ -0000ee20: 6669 6e65 536f 6c73 2c20 7769 6c6c 206f fineSols, will o │ │ │ │ -0000ee30: 6e6c 7920 7265 6669 6e65 206e 6f6e 2d73 nly refine non-s │ │ │ │ -0000ee40: 696e 6775 6c61 720a 736f 6c75 7469 6f6e ingular.solution │ │ │ │ -0000ee50: 7320 616e 6420 646f 6573 206e 6f74 2063 s and does not c │ │ │ │ -0000ee60: 7572 7265 6e74 6c79 2077 6f72 6b20 666f urrently work fo │ │ │ │ -0000ee70: 7220 686f 6d6f 6765 6e65 6f75 7320 7379 r homogeneous sy │ │ │ │ -0000ee80: 7374 656d 732e 0a0a 5761 7973 2074 6f20 stems...Ways to │ │ │ │ -0000ee90: 7573 6520 6265 7274 696e 6952 6566 696e use bertiniRefin │ │ │ │ -0000eea0: 6553 6f6c 733a 0a3d 3d3d 3d3d 3d3d 3d3d eSols:.========= │ │ │ │ +0000edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a2a ------------+..* │ │ │ │ +0000ee00: 6e6f 7465 2062 6572 7469 6e69 5265 6669 note bertiniRefi │ │ │ │ +0000ee10: 6e65 536f 6c73 3a20 6265 7274 696e 6952 neSols: bertiniR │ │ │ │ +0000ee20: 6566 696e 6553 6f6c 732c 2077 696c 6c20 efineSols, will │ │ │ │ +0000ee30: 6f6e 6c79 2072 6566 696e 6520 6e6f 6e2d only refine non- │ │ │ │ +0000ee40: 7369 6e67 756c 6172 0a73 6f6c 7574 696f singular.solutio │ │ │ │ +0000ee50: 6e73 2061 6e64 2064 6f65 7320 6e6f 7420 ns and does not │ │ │ │ +0000ee60: 6375 7272 656e 746c 7920 776f 726b 2066 currently work f │ │ │ │ +0000ee70: 6f72 2068 6f6d 6f67 656e 656f 7573 2073 or homogeneous s │ │ │ │ +0000ee80: 7973 7465 6d73 2e0a 0a57 6179 7320 746f ystems...Ways to │ │ │ │ +0000ee90: 2075 7365 2062 6572 7469 6e69 5265 6669 use bertiniRefi │ │ │ │ +0000eea0: 6e65 536f 6c73 3a0a 3d3d 3d3d 3d3d 3d3d neSols:.======== │ │ │ │ 0000eeb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000eec0: 3d3d 3d3d 3d0a 0a20 202a 2022 6265 7274 =====.. * "bert │ │ │ │ -0000eed0: 696e 6952 6566 696e 6553 6f6c 7328 5374 iniRefineSols(St │ │ │ │ -0000eee0: 7269 6e67 2c5a 5a2c 4c69 7374 2c53 7472 ring,ZZ,List,Str │ │ │ │ -0000eef0: 696e 6729 220a 2020 2a20 2262 6572 7469 ing)". * "berti │ │ │ │ -0000ef00: 6e69 5265 6669 6e65 536f 6c73 285a 5a2c niRefineSols(ZZ, │ │ │ │ -0000ef10: 4c69 7374 2922 0a20 202a 2022 6265 7274 List)". * "bert │ │ │ │ -0000ef20: 696e 6952 6566 696e 6553 6f6c 7328 5a5a iniRefineSols(ZZ │ │ │ │ -0000ef30: 2c4c 6973 742c 5374 7269 6e67 2922 0a0a ,List,String)".. │ │ │ │ -0000ef40: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0000ef50: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0000ef60: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0000ef70: 7420 2a6e 6f74 6520 6265 7274 696e 6952 t *note bertiniR │ │ │ │ -0000ef80: 6566 696e 6553 6f6c 733a 2062 6572 7469 efineSols: berti │ │ │ │ -0000ef90: 6e69 5265 6669 6e65 536f 6c73 2c20 6973 niRefineSols, is │ │ │ │ -0000efa0: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ -0000efb0: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ -0000efc0: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ -0000efd0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0000efe0: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ -0000eff0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0000eec0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2262 6572 ======.. * "ber │ │ │ │ +0000eed0: 7469 6e69 5265 6669 6e65 536f 6c73 2853 tiniRefineSols(S │ │ │ │ +0000eee0: 7472 696e 672c 5a5a 2c4c 6973 742c 5374 tring,ZZ,List,St │ │ │ │ +0000eef0: 7269 6e67 2922 0a20 202a 2022 6265 7274 ring)". * "bert │ │ │ │ +0000ef00: 696e 6952 6566 696e 6553 6f6c 7328 5a5a iniRefineSols(ZZ │ │ │ │ +0000ef10: 2c4c 6973 7429 220a 2020 2a20 2262 6572 ,List)". * "ber │ │ │ │ +0000ef20: 7469 6e69 5265 6669 6e65 536f 6c73 285a tiniRefineSols(Z │ │ │ │ +0000ef30: 5a2c 4c69 7374 2c53 7472 696e 6729 220a Z,List,String)". │ │ │ │ +0000ef40: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0000ef50: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0000ef60: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0000ef70: 6374 202a 6e6f 7465 2062 6572 7469 6e69 ct *note bertini │ │ │ │ +0000ef80: 5265 6669 6e65 536f 6c73 3a20 6265 7274 RefineSols: bert │ │ │ │ +0000ef90: 696e 6952 6566 696e 6553 6f6c 732c 2069 iniRefineSols, i │ │ │ │ +0000efa0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0000efb0: 0a66 756e 6374 696f 6e20 7769 7468 206f .function with o │ │ │ │ +0000efc0: 7074 696f 6e73 3a20 284d 6163 6175 6c61 ptions: (Macaula │ │ │ │ +0000efd0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0000efe0: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +0000eff0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0000f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f040: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0000f050: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0000f060: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0000f070: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0000f080: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ -0000f090: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0000f0a0: 7061 636b 6167 6573 2f42 6572 7469 6e69 packages/Bertini │ │ │ │ -0000f0b0: 2e6d 323a 0a32 3936 353a 302e 0a1f 0a46 .m2:.2965:0....F │ │ │ │ -0000f0c0: 696c 653a 2042 6572 7469 6e69 2e69 6e66 ile: Bertini.inf │ │ │ │ -0000f0d0: 6f2c 204e 6f64 653a 2062 6572 7469 6e69 o, Node: bertini │ │ │ │ -0000f0e0: 5361 6d70 6c65 2c20 4e65 7874 3a20 6265 Sample, Next: be │ │ │ │ -0000f0f0: 7274 696e 6954 7261 636b 486f 6d6f 746f rtiniTrackHomoto │ │ │ │ -0000f100: 7079 2c20 5072 6576 3a20 6265 7274 696e py, Prev: bertin │ │ │ │ -0000f110: 6952 6566 696e 6553 6f6c 732c 2055 703a iRefineSols, Up: │ │ │ │ -0000f120: 2054 6f70 0a0a 6265 7274 696e 6953 616d Top..bertiniSam │ │ │ │ -0000f130: 706c 6520 2d2d 2061 206d 6169 6e20 6d65 ple -- a main me │ │ │ │ -0000f140: 7468 6f64 2074 6f20 7361 6d70 6c65 2070 thod to sample p │ │ │ │ -0000f150: 6f69 6e74 7320 6672 6f6d 2061 6e20 6972 oints from an ir │ │ │ │ -0000f160: 7265 6475 6369 626c 6520 636f 6d70 6f6e reducible compon │ │ │ │ -0000f170: 656e 7420 6f66 2061 2076 6172 6965 7479 ent of a variety │ │ │ │ -0000f180: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0000f040: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0000f050: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0000f060: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0000f070: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0000f080: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +0000f090: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0000f0a0: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ +0000f0b0: 692e 6d32 3a0a 3239 3635 3a30 2e0a 1f0a i.m2:.2965:0.... │ │ │ │ +0000f0c0: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ +0000f0d0: 666f 2c20 4e6f 6465 3a20 6265 7274 696e fo, Node: bertin │ │ │ │ +0000f0e0: 6953 616d 706c 652c 204e 6578 743a 2062 iSample, Next: b │ │ │ │ +0000f0f0: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74 ertiniTrackHomot │ │ │ │ +0000f100: 6f70 792c 2050 7265 763a 2062 6572 7469 opy, Prev: berti │ │ │ │ +0000f110: 6e69 5265 6669 6e65 536f 6c73 2c20 5570 niRefineSols, Up │ │ │ │ +0000f120: 3a20 546f 700a 0a62 6572 7469 6e69 5361 : Top..bertiniSa │ │ │ │ +0000f130: 6d70 6c65 202d 2d20 6120 6d61 696e 206d mple -- a main m │ │ │ │ +0000f140: 6574 686f 6420 746f 2073 616d 706c 6520 ethod to sample │ │ │ │ +0000f150: 706f 696e 7473 2066 726f 6d20 616e 2069 points from an i │ │ │ │ +0000f160: 7272 6564 7563 6962 6c65 2063 6f6d 706f rreducible compo │ │ │ │ +0000f170: 6e65 6e74 206f 6620 6120 7661 7269 6574 nent of a variet │ │ │ │ +0000f180: 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a y.************** │ │ │ │ 0000f190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000f1a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000f1b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0000f1c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000f1d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0000f1e0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0000f1f0: 2056 203d 2062 6572 7469 6e69 5361 6d70 V = bertiniSamp │ │ │ │ -0000f200: 6c65 2028 6e2c 2057 290a 2020 2a20 496e le (n, W). * In │ │ │ │ -0000f210: 7075 7473 3a0a 2020 2020 2020 2a20 6e2c puts:. * n, │ │ │ │ -0000f220: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ -0000f230: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ -0000f240: 295a 5a2c 2c20 616e 2069 6e74 6567 6572 )ZZ,, an integer │ │ │ │ -0000f250: 2073 7065 6369 6679 696e 6720 7468 650a specifying the. │ │ │ │ -0000f260: 2020 2020 2020 2020 6e75 6d62 6572 206f number o │ │ │ │ -0000f270: 6620 6465 7369 7265 6420 7361 6d70 6c65 f desired sample │ │ │ │ -0000f280: 2070 6f69 6e74 730a 2020 2020 2020 2a20 points. * │ │ │ │ -0000f290: 572c 2061 202a 6e6f 7465 2077 6974 6e65 W, a *note witne │ │ │ │ -0000f2a0: 7373 2073 6574 3a20 284e 4147 7479 7065 ss set: (NAGtype │ │ │ │ -0000f2b0: 7329 5769 746e 6573 7353 6574 2c2c 2061 s)WitnessSet,, a │ │ │ │ -0000f2c0: 2077 6974 6e65 7373 2073 6574 2066 6f72 witness set for │ │ │ │ -0000f2d0: 2061 6e0a 2020 2020 2020 2020 6972 7265 an. irre │ │ │ │ -0000f2e0: 6475 6369 626c 6520 636f 6d70 6f6e 656e ducible componen │ │ │ │ -0000f2f0: 740a 2020 2a20 2a6e 6f74 6520 4f70 7469 t. * *note Opti │ │ │ │ -0000f300: 6f6e 616c 2069 6e70 7574 733a 2028 4d61 onal inputs: (Ma │ │ │ │ -0000f310: 6361 756c 6179 3244 6f63 2975 7369 6e67 caulay2Doc)using │ │ │ │ -0000f320: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ -0000f330: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ -0000f340: 3a0a 2020 2020 2020 2a20 4265 7274 696e :. * Bertin │ │ │ │ -0000f350: 6949 6e70 7574 436f 6e66 6967 7572 6174 iInputConfigurat │ │ │ │ -0000f360: 696f 6e20 286d 6973 7369 6e67 2064 6f63 ion (missing doc │ │ │ │ -0000f370: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -0000f380: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -0000f390: 650a 2020 2020 2020 2020 7b7d 2c0a 2020 e. {},. │ │ │ │ -0000f3a0: 2020 2020 2a20 2a6e 6f74 6520 4973 5072 * *note IsPr │ │ │ │ -0000f3b0: 6f6a 6563 7469 7665 3a20 4973 5072 6f6a ojective: IsProj │ │ │ │ -0000f3c0: 6563 7469 7665 2c20 3d3e 202e 2e2e 2c20 ective, => ..., │ │ │ │ -0000f3d0: 6465 6661 756c 7420 7661 6c75 6520 2d31 default value -1 │ │ │ │ -0000f3e0: 2c20 6f70 7469 6f6e 616c 0a20 2020 2020 , optional. │ │ │ │ -0000f3f0: 2020 2061 7267 756d 656e 7420 746f 2073 argument to s │ │ │ │ -0000f400: 7065 6369 6679 2077 6865 7468 6572 2074 pecify whether t │ │ │ │ -0000f410: 6f20 7573 6520 686f 6d6f 6765 6e65 6f75 o use homogeneou │ │ │ │ -0000f420: 7320 636f 6f72 6469 6e61 7465 730a 2020 s coordinates. │ │ │ │ -0000f430: 2020 2020 2a20 2a6e 6f74 6520 5665 7262 * *note Verb │ │ │ │ -0000f440: 6f73 653a 2062 6572 7469 6e69 5472 6163 ose: bertiniTrac │ │ │ │ -0000f450: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f kHomotopy_lp_pd_ │ │ │ │ -0000f460: 7064 5f70 645f 636d 5665 7262 6f73 653d pd_pd_cmVerbose= │ │ │ │ -0000f470: 3e5f 7064 5f70 645f 7064 5f72 700a 2020 >_pd_pd_pd_rp. │ │ │ │ -0000f480: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20 , => ..., │ │ │ │ -0000f490: 6465 6661 756c 7420 7661 6c75 6520 6661 default value fa │ │ │ │ -0000f4a0: 6c73 652c 204f 7074 696f 6e20 746f 2073 lse, Option to s │ │ │ │ -0000f4b0: 696c 656e 6365 2061 6464 6974 696f 6e61 ilence additiona │ │ │ │ -0000f4c0: 6c20 6f75 7470 7574 0a20 202a 204f 7574 l output. * Out │ │ │ │ -0000f4d0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ -0000f4e0: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ -0000f4f0: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -0000f500: 742c 2c20 6120 6c69 7374 206f 6620 7361 t,, a list of sa │ │ │ │ -0000f510: 6d70 6c65 2070 6f69 6e74 730a 0a44 6573 mple points..Des │ │ │ │ -0000f520: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0000f530: 3d3d 3d3d 0a0a 5361 6d70 6c65 7320 706f ====..Samples po │ │ │ │ -0000f540: 696e 7473 2066 726f 6d20 616e 2069 7272 ints from an irr │ │ │ │ -0000f550: 6564 7563 6962 6c65 2063 6f6d 706f 6e65 educible compone │ │ │ │ -0000f560: 6e74 206f 6620 6120 7661 7269 6574 7920 nt of a variety │ │ │ │ -0000f570: 7573 696e 6720 4265 7274 696e 692e 2020 using Bertini. │ │ │ │ -0000f580: 5468 650a 6972 7265 6475 6369 626c 6520 The.irreducible │ │ │ │ -0000f590: 636f 6d70 6f6e 656e 7420 6e65 6564 7320 component needs │ │ │ │ -0000f5a0: 746f 2062 6520 696e 2069 7473 206e 756d to be in its num │ │ │ │ -0000f5b0: 6572 6963 616c 2066 6f72 6d20 6173 2061 erical form as a │ │ │ │ -0000f5c0: 202a 6e6f 7465 2057 6974 6e65 7373 5365 *note WitnessSe │ │ │ │ -0000f5d0: 743a 0a28 4e41 4774 7970 6573 2957 6974 t:.(NAGtypes)Wit │ │ │ │ -0000f5e0: 6e65 7373 5365 742c 2e20 2054 6865 206d nessSet,. The m │ │ │ │ -0000f5f0: 6574 686f 6420 2a6e 6f74 6520 6265 7274 ethod *note bert │ │ │ │ -0000f600: 696e 6950 6f73 4469 6d53 6f6c 7665 3a0a iniPosDimSolve:. │ │ │ │ -0000f610: 6265 7274 696e 6950 6f73 4469 6d53 6f6c bertiniPosDimSol │ │ │ │ -0000f620: 7665 2c20 6361 6e20 6265 2075 7365 6420 ve, can be used │ │ │ │ -0000f630: 746f 2067 656e 6572 6174 6520 6120 7769 to generate a wi │ │ │ │ -0000f640: 746e 6573 7320 7365 7420 666f 7220 7468 tness set for th │ │ │ │ -0000f650: 6520 636f 6d70 6f6e 656e 742e 0a42 6572 e component..Ber │ │ │ │ -0000f660: 7469 6e69 2028 3129 2077 7269 7465 7320 tini (1) writes │ │ │ │ -0000f670: 7468 6520 7769 746e 6573 7320 7365 7420 the witness set │ │ │ │ -0000f680: 746f 2061 2074 656d 706f 7261 7279 2066 to a temporary f │ │ │ │ -0000f690: 696c 652c 2028 3229 2069 6e76 6f6b 6573 ile, (2) invokes │ │ │ │ -0000f6a0: 2042 6572 7469 6e69 2773 0a73 6f6c 7665 Bertini's.solve │ │ │ │ -0000f6b0: 7220 7769 7468 206f 7074 696f 6e20 5472 r with option Tr │ │ │ │ -0000f6c0: 6163 6b54 7970 6520 3d3e 2032 2c20 616e ackType => 2, an │ │ │ │ -0000f6d0: 6420 2833 206d 6f76 6573 2074 6865 2068 d (3 moves the h │ │ │ │ -0000f6e0: 7970 6572 706c 616e 6573 2064 6566 696e yperplanes defin │ │ │ │ -0000f6f0: 6564 2069 6e20 7468 650a 2a6e 6f74 6520 ed in the.*note │ │ │ │ -0000f700: 5769 746e 6573 7353 6574 3a20 284e 4147 WitnessSet: (NAG │ │ │ │ -0000f710: 7479 7065 7329 5769 746e 6573 7353 6574 types)WitnessSet │ │ │ │ -0000f720: 2c20 5720 7769 7468 696e 2074 6865 2073 , W within the s │ │ │ │ -0000f730: 7061 6365 2075 6e74 696c 2074 6865 2064 pace until the d │ │ │ │ -0000f740: 6573 6972 6564 0a70 6f69 6e74 7320 6172 esired.points ar │ │ │ │ -0000f750: 6520 7361 6d70 6c65 642c 2028 3429 2073 e sampled, (4) s │ │ │ │ -0000f760: 746f 7265 7320 7468 6520 6f75 7470 7574 tores the output │ │ │ │ -0000f770: 206f 6620 4265 7274 696e 6920 696e 2061 of Bertini in a │ │ │ │ -0000f780: 2074 656d 706f 7261 7279 2066 696c 652c temporary file, │ │ │ │ -0000f790: 2061 6e64 0a66 696e 616c 6c79 2028 3529 and.finally (5) │ │ │ │ -0000f7a0: 2070 6172 7365 7320 616e 6420 6f75 7470 parses and outp │ │ │ │ -0000f7b0: 7574 7320 7468 6520 736f 6c75 7469 6f6e uts the solution │ │ │ │ -0000f7c0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ +0000f1d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0000f1e0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +0000f1f0: 2020 5620 3d20 6265 7274 696e 6953 616d V = bertiniSam │ │ │ │ +0000f200: 706c 6520 286e 2c20 5729 0a20 202a 2049 ple (n, W). * I │ │ │ │ +0000f210: 6e70 7574 733a 0a20 2020 2020 202a 206e nputs:. * n │ │ │ │ +0000f220: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ +0000f230: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ +0000f240: 6329 5a5a 2c2c 2061 6e20 696e 7465 6765 c)ZZ,, an intege │ │ │ │ +0000f250: 7220 7370 6563 6966 7969 6e67 2074 6865 r specifying the │ │ │ │ +0000f260: 0a20 2020 2020 2020 206e 756d 6265 7220 . number │ │ │ │ +0000f270: 6f66 2064 6573 6972 6564 2073 616d 706c of desired sampl │ │ │ │ +0000f280: 6520 706f 696e 7473 0a20 2020 2020 202a e points. * │ │ │ │ +0000f290: 2057 2c20 6120 2a6e 6f74 6520 7769 746e W, a *note witn │ │ │ │ +0000f2a0: 6573 7320 7365 743a 2028 4e41 4774 7970 ess set: (NAGtyp │ │ │ │ +0000f2b0: 6573 2957 6974 6e65 7373 5365 742c 2c20 es)WitnessSet,, │ │ │ │ +0000f2c0: 6120 7769 746e 6573 7320 7365 7420 666f a witness set fo │ │ │ │ +0000f2d0: 7220 616e 0a20 2020 2020 2020 2069 7272 r an. irr │ │ │ │ +0000f2e0: 6564 7563 6962 6c65 2063 6f6d 706f 6e65 educible compone │ │ │ │ +0000f2f0: 6e74 0a20 202a 202a 6e6f 7465 204f 7074 nt. * *note Opt │ │ │ │ +0000f300: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ +0000f310: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ +0000f320: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ +0000f330: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ +0000f340: 2c3a 0a20 2020 2020 202a 2042 6572 7469 ,:. * Berti │ │ │ │ +0000f350: 6e69 496e 7075 7443 6f6e 6669 6775 7261 niInputConfigura │ │ │ │ +0000f360: 7469 6f6e 2028 6d69 7373 696e 6720 646f tion (missing do │ │ │ │ +0000f370: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +0000f380: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +0000f390: 7565 0a20 2020 2020 2020 207b 7d2c 0a20 ue. {},. │ │ │ │ +0000f3a0: 2020 2020 202a 202a 6e6f 7465 2049 7350 * *note IsP │ │ │ │ +0000f3b0: 726f 6a65 6374 6976 653a 2049 7350 726f rojective: IsPro │ │ │ │ +0000f3c0: 6a65 6374 6976 652c 203d 3e20 2e2e 2e2c jective, => ..., │ │ │ │ +0000f3d0: 2064 6566 6175 6c74 2076 616c 7565 202d default value - │ │ │ │ +0000f3e0: 312c 206f 7074 696f 6e61 6c0a 2020 2020 1, optional. │ │ │ │ +0000f3f0: 2020 2020 6172 6775 6d65 6e74 2074 6f20 argument to │ │ │ │ +0000f400: 7370 6563 6966 7920 7768 6574 6865 7220 specify whether │ │ │ │ +0000f410: 746f 2075 7365 2068 6f6d 6f67 656e 656f to use homogeneo │ │ │ │ +0000f420: 7573 2063 6f6f 7264 696e 6174 6573 0a20 us coordinates. │ │ │ │ +0000f430: 2020 2020 202a 202a 6e6f 7465 2056 6572 * *note Ver │ │ │ │ +0000f440: 626f 7365 3a20 6265 7274 696e 6954 7261 bose: bertiniTra │ │ │ │ +0000f450: 636b 486f 6d6f 746f 7079 5f6c 705f 7064 ckHomotopy_lp_pd │ │ │ │ +0000f460: 5f70 645f 7064 5f63 6d56 6572 626f 7365 _pd_pd_cmVerbose │ │ │ │ +0000f470: 3d3e 5f70 645f 7064 5f70 645f 7270 0a20 =>_pd_pd_pd_rp. │ │ │ │ +0000f480: 2020 2020 2020 202c 203d 3e20 2e2e 2e2c , => ..., │ │ │ │ +0000f490: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +0000f4a0: 616c 7365 2c20 4f70 7469 6f6e 2074 6f20 alse, Option to │ │ │ │ +0000f4b0: 7369 6c65 6e63 6520 6164 6469 7469 6f6e silence addition │ │ │ │ +0000f4c0: 616c 206f 7574 7075 740a 2020 2a20 4f75 al output. * Ou │ │ │ │ +0000f4d0: 7470 7574 733a 0a20 2020 2020 202a 204c tputs:. * L │ │ │ │ +0000f4e0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +0000f4f0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +0000f500: 7374 2c2c 2061 206c 6973 7420 6f66 2073 st,, a list of s │ │ │ │ +0000f510: 616d 706c 6520 706f 696e 7473 0a0a 4465 ample points..De │ │ │ │ +0000f520: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0000f530: 3d3d 3d3d 3d0a 0a53 616d 706c 6573 2070 =====..Samples p │ │ │ │ +0000f540: 6f69 6e74 7320 6672 6f6d 2061 6e20 6972 oints from an ir │ │ │ │ +0000f550: 7265 6475 6369 626c 6520 636f 6d70 6f6e reducible compon │ │ │ │ +0000f560: 656e 7420 6f66 2061 2076 6172 6965 7479 ent of a variety │ │ │ │ +0000f570: 2075 7369 6e67 2042 6572 7469 6e69 2e20 using Bertini. │ │ │ │ +0000f580: 2054 6865 0a69 7272 6564 7563 6962 6c65 The.irreducible │ │ │ │ +0000f590: 2063 6f6d 706f 6e65 6e74 206e 6565 6473 component needs │ │ │ │ +0000f5a0: 2074 6f20 6265 2069 6e20 6974 7320 6e75 to be in its nu │ │ │ │ +0000f5b0: 6d65 7269 6361 6c20 666f 726d 2061 7320 merical form as │ │ │ │ +0000f5c0: 6120 2a6e 6f74 6520 5769 746e 6573 7353 a *note WitnessS │ │ │ │ +0000f5d0: 6574 3a0a 284e 4147 7479 7065 7329 5769 et:.(NAGtypes)Wi │ │ │ │ +0000f5e0: 746e 6573 7353 6574 2c2e 2020 5468 6520 tnessSet,. The │ │ │ │ +0000f5f0: 6d65 7468 6f64 202a 6e6f 7465 2062 6572 method *note ber │ │ │ │ +0000f600: 7469 6e69 506f 7344 696d 536f 6c76 653a tiniPosDimSolve: │ │ │ │ +0000f610: 0a62 6572 7469 6e69 506f 7344 696d 536f .bertiniPosDimSo │ │ │ │ +0000f620: 6c76 652c 2063 616e 2062 6520 7573 6564 lve, can be used │ │ │ │ +0000f630: 2074 6f20 6765 6e65 7261 7465 2061 2077 to generate a w │ │ │ │ +0000f640: 6974 6e65 7373 2073 6574 2066 6f72 2074 itness set for t │ │ │ │ +0000f650: 6865 2063 6f6d 706f 6e65 6e74 2e0a 4265 he component..Be │ │ │ │ +0000f660: 7274 696e 6920 2831 2920 7772 6974 6573 rtini (1) writes │ │ │ │ +0000f670: 2074 6865 2077 6974 6e65 7373 2073 6574 the witness set │ │ │ │ +0000f680: 2074 6f20 6120 7465 6d70 6f72 6172 7920 to a temporary │ │ │ │ +0000f690: 6669 6c65 2c20 2832 2920 696e 766f 6b65 file, (2) invoke │ │ │ │ +0000f6a0: 7320 4265 7274 696e 6927 730a 736f 6c76 s Bertini's.solv │ │ │ │ +0000f6b0: 6572 2077 6974 6820 6f70 7469 6f6e 2054 er with option T │ │ │ │ +0000f6c0: 7261 636b 5479 7065 203d 3e20 322c 2061 rackType => 2, a │ │ │ │ +0000f6d0: 6e64 2028 3320 6d6f 7665 7320 7468 6520 nd (3 moves the │ │ │ │ +0000f6e0: 6879 7065 7270 6c61 6e65 7320 6465 6669 hyperplanes defi │ │ │ │ +0000f6f0: 6e65 6420 696e 2074 6865 0a2a 6e6f 7465 ned in the.*note │ │ │ │ +0000f700: 2057 6974 6e65 7373 5365 743a 2028 4e41 WitnessSet: (NA │ │ │ │ +0000f710: 4774 7970 6573 2957 6974 6e65 7373 5365 Gtypes)WitnessSe │ │ │ │ +0000f720: 742c 2057 2077 6974 6869 6e20 7468 6520 t, W within the │ │ │ │ +0000f730: 7370 6163 6520 756e 7469 6c20 7468 6520 space until the │ │ │ │ +0000f740: 6465 7369 7265 640a 706f 696e 7473 2061 desired.points a │ │ │ │ +0000f750: 7265 2073 616d 706c 6564 2c20 2834 2920 re sampled, (4) │ │ │ │ +0000f760: 7374 6f72 6573 2074 6865 206f 7574 7075 stores the outpu │ │ │ │ +0000f770: 7420 6f66 2042 6572 7469 6e69 2069 6e20 t of Bertini in │ │ │ │ +0000f780: 6120 7465 6d70 6f72 6172 7920 6669 6c65 a temporary file │ │ │ │ +0000f790: 2c20 616e 640a 6669 6e61 6c6c 7920 2835 , and.finally (5 │ │ │ │ +0000f7a0: 2920 7061 7273 6573 2061 6e64 206f 7574 ) parses and out │ │ │ │ +0000f7b0: 7075 7473 2074 6865 2073 6f6c 7574 696f puts the solutio │ │ │ │ +0000f7c0: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ns...+---------- │ │ │ │ 0000f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f810: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 4343 --+.|i1 : R = CC │ │ │ │ -0000f820: 5b78 2c79 2c7a 5d20 2020 2020 2020 2020 [x,y,z] │ │ │ │ +0000f810: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2043 ---+.|i1 : R = C │ │ │ │ +0000f820: 435b 782c 792c 7a5d 2020 2020 2020 2020 C[x,y,z] │ │ │ │ 0000f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f860: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000f860: 2020 207c 0a7c 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 │ │ │ │ 0000f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f8b0: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ +0000f8b0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ 0000f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f900: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000f900: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000f910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f950: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f |.|o1 : Polyno │ │ │ │ -0000f960: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0000f950: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ +0000f960: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ 0000f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f9a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000f9a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f9f0: 2d2d 2b0a 7c69 3220 3a20 4620 3d20 7b20 --+.|i2 : F = { │ │ │ │ -0000fa00: 2879 5e32 2b78 5e32 2b7a 5e32 2d31 292a (y^2+x^2+z^2-1)* │ │ │ │ -0000fa10: 782c 2028 795e 322b 785e 322b 7a5e 322d x, (y^2+x^2+z^2- │ │ │ │ -0000fa20: 3129 2a79 207d 2020 2020 2020 2020 2020 1)*y } │ │ │ │ +0000f9f0: 2d2d 2d2b 0a7c 6932 203a 2046 203d 207b ---+.|i2 : F = { │ │ │ │ +0000fa00: 2028 795e 322b 785e 322b 7a5e 322d 3129 (y^2+x^2+z^2-1) │ │ │ │ +0000fa10: 2a78 2c20 2879 5e32 2b78 5e32 2b7a 5e32 *x, (y^2+x^2+z^2 │ │ │ │ +0000fa20: 2d31 292a 7920 7d20 2020 2020 2020 2020 -1)*y } │ │ │ │ 0000fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fa40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fa40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fa90: 2020 7c0a 7c20 2020 2020 2020 3320 2020 |.| 3 │ │ │ │ -0000faa0: 2020 2032 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -0000fab0: 2020 3220 2020 2020 3320 2020 2020 2032 2 3 2 │ │ │ │ -0000fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000fa90: 2020 207c 0a7c 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ +0000faa0: 2020 2020 3220 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0000fab0: 2020 2032 2020 2020 2033 2020 2020 2020 2 3 │ │ │ │ +0000fac0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fae0: 2020 7c0a 7c6f 3220 3d20 7b78 2020 2b20 |.|o2 = {x + │ │ │ │ -0000faf0: 782a 7920 202b 2078 2a7a 2020 2d20 782c x*y + x*z - x, │ │ │ │ -0000fb00: 2078 2079 202b 2079 2020 2b20 792a 7a20 x y + y + y*z │ │ │ │ -0000fb10: 202d 2079 7d20 2020 2020 2020 2020 2020 - y} │ │ │ │ +0000fae0: 2020 207c 0a7c 6f32 203d 207b 7820 202b |.|o2 = {x + │ │ │ │ +0000faf0: 2078 2a79 2020 2b20 782a 7a20 202d 2078 x*y + x*z - x │ │ │ │ +0000fb00: 2c20 7820 7920 2b20 7920 202b 2079 2a7a , x y + y + y*z │ │ │ │ +0000fb10: 2020 2d20 797d 2020 2020 2020 2020 2020 - y} │ │ │ │ 0000fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fb30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb80: 2020 7c0a 7c6f 3220 3a20 4c69 7374 2020 |.|o2 : List │ │ │ │ +0000fb80: 2020 207c 0a7c 6f32 203a 204c 6973 7420 |.|o2 : List │ │ │ │ 0000fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fba0: 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000fbd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fc20: 2d2d 2b0a 7c69 3320 3a20 4e56 203d 2062 --+.|i3 : NV = b │ │ │ │ -0000fc30: 6572 7469 6e69 506f 7344 696d 536f 6c76 ertiniPosDimSolv │ │ │ │ -0000fc40: 6528 4629 2020 2020 2020 2020 2020 2020 e(F) │ │ │ │ +0000fc20: 2d2d 2d2b 0a7c 6933 203a 204e 5620 3d20 ---+.|i3 : NV = │ │ │ │ +0000fc30: 6265 7274 696e 6950 6f73 4469 6d53 6f6c bertiniPosDimSol │ │ │ │ +0000fc40: 7665 2846 2920 2020 2020 2020 2020 2020 ve(F) │ │ │ │ 0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fc70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fc70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fcc0: 2020 7c0a 7c6f 3320 3d20 4e56 2020 2020 |.|o3 = NV │ │ │ │ +0000fcc0: 2020 207c 0a7c 6f33 203d 204e 5620 2020 |.|o3 = NV │ │ │ │ 0000fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fd10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd30: 2020 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 7c0a 7c6f 3320 3a20 4e75 6d65 7269 |.|o3 : Numeri │ │ │ │ -0000fd70: 6361 6c56 6172 6965 7479 2020 2020 2020 calVariety │ │ │ │ +0000fd60: 2020 207c 0a7c 6f33 203a 204e 756d 6572 |.|o3 : Numer │ │ │ │ +0000fd70: 6963 616c 5661 7269 6574 7920 2020 2020 icalVariety │ │ │ │ 0000fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fdb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000fdb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fe00: 2d2d 2b0a 7c69 3420 3a20 5720 3d20 4e56 --+.|i4 : W = NV │ │ │ │ -0000fe10: 2331 5f30 202d 2d7a 2d61 7869 7320 2020 #1_0 --z-axis │ │ │ │ +0000fe00: 2d2d 2d2b 0a7c 6934 203a 2057 203d 204e ---+.|i4 : W = N │ │ │ │ +0000fe10: 5623 315f 3020 2d2d 7a2d 6178 6973 2020 V#1_0 --z-axis │ │ │ │ 0000fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fe50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fe50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fea0: 2020 7c0a 7c6f 3420 3d20 5720 2020 2020 |.|o4 = W │ │ │ │ +0000fea0: 2020 207c 0a7c 6f34 203d 2057 2020 2020 |.|o4 = W │ │ │ │ 0000feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fef0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000fef0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ff40: 2020 7c0a 7c6f 3420 3a20 5769 746e 6573 |.|o4 : Witnes │ │ │ │ -0000ff50: 7353 6574 2020 2020 2020 2020 2020 2020 sSet │ │ │ │ +0000ff40: 2020 207c 0a7c 6f34 203a 2057 6974 6e65 |.|o4 : Witne │ │ │ │ +0000ff50: 7373 5365 7420 2020 2020 2020 2020 2020 ssSet │ │ │ │ 0000ff60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ff90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0000ff90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0000ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ffe0: 2d2d 2b0a 7c69 3520 3a20 6265 7274 696e --+.|i5 : bertin │ │ │ │ -0000fff0: 6953 616d 706c 6528 342c 2057 2920 2020 iSample(4, W) │ │ │ │ +0000ffe0: 2d2d 2d2b 0a7c 6935 203a 2062 6572 7469 ---+.|i5 : berti │ │ │ │ +0000fff0: 6e69 5361 6d70 6c65 2834 2c20 5729 2020 niSample(4, W) │ │ │ │ 00010000: 2020 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 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00010030: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00010040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010080: 2020 7c0a 7c6f 3520 3d20 7b7b 2d33 2e39 |.|o5 = {{-3.9 │ │ │ │ -00010090: 3838 3935 652d 3231 2d32 2e36 3639 3033 8895e-21-2.66903 │ │ │ │ -000100a0: 652d 3231 2a69 692c 202d 312e 3039 3034 e-21*ii, -1.0904 │ │ │ │ -000100b0: 3965 2d32 302b 322e 3339 3035 3765 2d32 9e-20+2.39057e-2 │ │ │ │ -000100c0: 312a 6969 2c20 2020 2020 2020 2020 2020 1*ii, │ │ │ │ -000100d0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00010080: 2020 207c 0a7c 6f35 203d 207b 7b2d 332e |.|o5 = {{-3. │ │ │ │ +00010090: 3938 3839 3565 2d32 312d 322e 3636 3930 98895e-21-2.6690 │ │ │ │ +000100a0: 3365 2d32 312a 6969 2c20 2d31 2e30 3930 3e-21*ii, -1.090 │ │ │ │ +000100b0: 3439 652d 3230 2b32 2e33 3930 3537 652d 49e-20+2.39057e- │ │ │ │ +000100c0: 3231 2a69 692c 2020 2020 2020 2020 2020 21*ii, │ │ │ │ +000100d0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 000100e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010120: 2d2d 7c0a 7c20 2020 2020 2e33 3335 3439 --|.| .33549 │ │ │ │ -00010130: 372b 2e30 3837 3736 3734 2a69 697d 2c20 7+.0877674*ii}, │ │ │ │ -00010140: 7b2d 322e 3338 3835 3165 2d32 312b 372e {-2.38851e-21+7. │ │ │ │ -00010150: 3433 3036 3665 2d32 312a 6969 2c20 2020 43066e-21*ii, │ │ │ │ +00010120: 2d2d 2d7c 0a7c 2020 2020 202e 3333 3534 ---|.| .3354 │ │ │ │ +00010130: 3937 2b2e 3038 3737 3637 342a 6969 7d2c 97+.0877674*ii}, │ │ │ │ +00010140: 207b 2d32 2e33 3838 3531 652d 3231 2b37 {-2.38851e-21+7 │ │ │ │ +00010150: 2e34 3330 3636 652d 3231 2a69 692c 2020 .43066e-21*ii, │ │ │ │ 00010160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010170: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00010170: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00010180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000101a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000101b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000101c0: 2d2d 7c0a 7c20 2020 2020 372e 3034 3838 --|.| 7.0488 │ │ │ │ -000101d0: 3565 2d32 312d 332e 3331 3433 3565 2d32 5e-21-3.31435e-2 │ │ │ │ -000101e0: 312a 6969 2c20 2e32 3335 3136 342b 2e30 1*ii, .235164+.0 │ │ │ │ -000101f0: 3637 3933 3136 2a69 697d 2c20 2020 2020 679316*ii}, │ │ │ │ +000101c0: 2d2d 2d7c 0a7c 2020 2020 2037 2e30 3438 ---|.| 7.048 │ │ │ │ +000101d0: 3835 652d 3231 2d33 2e33 3134 3335 652d 85e-21-3.31435e- │ │ │ │ +000101e0: 3231 2a69 692c 202e 3233 3531 3634 2b2e 21*ii, .235164+. │ │ │ │ +000101f0: 3036 3739 3331 362a 6969 7d2c 2020 2020 0679316*ii}, │ │ │ │ 00010200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010210: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00010210: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00010220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010260: 2d2d 7c0a 7c20 2020 2020 7b2d 372e 3136 --|.| {-7.16 │ │ │ │ -00010270: 3431 3865 2d32 312b 332e 3430 3739 3365 418e-21+3.40793e │ │ │ │ -00010280: 2d32 312a 6969 2c20 382e 3638 3636 3565 -21*ii, 8.68665e │ │ │ │ -00010290: 2d32 312b 312e 3335 3435 3465 2d32 302a -21+1.35454e-20* │ │ │ │ -000102a0: 6969 2c20 2020 2020 2020 2020 2020 2020 ii, │ │ │ │ -000102b0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00010260: 2d2d 2d7c 0a7c 2020 2020 207b 2d37 2e31 ---|.| {-7.1 │ │ │ │ +00010270: 3634 3138 652d 3231 2b33 2e34 3037 3933 6418e-21+3.40793 │ │ │ │ +00010280: 652d 3231 2a69 692c 2038 2e36 3836 3635 e-21*ii, 8.68665 │ │ │ │ +00010290: 652d 3231 2b31 2e33 3534 3534 652d 3230 e-21+1.35454e-20 │ │ │ │ +000102a0: 2a69 692c 2020 2020 2020 2020 2020 2020 *ii, │ │ │ │ +000102b0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 000102c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010300: 2d2d 7c0a 7c20 2020 2020 2e33 3736 3339 --|.| .37639 │ │ │ │ -00010310: 362b 2e31 3630 3138 382a 6969 7d2c 207b 6+.160188*ii}, { │ │ │ │ -00010320: 372e 3234 3538 3365 2d32 312b 322e 3232 7.24583e-21+2.22 │ │ │ │ -00010330: 3631 3665 2d32 312a 6969 2c20 2020 2020 616e-21*ii, │ │ │ │ +00010300: 2d2d 2d7c 0a7c 2020 2020 202e 3337 3633 ---|.| .3763 │ │ │ │ +00010310: 3936 2b2e 3136 3031 3838 2a69 697d 2c20 96+.160188*ii}, │ │ │ │ +00010320: 7b37 2e32 3435 3833 652d 3231 2b32 2e32 {7.24583e-21+2.2 │ │ │ │ +00010330: 3236 3136 652d 3231 2a69 692c 2020 2020 2616e-21*ii, │ │ │ │ 00010340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010350: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00010350: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00010360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000103a0: 2d2d 7c0a 7c20 2020 2020 352e 3234 3038 --|.| 5.2408 │ │ │ │ -000103b0: 3365 2d32 312b 352e 3830 3839 3965 2d32 3e-21+5.80899e-2 │ │ │ │ -000103c0: 312a 6969 2c20 2e33 3330 3038 332b 2e32 1*ii, .330083+.2 │ │ │ │ -000103d0: 3635 3632 342a 6969 7d7d 2020 2020 2020 65624*ii}} │ │ │ │ +000103a0: 2d2d 2d7c 0a7c 2020 2020 2035 2e32 3430 ---|.| 5.240 │ │ │ │ +000103b0: 3833 652d 3231 2b35 2e38 3038 3939 652d 83e-21+5.80899e- │ │ │ │ +000103c0: 3231 2a69 692c 202e 3333 3030 3833 2b2e 21*ii, .330083+. │ │ │ │ +000103d0: 3236 3536 3234 2a69 697d 7d20 2020 2020 265624*ii}} │ │ │ │ 000103e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000103f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000103f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00010400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010440: 2020 7c0a 7c6f 3520 3a20 4c69 7374 2020 |.|o5 : List │ │ │ │ +00010440: 2020 207c 0a7c 6f35 203a 204c 6973 7420 |.|o5 : List │ │ │ │ 00010450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010490: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00010490: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000104a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000104b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000104c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000104d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000104e0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -000104f0: 2062 6572 7469 6e69 5361 6d70 6c65 3a0a bertiniSample:. │ │ │ │ -00010500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00010510: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00010520: 2262 6572 7469 6e69 5361 6d70 6c65 285a "bertiniSample(Z │ │ │ │ -00010530: 5a2c 5769 746e 6573 7353 6574 2922 0a0a Z,WitnessSet)".. │ │ │ │ -00010540: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00010550: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00010560: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00010570: 7420 2a6e 6f74 6520 6265 7274 696e 6953 t *note bertiniS │ │ │ │ -00010580: 616d 706c 653a 2062 6572 7469 6e69 5361 ample: bertiniSa │ │ │ │ -00010590: 6d70 6c65 2c20 6973 2061 202a 6e6f 7465 mple, is a *note │ │ │ │ -000105a0: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -000105b0: 2077 6974 680a 6f70 7469 6f6e 733a 2028 with.options: ( │ │ │ │ -000105c0: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -000105d0: 686f 6446 756e 6374 696f 6e57 6974 684f hodFunctionWithO │ │ │ │ -000105e0: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d ptions,...------ │ │ │ │ +000104e0: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573 ---+..Ways to us │ │ │ │ +000104f0: 6520 6265 7274 696e 6953 616d 706c 653a e bertiniSample: │ │ │ │ +00010500: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00010510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00010520: 2022 6265 7274 696e 6953 616d 706c 6528 "bertiniSample( │ │ │ │ +00010530: 5a5a 2c57 6974 6e65 7373 5365 7429 220a ZZ,WitnessSet)". │ │ │ │ +00010540: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00010550: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00010560: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00010570: 6374 202a 6e6f 7465 2062 6572 7469 6e69 ct *note bertini │ │ │ │ +00010580: 5361 6d70 6c65 3a20 6265 7274 696e 6953 Sample: bertiniS │ │ │ │ +00010590: 616d 706c 652c 2069 7320 6120 2a6e 6f74 ample, is a *not │ │ │ │ +000105a0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +000105b0: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ +000105c0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +000105d0: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +000105e0: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ 000105f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010630: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00010640: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00010650: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00010660: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00010670: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00010680: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ -00010690: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -000106a0: 2f42 6572 7469 6e69 2e6d 323a 0a32 3739 /Bertini.m2:.279 │ │ │ │ -000106b0: 303a 302e 0a1f 0a46 696c 653a 2042 6572 0:0....File: Ber │ │ │ │ -000106c0: 7469 6e69 2e69 6e66 6f2c 204e 6f64 653a tini.info, Node: │ │ │ │ -000106d0: 2062 6572 7469 6e69 5472 6163 6b48 6f6d bertiniTrackHom │ │ │ │ -000106e0: 6f74 6f70 792c 204e 6578 743a 2062 6572 otopy, Next: ber │ │ │ │ -000106f0: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70 tiniTrackHomotop │ │ │ │ -00010700: 795f 6c70 5f70 645f 7064 5f70 645f 636d y_lp_pd_pd_pd_cm │ │ │ │ -00010710: 5665 7262 6f73 653d 3e5f 7064 5f70 645f Verbose=>_pd_pd_ │ │ │ │ -00010720: 7064 5f72 702c 2050 7265 763a 2062 6572 pd_rp, Prev: ber │ │ │ │ -00010730: 7469 6e69 5361 6d70 6c65 2c20 5570 3a20 tiniSample, Up: │ │ │ │ -00010740: 546f 700a 0a62 6572 7469 6e69 5472 6163 Top..bertiniTrac │ │ │ │ -00010750: 6b48 6f6d 6f74 6f70 7920 2d2d 2061 206d kHomotopy -- a m │ │ │ │ -00010760: 6169 6e20 6d65 7468 6f64 2074 6f20 7472 ain method to tr │ │ │ │ -00010770: 6163 6b20 7573 696e 6720 6120 7573 6572 ack using a user │ │ │ │ -00010780: 2d64 6566 696e 6564 2068 6f6d 6f74 6f70 -defined homotop │ │ │ │ -00010790: 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a y.************** │ │ │ │ +00010630: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00010640: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00010650: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00010660: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00010670: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00010680: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ +00010690: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +000106a0: 732f 4265 7274 696e 692e 6d32 3a0a 3237 s/Bertini.m2:.27 │ │ │ │ +000106b0: 3930 3a30 2e0a 1f0a 4669 6c65 3a20 4265 90:0....File: Be │ │ │ │ +000106c0: 7274 696e 692e 696e 666f 2c20 4e6f 6465 rtini.info, Node │ │ │ │ +000106d0: 3a20 6265 7274 696e 6954 7261 636b 486f : bertiniTrackHo │ │ │ │ +000106e0: 6d6f 746f 7079 2c20 4e65 7874 3a20 6265 motopy, Next: be │ │ │ │ +000106f0: 7274 696e 6954 7261 636b 486f 6d6f 746f rtiniTrackHomoto │ │ │ │ +00010700: 7079 5f6c 705f 7064 5f70 645f 7064 5f63 py_lp_pd_pd_pd_c │ │ │ │ +00010710: 6d56 6572 626f 7365 3d3e 5f70 645f 7064 mVerbose=>_pd_pd │ │ │ │ +00010720: 5f70 645f 7270 2c20 5072 6576 3a20 6265 _pd_rp, Prev: be │ │ │ │ +00010730: 7274 696e 6953 616d 706c 652c 2055 703a rtiniSample, Up: │ │ │ │ +00010740: 2054 6f70 0a0a 6265 7274 696e 6954 7261 Top..bertiniTra │ │ │ │ +00010750: 636b 486f 6d6f 746f 7079 202d 2d20 6120 ckHomotopy -- a │ │ │ │ +00010760: 6d61 696e 206d 6574 686f 6420 746f 2074 main method to t │ │ │ │ +00010770: 7261 636b 2075 7369 6e67 2061 2075 7365 rack using a use │ │ │ │ +00010780: 722d 6465 6669 6e65 6420 686f 6d6f 746f r-defined homoto │ │ │ │ +00010790: 7079 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a py.************* │ │ │ │ 000107a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000107b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000107c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000107d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -000107e0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -000107f0: 2020 2020 5330 3d62 6572 7469 6e69 5472 S0=bertiniTr │ │ │ │ -00010800: 6163 6b48 6f6d 6f74 6f70 7928 742c 2048 ackHomotopy(t, H │ │ │ │ -00010810: 2c20 5331 290a 2020 2a20 496e 7075 7473 , S1). * Inputs │ │ │ │ -00010820: 3a0a 2020 2020 2020 2a20 742c 2061 202a :. * t, a * │ │ │ │ -00010830: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ -00010840: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -00010850: 2952 696e 6745 6c65 6d65 6e74 2c2c 2061 )RingElement,, a │ │ │ │ -00010860: 2070 6174 6820 7661 7269 6162 6c65 0a20 path variable. │ │ │ │ -00010870: 2020 2020 202a 2048 2c20 6120 2a6e 6f74 * H, a *not │ │ │ │ -00010880: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -00010890: 7932 446f 6329 4c69 7374 2c2c 2061 206c y2Doc)List,, a l │ │ │ │ -000108a0: 6973 7420 706f 6c79 6e6f 6d69 616c 7320 ist polynomials │ │ │ │ -000108b0: 7468 6174 2064 6566 696e 650a 2020 2020 that define. │ │ │ │ -000108c0: 2020 2020 7468 6520 686f 6d6f 746f 7079 the homotopy │ │ │ │ -000108d0: 2077 6974 6820 7265 7370 6563 7420 746f with respect to │ │ │ │ -000108e0: 2074 6865 2070 6174 6820 7661 7269 6162 the path variab │ │ │ │ -000108f0: 6c65 0a20 2020 2020 202a 2053 312c 2061 le. * S1, a │ │ │ │ -00010900: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ -00010910: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ -00010920: 2c20 6120 6c69 7374 206f 6620 736f 6c75 , a list of solu │ │ │ │ -00010930: 7469 6f6e 7320 746f 2074 6865 2073 7461 tions to the sta │ │ │ │ -00010940: 7274 0a20 2020 2020 2020 2073 7973 7465 rt. syste │ │ │ │ -00010950: 6d0a 2020 2a20 2a6e 6f74 6520 4f70 7469 m. * *note Opti │ │ │ │ -00010960: 6f6e 616c 2069 6e70 7574 733a 2028 4d61 onal inputs: (Ma │ │ │ │ -00010970: 6361 756c 6179 3244 6f63 2975 7369 6e67 caulay2Doc)using │ │ │ │ -00010980: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ -00010990: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ -000109a0: 3a0a 2020 2020 2020 2a20 4265 7274 696e :. * Bertin │ │ │ │ -000109b0: 6949 6e70 7574 436f 6e66 6967 7572 6174 iInputConfigurat │ │ │ │ -000109c0: 696f 6e20 286d 6973 7369 6e67 2064 6f63 ion (missing doc │ │ │ │ -000109d0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -000109e0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -000109f0: 650a 2020 2020 2020 2020 7b7d 2c0a 2020 e. {},. │ │ │ │ -00010a00: 2020 2020 2a20 2a6e 6f74 6520 4973 5072 * *note IsPr │ │ │ │ -00010a10: 6f6a 6563 7469 7665 3a20 4973 5072 6f6a ojective: IsProj │ │ │ │ -00010a20: 6563 7469 7665 2c20 3d3e 202e 2e2e 2c20 ective, => ..., │ │ │ │ -00010a30: 6465 6661 756c 7420 7661 6c75 6520 2d31 default value -1 │ │ │ │ -00010a40: 2c20 6f70 7469 6f6e 616c 0a20 2020 2020 , optional. │ │ │ │ -00010a50: 2020 2061 7267 756d 656e 7420 746f 2073 argument to s │ │ │ │ -00010a60: 7065 6369 6679 2077 6865 7468 6572 2074 pecify whether t │ │ │ │ -00010a70: 6f20 7573 6520 686f 6d6f 6765 6e65 6f75 o use homogeneou │ │ │ │ -00010a80: 7320 636f 6f72 6469 6e61 7465 730a 2020 s coordinates. │ │ │ │ -00010a90: 2020 2020 2a20 2a6e 6f74 6520 5665 7262 * *note Verb │ │ │ │ -00010aa0: 6f73 653a 2062 6572 7469 6e69 5472 6163 ose: bertiniTrac │ │ │ │ -00010ab0: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f kHomotopy_lp_pd_ │ │ │ │ -00010ac0: 7064 5f70 645f 636d 5665 7262 6f73 653d pd_pd_cmVerbose= │ │ │ │ -00010ad0: 3e5f 7064 5f70 645f 7064 5f72 700a 2020 >_pd_pd_pd_rp. │ │ │ │ -00010ae0: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20 , => ..., │ │ │ │ -00010af0: 6465 6661 756c 7420 7661 6c75 6520 6661 default value fa │ │ │ │ -00010b00: 6c73 652c 204f 7074 696f 6e20 746f 2073 lse, Option to s │ │ │ │ -00010b10: 696c 656e 6365 2061 6464 6974 696f 6e61 ilence additiona │ │ │ │ -00010b20: 6c20 6f75 7470 7574 0a20 202a 204f 7574 l output. * Out │ │ │ │ -00010b30: 7075 7473 3a0a 2020 2020 2020 2a20 5330 puts:. * S0 │ │ │ │ -00010b40: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00010b50: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00010b60: 7374 2c2c 2061 206c 6973 7420 6f66 2073 st,, a list of s │ │ │ │ -00010b70: 6f6c 7574 696f 6e73 2074 6f20 7468 650a olutions to the. │ │ │ │ -00010b80: 2020 2020 2020 2020 7461 7267 6574 2073 target s │ │ │ │ -00010b90: 7973 7465 6d0a 0a44 6573 6372 6970 7469 ystem..Descripti │ │ │ │ -00010ba0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00010bb0: 5468 6973 206d 6574 686f 6420 6361 6c6c This method call │ │ │ │ -00010bc0: 7320 4265 7274 696e 6920 746f 2074 7261 s Bertini to tra │ │ │ │ -00010bd0: 636b 2061 2075 7365 722d 6465 6669 6e65 ck a user-define │ │ │ │ -00010be0: 6420 686f 6d6f 746f 7079 2e20 2054 6865 d homotopy. The │ │ │ │ -00010bf0: 2075 7365 7220 6e65 6564 7320 746f 0a73 user needs to.s │ │ │ │ -00010c00: 7065 6369 6679 2074 6865 2068 6f6d 6f74 pecify the homot │ │ │ │ -00010c10: 6f70 7920 482c 2074 6865 2070 6174 6820 opy H, the path │ │ │ │ -00010c20: 7661 7269 6162 6c65 2074 2c20 616e 6420 variable t, and │ │ │ │ -00010c30: 6120 6c69 7374 206f 6620 7374 6172 7420 a list of start │ │ │ │ -00010c40: 736f 6c75 7469 6f6e 7320 5331 2e0a 4265 solutions S1..Be │ │ │ │ -00010c50: 7274 696e 6920 2831 2920 7772 6974 6573 rtini (1) writes │ │ │ │ -00010c60: 2074 6865 2068 6f6d 6f74 6f70 7920 616e the homotopy an │ │ │ │ -00010c70: 6420 7374 6172 7420 736f 6c75 7469 6f6e d start solution │ │ │ │ -00010c80: 7320 746f 2074 656d 706f 7261 7279 2066 s to temporary f │ │ │ │ -00010c90: 696c 6573 2c20 2832 290a 696e 766f 6b65 iles, (2).invoke │ │ │ │ -00010ca0: 7320 4265 7274 696e 6927 7320 736f 6c76 s Bertini's solv │ │ │ │ -00010cb0: 6572 2077 6974 6820 636f 6e66 6967 7572 er with configur │ │ │ │ -00010cc0: 6174 696f 6e20 6b65 7977 6f72 6420 5573 ation keyword Us │ │ │ │ -00010cd0: 6572 486f 6d6f 746f 7079 203d 3e20 3120 erHomotopy => 1 │ │ │ │ -00010ce0: 696e 2074 6865 0a61 6666 696e 6520 6361 in the.affine ca │ │ │ │ -00010cf0: 7365 2061 6e64 2055 7365 7248 6f6d 6f74 se and UserHomot │ │ │ │ -00010d00: 6f70 7920 3d3e 2032 2069 6e20 7468 6520 opy => 2 in the │ │ │ │ -00010d10: 7072 6f6a 6563 7469 7665 2073 6974 7561 projective situa │ │ │ │ -00010d20: 7469 6f6e 2c20 2833 2920 7374 6f72 6573 tion, (3) stores │ │ │ │ -00010d30: 2074 6865 0a6f 7574 7075 7420 6f66 2042 the.output of B │ │ │ │ -00010d40: 6572 7469 6e69 2069 6e20 6120 7465 6d70 ertini in a temp │ │ │ │ -00010d50: 6f72 6172 7920 6669 6c65 2c20 616e 6420 orary file, and │ │ │ │ -00010d60: 2834 2920 7061 7273 6573 2061 206d 6163 (4) parses a mac │ │ │ │ -00010d70: 6869 6e65 2072 6561 6461 626c 6520 6669 hine readable fi │ │ │ │ -00010d80: 6c65 0a74 6f20 6f75 7470 7574 2061 206c le.to output a l │ │ │ │ -00010d90: 6973 7420 6f66 2073 6f6c 7574 696f 6e73 ist of solutions │ │ │ │ -00010da0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +000107d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +000107e0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000107f0: 2020 2020 2053 303d 6265 7274 696e 6954 S0=bertiniT │ │ │ │ +00010800: 7261 636b 486f 6d6f 746f 7079 2874 2c20 rackHomotopy(t, │ │ │ │ +00010810: 482c 2053 3129 0a20 202a 2049 6e70 7574 H, S1). * Input │ │ │ │ +00010820: 733a 0a20 2020 2020 202a 2074 2c20 6120 s:. * t, a │ │ │ │ +00010830: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ +00010840: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ +00010850: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ +00010860: 6120 7061 7468 2076 6172 6961 626c 650a a path variable. │ │ │ │ +00010870: 2020 2020 2020 2a20 482c 2061 202a 6e6f * H, a *no │ │ │ │ +00010880: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ +00010890: 6179 3244 6f63 294c 6973 742c 2c20 6120 ay2Doc)List,, a │ │ │ │ +000108a0: 6c69 7374 2070 6f6c 796e 6f6d 6961 6c73 list polynomials │ │ │ │ +000108b0: 2074 6861 7420 6465 6669 6e65 0a20 2020 that define. │ │ │ │ +000108c0: 2020 2020 2074 6865 2068 6f6d 6f74 6f70 the homotop │ │ │ │ +000108d0: 7920 7769 7468 2072 6573 7065 6374 2074 y with respect t │ │ │ │ +000108e0: 6f20 7468 6520 7061 7468 2076 6172 6961 o the path varia │ │ │ │ +000108f0: 626c 650a 2020 2020 2020 2a20 5331 2c20 ble. * S1, │ │ │ │ +00010900: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ +00010910: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ +00010920: 2c2c 2061 206c 6973 7420 6f66 2073 6f6c ,, a list of sol │ │ │ │ +00010930: 7574 696f 6e73 2074 6f20 7468 6520 7374 utions to the st │ │ │ │ +00010940: 6172 740a 2020 2020 2020 2020 7379 7374 art. syst │ │ │ │ +00010950: 656d 0a20 202a 202a 6e6f 7465 204f 7074 em. * *note Opt │ │ │ │ +00010960: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ +00010970: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ +00010980: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ +00010990: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ +000109a0: 2c3a 0a20 2020 2020 202a 2042 6572 7469 ,:. * Berti │ │ │ │ +000109b0: 6e69 496e 7075 7443 6f6e 6669 6775 7261 niInputConfigura │ │ │ │ +000109c0: 7469 6f6e 2028 6d69 7373 696e 6720 646f tion (missing do │ │ │ │ +000109d0: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +000109e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +000109f0: 7565 0a20 2020 2020 2020 207b 7d2c 0a20 ue. {},. │ │ │ │ +00010a00: 2020 2020 202a 202a 6e6f 7465 2049 7350 * *note IsP │ │ │ │ +00010a10: 726f 6a65 6374 6976 653a 2049 7350 726f rojective: IsPro │ │ │ │ +00010a20: 6a65 6374 6976 652c 203d 3e20 2e2e 2e2c jective, => ..., │ │ │ │ +00010a30: 2064 6566 6175 6c74 2076 616c 7565 202d default value - │ │ │ │ +00010a40: 312c 206f 7074 696f 6e61 6c0a 2020 2020 1, optional. │ │ │ │ +00010a50: 2020 2020 6172 6775 6d65 6e74 2074 6f20 argument to │ │ │ │ +00010a60: 7370 6563 6966 7920 7768 6574 6865 7220 specify whether │ │ │ │ +00010a70: 746f 2075 7365 2068 6f6d 6f67 656e 656f to use homogeneo │ │ │ │ +00010a80: 7573 2063 6f6f 7264 696e 6174 6573 0a20 us coordinates. │ │ │ │ +00010a90: 2020 2020 202a 202a 6e6f 7465 2056 6572 * *note Ver │ │ │ │ +00010aa0: 626f 7365 3a20 6265 7274 696e 6954 7261 bose: bertiniTra │ │ │ │ +00010ab0: 636b 486f 6d6f 746f 7079 5f6c 705f 7064 ckHomotopy_lp_pd │ │ │ │ +00010ac0: 5f70 645f 7064 5f63 6d56 6572 626f 7365 _pd_pd_cmVerbose │ │ │ │ +00010ad0: 3d3e 5f70 645f 7064 5f70 645f 7270 0a20 =>_pd_pd_pd_rp. │ │ │ │ +00010ae0: 2020 2020 2020 202c 203d 3e20 2e2e 2e2c , => ..., │ │ │ │ +00010af0: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +00010b00: 616c 7365 2c20 4f70 7469 6f6e 2074 6f20 alse, Option to │ │ │ │ +00010b10: 7369 6c65 6e63 6520 6164 6469 7469 6f6e silence addition │ │ │ │ +00010b20: 616c 206f 7574 7075 740a 2020 2a20 4f75 al output. * Ou │ │ │ │ +00010b30: 7470 7574 733a 0a20 2020 2020 202a 2053 tputs:. * S │ │ │ │ +00010b40: 302c 2061 202a 6e6f 7465 206c 6973 743a 0, a *note list: │ │ │ │ +00010b50: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ +00010b60: 6973 742c 2c20 6120 6c69 7374 206f 6620 ist,, a list of │ │ │ │ +00010b70: 736f 6c75 7469 6f6e 7320 746f 2074 6865 solutions to the │ │ │ │ +00010b80: 0a20 2020 2020 2020 2074 6172 6765 7420 . target │ │ │ │ +00010b90: 7379 7374 656d 0a0a 4465 7363 7269 7074 system..Descript │ │ │ │ +00010ba0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00010bb0: 0a54 6869 7320 6d65 7468 6f64 2063 616c .This method cal │ │ │ │ +00010bc0: 6c73 2042 6572 7469 6e69 2074 6f20 7472 ls Bertini to tr │ │ │ │ +00010bd0: 6163 6b20 6120 7573 6572 2d64 6566 696e ack a user-defin │ │ │ │ +00010be0: 6564 2068 6f6d 6f74 6f70 792e 2020 5468 ed homotopy. Th │ │ │ │ +00010bf0: 6520 7573 6572 206e 6565 6473 2074 6f0a e user needs to. │ │ │ │ +00010c00: 7370 6563 6966 7920 7468 6520 686f 6d6f specify the homo │ │ │ │ +00010c10: 746f 7079 2048 2c20 7468 6520 7061 7468 topy H, the path │ │ │ │ +00010c20: 2076 6172 6961 626c 6520 742c 2061 6e64 variable t, and │ │ │ │ +00010c30: 2061 206c 6973 7420 6f66 2073 7461 7274 a list of start │ │ │ │ +00010c40: 2073 6f6c 7574 696f 6e73 2053 312e 0a42 solutions S1..B │ │ │ │ +00010c50: 6572 7469 6e69 2028 3129 2077 7269 7465 ertini (1) write │ │ │ │ +00010c60: 7320 7468 6520 686f 6d6f 746f 7079 2061 s the homotopy a │ │ │ │ +00010c70: 6e64 2073 7461 7274 2073 6f6c 7574 696f nd start solutio │ │ │ │ +00010c80: 6e73 2074 6f20 7465 6d70 6f72 6172 7920 ns to temporary │ │ │ │ +00010c90: 6669 6c65 732c 2028 3229 0a69 6e76 6f6b files, (2).invok │ │ │ │ +00010ca0: 6573 2042 6572 7469 6e69 2773 2073 6f6c es Bertini's sol │ │ │ │ +00010cb0: 7665 7220 7769 7468 2063 6f6e 6669 6775 ver with configu │ │ │ │ +00010cc0: 7261 7469 6f6e 206b 6579 776f 7264 2055 ration keyword U │ │ │ │ +00010cd0: 7365 7248 6f6d 6f74 6f70 7920 3d3e 2031 serHomotopy => 1 │ │ │ │ +00010ce0: 2069 6e20 7468 650a 6166 6669 6e65 2063 in the.affine c │ │ │ │ +00010cf0: 6173 6520 616e 6420 5573 6572 486f 6d6f ase and UserHomo │ │ │ │ +00010d00: 746f 7079 203d 3e20 3220 696e 2074 6865 topy => 2 in the │ │ │ │ +00010d10: 2070 726f 6a65 6374 6976 6520 7369 7475 projective situ │ │ │ │ +00010d20: 6174 696f 6e2c 2028 3329 2073 746f 7265 ation, (3) store │ │ │ │ +00010d30: 7320 7468 650a 6f75 7470 7574 206f 6620 s the.output of │ │ │ │ +00010d40: 4265 7274 696e 6920 696e 2061 2074 656d Bertini in a tem │ │ │ │ +00010d50: 706f 7261 7279 2066 696c 652c 2061 6e64 porary file, and │ │ │ │ +00010d60: 2028 3429 2070 6172 7365 7320 6120 6d61 (4) parses a ma │ │ │ │ +00010d70: 6368 696e 6520 7265 6164 6162 6c65 2066 chine readable f │ │ │ │ +00010d80: 696c 650a 746f 206f 7574 7075 7420 6120 ile.to output a │ │ │ │ +00010d90: 6c69 7374 206f 6620 736f 6c75 7469 6f6e list of solution │ │ │ │ +00010da0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ 00010db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010de0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 -------+.|i1 : R │ │ │ │ -00010df0: 203d 2043 435b 782c 745d 3b20 2d2d 2069 = CC[x,t]; -- i │ │ │ │ -00010e00: 6e63 6c75 6465 2074 6865 2070 6174 6820 nclude the path │ │ │ │ -00010e10: 7661 7269 6162 6c65 2069 6e20 7468 6520 variable in the │ │ │ │ -00010e20: 7269 6e67 2020 2020 2020 2020 207c 0a2b ring |.+ │ │ │ │ -00010e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010de0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00010df0: 5220 3d20 4343 5b78 2c74 5d3b 202d 2d20 R = CC[x,t]; -- │ │ │ │ +00010e00: 696e 636c 7564 6520 7468 6520 7061 7468 include the path │ │ │ │ +00010e10: 2076 6172 6961 626c 6520 696e 2074 6865 variable in the │ │ │ │ +00010e20: 2072 696e 6720 2020 2020 2020 2020 7c0a ring |. │ │ │ │ +00010e30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00010e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010e70: 2d2d 2d2b 0a7c 6932 203a 2048 203d 207b ---+.|i2 : H = { │ │ │ │ -00010e80: 2028 785e 322d 3129 2a74 202b 2028 785e (x^2-1)*t + (x^ │ │ │ │ -00010e90: 322d 3229 2a28 312d 7429 7d3b 2020 2020 2-2)*(1-t)}; │ │ │ │ +00010e70: 2d2d 2d2d 2b0a 7c69 3220 3a20 4820 3d20 ----+.|i2 : H = │ │ │ │ +00010e80: 7b20 2878 5e32 2d31 292a 7420 2b20 2878 { (x^2-1)*t + (x │ │ │ │ +00010e90: 5e32 2d32 292a 2831 2d74 297d 3b20 2020 ^2-2)*(1-t)}; │ │ │ │ 00010ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010eb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00010eb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00010ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00010f00: 0a7c 6933 203a 2073 6f6c 3120 3d20 706f .|i3 : sol1 = po │ │ │ │ -00010f10: 696e 7420 7b7b 317d 7d3b 2020 2020 2020 int {{1}}; │ │ │ │ +00010ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010f00: 2b0a 7c69 3320 3a20 736f 6c31 203d 2070 +.|i3 : sol1 = p │ │ │ │ +00010f10: 6f69 6e74 207b 7b31 7d7d 3b20 2020 2020 oint {{1}}; │ │ │ │ 00010f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010f40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00010f40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -00010f90: 203a 2073 6f6c 3220 3d20 706f 696e 7420 : sol2 = point │ │ │ │ -00010fa0: 7b7b 2d31 7d7d 3b20 2020 2020 2020 2020 {{-1}}; │ │ │ │ +00010f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00010f90: 3420 3a20 736f 6c32 203d 2070 6f69 6e74 4 : sol2 = point │ │ │ │ +00010fa0: 207b 7b2d 317d 7d3b 2020 2020 2020 2020 {{-1}}; │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010fd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00010fd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00010fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011010: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2053 -------+.|i5 : S │ │ │ │ -00011020: 313d 207b 2073 6f6c 312c 2073 6f6c 3220 1= { sol1, sol2 │ │ │ │ -00011030: 207d 3b2d 2d73 6f6c 7574 696f 6e73 2074 };--solutions t │ │ │ │ -00011040: 6f20 4820 7768 656e 2074 3d31 2020 2020 o H when t=1 │ │ │ │ -00011050: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00011060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011010: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ +00011020: 5331 3d20 7b20 736f 6c31 2c20 736f 6c32 S1= { sol1, sol2 │ │ │ │ +00011030: 2020 7d3b 2d2d 736f 6c75 7469 6f6e 7320 };--solutions │ │ │ │ +00011040: 746f 2048 2077 6865 6e20 743d 3120 2020 to H when t=1 │ │ │ │ +00011050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00011060: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00011070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000110a0: 2d2d 2d2b 0a7c 6936 203a 2053 3020 3d20 ---+.|i6 : S0 = │ │ │ │ -000110b0: 6265 7274 696e 6954 7261 636b 486f 6d6f bertiniTrackHomo │ │ │ │ -000110c0: 746f 7079 2028 742c 2048 2c20 5331 2920 topy (t, H, S1) │ │ │ │ -000110d0: 2d2d 736f 6c75 7469 6f6e 7320 746f 2048 --solutions to H │ │ │ │ -000110e0: 2077 6865 6e20 743d 307c 0a7c 2020 2020 when t=0|.| │ │ │ │ +000110a0: 2d2d 2d2d 2b0a 7c69 3620 3a20 5330 203d ----+.|i6 : S0 = │ │ │ │ +000110b0: 2062 6572 7469 6e69 5472 6163 6b48 6f6d bertiniTrackHom │ │ │ │ +000110c0: 6f74 6f70 7920 2874 2c20 482c 2053 3129 otopy (t, H, S1) │ │ │ │ +000110d0: 202d 2d73 6f6c 7574 696f 6e73 2074 6f20 --solutions to │ │ │ │ +000110e0: 4820 7768 656e 2074 3d30 7c0a 7c20 2020 H when t=0|.| │ │ │ │ 000110f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011130: 0a7c 6f36 203d 207b 7b2d 312e 3431 3432 .|o6 = {{-1.4142 │ │ │ │ -00011140: 317d 2c20 7b31 2e34 3134 3231 7d7d 2020 1}, {1.41421}} │ │ │ │ +00011120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011130: 7c0a 7c6f 3620 3d20 7b7b 2d31 2e34 3134 |.|o6 = {{-1.414 │ │ │ │ +00011140: 3231 7d2c 207b 312e 3431 3432 317d 7d20 21}, {1.41421}} │ │ │ │ 00011150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011170: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00011170: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00011180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000111a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000111b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -000111c0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +000111b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000111c0: 3620 3a20 4c69 7374 2020 2020 2020 2020 6 : List │ │ │ │ 000111d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000111e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000111f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011200: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00011200: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00011210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011240: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 -------+.|i7 : p │ │ │ │ -00011250: 6565 6b20 5330 5f30 2020 2020 2020 2020 eek S0_0 │ │ │ │ +00011240: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ +00011250: 7065 656b 2053 305f 3020 2020 2020 2020 peek S0_0 │ │ │ │ 00011260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00011290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00011290: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000112a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000112b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000112c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000112d0: 2020 207c 0a7c 6f37 203d 2050 6f69 6e74 |.|o7 = Point │ │ │ │ -000112e0: 7b63 6163 6865 203d 3e20 4361 6368 6554 {cache => CacheT │ │ │ │ -000112f0: 6162 6c65 7b2e 2e2e 392e 2e2e 7d7d 2020 able{...9...}} │ │ │ │ +000112d0: 2020 2020 7c0a 7c6f 3720 3d20 506f 696e |.|o7 = Poin │ │ │ │ +000112e0: 747b 6361 6368 6520 3d3e 2043 6163 6865 t{cache => Cache │ │ │ │ +000112f0: 5461 626c 657b 2e2e 2e39 2e2e 2e7d 7d20 Table{...9...}} │ │ │ │ 00011300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011310: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011320: 2020 2020 2020 2043 6f6f 7264 696e 6174 Coordinat │ │ │ │ -00011330: 6573 203d 3e20 7b2d 312e 3431 3432 317d es => {-1.41421} │ │ │ │ -00011340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00011310: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00011320: 2020 2020 2020 2020 436f 6f72 6469 6e61 Coordina │ │ │ │ +00011330: 7465 7320 3d3e 207b 2d31 2e34 3134 3231 tes => {-1.41421 │ │ │ │ +00011340: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00011350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00011370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000113a0: 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 2070 -----+..In the p │ │ │ │ -000113b0: 7265 7669 6f75 7320 6578 616d 706c 652c revious example, │ │ │ │ -000113c0: 2077 6520 736f 6c76 6564 2024 785e 322d we solved $x^2- │ │ │ │ -000113d0: 3224 2062 7920 6d6f 7669 6e67 2066 726f 2$ by moving fro │ │ │ │ -000113e0: 6d20 2478 5e32 2d31 2420 7769 7468 2061 m $x^2-1$ with a │ │ │ │ -000113f0: 206c 696e 6561 720a 686f 6d6f 746f 7079 linear.homotopy │ │ │ │ -00011400: 2e20 4265 7274 696e 6920 7472 6163 6b73 . Bertini tracks │ │ │ │ -00011410: 2068 6f6d 6f74 6f70 6965 7320 7374 6172 homotopies star │ │ │ │ -00011420: 7469 6e67 2061 7420 2474 3d31 2420 616e ting at $t=1$ an │ │ │ │ -00011430: 6420 656e 6469 6e67 2061 7420 2474 3d30 d ending at $t=0 │ │ │ │ -00011440: 242e 0a46 696e 616c 2073 6f6c 7574 696f $..Final solutio │ │ │ │ -00011450: 6e73 2061 7265 206f 6620 7468 6520 7479 ns are of the ty │ │ │ │ -00011460: 7065 2050 6f69 6e74 2e0a 0a2b 2d2d 2d2d pe Point...+---- │ │ │ │ +000113a0: 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 ------+..In the │ │ │ │ +000113b0: 7072 6576 696f 7573 2065 7861 6d70 6c65 previous example │ │ │ │ +000113c0: 2c20 7765 2073 6f6c 7665 6420 2478 5e32 , we solved $x^2 │ │ │ │ +000113d0: 2d32 2420 6279 206d 6f76 696e 6720 6672 -2$ by moving fr │ │ │ │ +000113e0: 6f6d 2024 785e 322d 3124 2077 6974 6820 om $x^2-1$ with │ │ │ │ +000113f0: 6120 6c69 6e65 6172 0a68 6f6d 6f74 6f70 a linear.homotop │ │ │ │ +00011400: 792e 2042 6572 7469 6e69 2074 7261 636b y. Bertini track │ │ │ │ +00011410: 7320 686f 6d6f 746f 7069 6573 2073 7461 s homotopies sta │ │ │ │ +00011420: 7274 696e 6720 6174 2024 743d 3124 2061 rting at $t=1$ a │ │ │ │ +00011430: 6e64 2065 6e64 696e 6720 6174 2024 743d nd ending at $t= │ │ │ │ +00011440: 3024 2e0a 4669 6e61 6c20 736f 6c75 7469 0$..Final soluti │ │ │ │ +00011450: 6f6e 7320 6172 6520 6f66 2074 6865 2074 ons are of the t │ │ │ │ +00011460: 7970 6520 506f 696e 742e 0a0a 2b2d 2d2d ype Point...+--- │ │ │ │ 00011470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000114a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000114b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ -000114c0: 2052 3d43 435b 782c 792c 745d 3b20 2d2d R=CC[x,y,t]; -- │ │ │ │ -000114d0: 2069 6e63 6c75 6465 2074 6865 2070 6174 include the pat │ │ │ │ -000114e0: 6820 7661 7269 6162 6c65 2069 6e20 7468 h variable in th │ │ │ │ -000114f0: 6520 7269 6e67 2020 2020 2020 2020 2020 e ring │ │ │ │ -00011500: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000114b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ +000114c0: 3a20 523d 4343 5b78 2c79 2c74 5d3b 202d : R=CC[x,y,t]; - │ │ │ │ +000114d0: 2d20 696e 636c 7564 6520 7468 6520 7061 - include the pa │ │ │ │ +000114e0: 7468 2076 6172 6961 626c 6520 696e 2074 th variable in t │ │ │ │ +000114f0: 6865 2072 696e 6720 2020 2020 2020 2020 he ring │ │ │ │ +00011500: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00011510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011550: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ -00011560: 2066 313d 2878 5e32 2d79 5e32 293b 2020 f1=(x^2-y^2); │ │ │ │ +00011550: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +00011560: 3a20 6631 3d28 785e 322d 795e 3229 3b20 : f1=(x^2-y^2); │ │ │ │ 00011570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000115a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000115a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 000115b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000115c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000115d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000115e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ -00011600: 3a20 6632 3d28 322a 785e 322d 332a 782a : f2=(2*x^2-3*x* │ │ │ │ -00011610: 792b 352a 795e 3229 3b20 2020 2020 2020 y+5*y^2); │ │ │ │ +000115f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ +00011600: 203a 2066 323d 2832 2a78 5e32 2d33 2a78 : f2=(2*x^2-3*x │ │ │ │ +00011610: 2a79 2b35 2a79 5e32 293b 2020 2020 2020 *y+5*y^2); │ │ │ │ 00011620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011640: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011640: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00011650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011690: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -000116a0: 3a20 4820 3d20 7b20 6631 2a74 202b 2066 : H = { f1*t + f │ │ │ │ -000116b0: 322a 2831 2d74 297d 3b20 2d2d 4820 6973 2*(1-t)}; --H is │ │ │ │ -000116c0: 2061 206c 6973 7420 6f66 2070 6f6c 796e a list of polyn │ │ │ │ -000116d0: 6f6d 6961 6c73 2069 6e20 782c 792c 7420 omials in x,y,t │ │ │ │ -000116e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011690: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ +000116a0: 203a 2048 203d 207b 2066 312a 7420 2b20 : H = { f1*t + │ │ │ │ +000116b0: 6632 2a28 312d 7429 7d3b 202d 2d48 2069 f2*(1-t)}; --H i │ │ │ │ +000116c0: 7320 6120 6c69 7374 206f 6620 706f 6c79 s a list of poly │ │ │ │ +000116d0: 6e6f 6d69 616c 7320 696e 2078 2c79 2c74 nomials in x,y,t │ │ │ │ +000116e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 000116f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ -00011740: 3a20 736f 6c31 3d20 2020 2070 6f69 6e74 : sol1= point │ │ │ │ -00011750: 7b7b 312c 317d 7d2d 2d7b 7b78 2c79 7d7d {{1,1}}--{{x,y}} │ │ │ │ -00011760: 2063 6f6f 7264 696e 6174 6573 2020 2020 coordinates │ │ │ │ +00011730: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ +00011740: 203a 2073 6f6c 313d 2020 2020 706f 696e : sol1= poin │ │ │ │ +00011750: 747b 7b31 2c31 7d7d 2d2d 7b7b 782c 797d t{{1,1}}--{{x,y} │ │ │ │ +00011760: 7d20 636f 6f72 6469 6e61 7465 7320 2020 } coordinates │ │ │ │ 00011770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011780: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011780: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00011790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000117a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000117b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000117c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000117d0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -000117e0: 3d20 736f 6c31 2020 2020 2020 2020 2020 = sol1 │ │ │ │ +000117d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ +000117e0: 203d 2073 6f6c 3120 2020 2020 2020 2020 = sol1 │ │ │ │ 000117f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011820: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011820: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00011830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011870: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00011880: 3a20 506f 696e 7420 2020 2020 2020 2020 : Point │ │ │ │ +00011870: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ +00011880: 203a 2050 6f69 6e74 2020 2020 2020 2020 : Point │ │ │ │ 00011890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000118c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000118c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 000118d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000118e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000118f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011910: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ -00011920: 3a20 736f 6c32 3d20 2020 2070 6f69 6e74 : sol2= point │ │ │ │ -00011930: 7b7b 202d 312c 317d 7d20 2020 2020 2020 {{ -1,1}} │ │ │ │ +00011910: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +00011920: 203a 2073 6f6c 323d 2020 2020 706f 696e : sol2= poin │ │ │ │ +00011930: 747b 7b20 2d31 2c31 7d7d 2020 2020 2020 t{{ -1,1}} │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011960: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011960: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00011970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000119a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000119b0: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ -000119c0: 3d20 736f 6c32 2020 2020 2020 2020 2020 = sol2 │ │ │ │ +000119b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +000119c0: 203d 2073 6f6c 3220 2020 2020 2020 2020 = sol2 │ │ │ │ 000119d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000119e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000119f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011a00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00011a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011a50: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ -00011a60: 3a20 506f 696e 7420 2020 2020 2020 2020 : Point │ │ │ │ +00011a50: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +00011a60: 203a 2050 6f69 6e74 2020 2020 2020 2020 : Point │ │ │ │ 00011a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011aa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011aa0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00011ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011af0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ -00011b00: 3a20 5331 3d7b 736f 6c31 2c73 6f6c 327d : S1={sol1,sol2} │ │ │ │ -00011b10: 2d2d 736f 6c75 7469 6f6e 7320 746f 2048 --solutions to H │ │ │ │ -00011b20: 2077 6865 6e20 743d 3120 2020 2020 2020 when t=1 │ │ │ │ +00011af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ +00011b00: 203a 2053 313d 7b73 6f6c 312c 736f 6c32 : S1={sol1,sol2 │ │ │ │ +00011b10: 7d2d 2d73 6f6c 7574 696f 6e73 2074 6f20 }--solutions to │ │ │ │ +00011b20: 4820 7768 656e 2074 3d31 2020 2020 2020 H when t=1 │ │ │ │ 00011b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00011b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b90: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ -00011ba0: 3d20 7b73 6f6c 312c 2073 6f6c 327d 2020 = {sol1, sol2} │ │ │ │ +00011b90: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +00011ba0: 203d 207b 736f 6c31 2c20 736f 6c32 7d20 = {sol1, sol2} │ │ │ │ 00011bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011be0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011be0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 │ │ │ │ -00011c30: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ -00011c40: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00011c30: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +00011c40: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 00011c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011c80: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011c80: 2020 2020 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011cd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 ---------+.|i15 │ │ │ │ -00011ce0: 3a20 5330 3d62 6572 7469 6e69 5472 6163 : S0=bertiniTrac │ │ │ │ -00011cf0: 6b48 6f6d 6f74 6f70 7928 742c 2048 2c20 kHomotopy(t, H, │ │ │ │ -00011d00: 5331 2c20 4973 5072 6f6a 6563 7469 7665 S1, IsProjective │ │ │ │ -00011d10: 3d3e 3129 202d 2d73 6f6c 7574 696f 6e73 =>1) --solutions │ │ │ │ -00011d20: 2074 6f20 4820 7768 657c 0a7c 2020 2020 to H whe|.| │ │ │ │ +00011cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 ----------+.|i15 │ │ │ │ +00011ce0: 203a 2053 303d 6265 7274 696e 6954 7261 : S0=bertiniTra │ │ │ │ +00011cf0: 636b 486f 6d6f 746f 7079 2874 2c20 482c ckHomotopy(t, H, │ │ │ │ +00011d00: 2053 312c 2049 7350 726f 6a65 6374 6976 S1, IsProjectiv │ │ │ │ +00011d10: 653d 3e31 2920 2d2d 736f 6c75 7469 6f6e e=>1) --solution │ │ │ │ +00011d20: 7320 746f 2048 2077 6865 7c0a 7c20 2020 s to H whe|.| │ │ │ │ 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 2020 2020 2020 2020 2020 │ │ │ │ -00011d70: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ -00011d80: 3d20 7b7b 2d2e 3438 3231 3134 2d2e 3634 = {{-.482114-.64 │ │ │ │ -00011d90: 3630 3039 2a69 692c 202e 3231 3530 3438 6009*ii, .215048 │ │ │ │ -00011da0: 2d2e 3436 3232 3333 2a69 697d 2c20 7b34 -.462233*ii}, {4 │ │ │ │ -00011db0: 2e31 3339 382d 342e 3732 3533 2a69 692c .1398-4.7253*ii, │ │ │ │ -00011dc0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011dd0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +00011d70: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ +00011d80: 203d 207b 7b2d 2e34 3832 3131 342d 2e36 = {{-.482114-.6 │ │ │ │ +00011d90: 3436 3030 392a 6969 2c20 2e32 3135 3034 46009*ii, .21504 │ │ │ │ +00011da0: 382d 2e34 3632 3233 332a 6969 7d2c 207b 8-.462233*ii}, { │ │ │ │ +00011db0: 342e 3133 3938 2d34 2e37 3235 332a 6969 4.1398-4.7253*ii │ │ │ │ +00011dc0: 2c20 2020 2020 2020 2020 7c0a 7c20 2020 , |.| │ │ │ │ +00011dd0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ 00011de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011e10: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ -00011e20: 2020 2d31 2e33 3839 2d33 2e37 3232 3533 -1.389-3.72253 │ │ │ │ -00011e30: 2a69 697d 7d20 2020 2020 2020 2020 2020 *ii}} │ │ │ │ +00011e10: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00011e20: 2020 202d 312e 3338 392d 332e 3732 3235 -1.389-3.7225 │ │ │ │ +00011e30: 332a 6969 7d7d 2020 2020 2020 2020 2020 3*ii}} │ │ │ │ 00011e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011e60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011e60: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 │ │ │ │ -00011eb0: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ -00011ec0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00011eb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ +00011ec0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 00011ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011f00: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ +00011f00: 2020 2020 2020 2020 2020 7c0a 7c2d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011f50: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6e20 743d ---------|.|n t= │ │ │ │ -00011f60: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00011f50: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e 2074 ----------|.|n t │ │ │ │ +00011f60: 3d30 2020 2020 2020 2020 2020 2020 2020 =0 │ │ │ │ 00011f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011fa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011fa0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011ff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5661 7269 ---------+..Vari │ │ │ │ -00012000: 6162 6c65 7320 6d75 7374 2062 6567 696e ables must begin │ │ │ │ -00012010: 2077 6974 6820 6120 6c65 7474 6572 2028 with a letter ( │ │ │ │ -00012020: 6c6f 7765 7263 6173 6520 6f72 2063 6170 lowercase or cap │ │ │ │ -00012030: 6974 616c 2920 616e 6420 6361 6e20 6f6e ital) and can on │ │ │ │ -00012040: 6c79 2063 6f6e 7461 696e 0a6c 6574 7465 ly contain.lette │ │ │ │ -00012050: 7273 2c20 6e75 6d62 6572 732c 2075 6e64 rs, numbers, und │ │ │ │ -00012060: 6572 7363 6f72 6573 2c20 616e 6420 7371 erscores, and sq │ │ │ │ -00012070: 7561 7265 2062 7261 636b 6574 732e 0a0a uare brackets... │ │ │ │ -00012080: 5761 7973 2074 6f20 7573 6520 6265 7274 Ways to use bert │ │ │ │ -00012090: 696e 6954 7261 636b 486f 6d6f 746f 7079 iniTrackHomotopy │ │ │ │ -000120a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00011ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a56 6172 ----------+..Var │ │ │ │ +00012000: 6961 626c 6573 206d 7573 7420 6265 6769 iables must begi │ │ │ │ +00012010: 6e20 7769 7468 2061 206c 6574 7465 7220 n with a letter │ │ │ │ +00012020: 286c 6f77 6572 6361 7365 206f 7220 6361 (lowercase or ca │ │ │ │ +00012030: 7069 7461 6c29 2061 6e64 2063 616e 206f pital) and can o │ │ │ │ +00012040: 6e6c 7920 636f 6e74 6169 6e0a 6c65 7474 nly contain.lett │ │ │ │ +00012050: 6572 732c 206e 756d 6265 7273 2c20 756e ers, numbers, un │ │ │ │ +00012060: 6465 7273 636f 7265 732c 2061 6e64 2073 derscores, and s │ │ │ │ +00012070: 7175 6172 6520 6272 6163 6b65 7473 2e0a quare brackets.. │ │ │ │ +00012080: 0a57 6179 7320 746f 2075 7365 2062 6572 .Ways to use ber │ │ │ │ +00012090: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70 tiniTrackHomotop │ │ │ │ +000120a0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d y:.============= │ │ │ │ 000120b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000120c0: 3d3d 3d0a 0a20 202a 2022 6265 7274 696e ===.. * "bertin │ │ │ │ -000120d0: 6954 7261 636b 486f 6d6f 746f 7079 2852 iTrackHomotopy(R │ │ │ │ -000120e0: 696e 6745 6c65 6d65 6e74 2c4c 6973 742c ingElement,List, │ │ │ │ -000120f0: 4c69 7374 2922 0a0a 466f 7220 7468 6520 List)"..For the │ │ │ │ -00012100: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00012110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00012120: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00012130: 6265 7274 696e 6954 7261 636b 486f 6d6f bertiniTrackHomo │ │ │ │ -00012140: 746f 7079 3a20 6265 7274 696e 6954 7261 topy: bertiniTra │ │ │ │ -00012150: 636b 486f 6d6f 746f 7079 2c20 6973 2061 ckHomotopy, is a │ │ │ │ -00012160: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00012170: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ -00012180: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -00012190: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -000121a0: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ -000121b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000120c0: 3d3d 3d3d 0a0a 2020 2a20 2262 6572 7469 ====.. * "berti │ │ │ │ +000120d0: 6e69 5472 6163 6b48 6f6d 6f74 6f70 7928 niTrackHomotopy( │ │ │ │ +000120e0: 5269 6e67 456c 656d 656e 742c 4c69 7374 RingElement,List │ │ │ │ +000120f0: 2c4c 6973 7429 220a 0a46 6f72 2074 6865 ,List)"..For the │ │ │ │ +00012100: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +00012110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00012120: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +00012130: 2062 6572 7469 6e69 5472 6163 6b48 6f6d bertiniTrackHom │ │ │ │ +00012140: 6f74 6f70 793a 2062 6572 7469 6e69 5472 otopy: bertiniTr │ │ │ │ +00012150: 6163 6b48 6f6d 6f74 6f70 792c 2069 7320 ackHomotopy, is │ │ │ │ +00012160: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00012170: 756e 6374 696f 6e20 7769 7468 206f 7074 unction with opt │ │ │ │ +00012180: 696f 6e73 3a20 284d 6163 6175 6c61 7932 ions: (Macaulay2 │ │ │ │ +00012190: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +000121a0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ +000121b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 000121c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000121d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000121e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00012200: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00012210: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00012220: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00012230: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00012240: 756c 6179 322d 312e 3235 2e30 362b 6473 ulay2-1.25.06+ds │ │ │ │ -00012250: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00012260: 636b 6167 6573 2f42 6572 7469 6e69 2e6d ckages/Bertini.m │ │ │ │ -00012270: 323a 0a32 3834 353a 302e 0a1f 0a46 696c 2:.2845:0....Fil │ │ │ │ -00012280: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c e: Bertini.info, │ │ │ │ -00012290: 204e 6f64 653a 2062 6572 7469 6e69 5472 Node: bertiniTr │ │ │ │ -000122a0: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70 ackHomotopy_lp_p │ │ │ │ -000122b0: 645f 7064 5f70 645f 636d 5665 7262 6f73 d_pd_pd_cmVerbos │ │ │ │ -000122c0: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ -000122d0: 204e 6578 743a 2062 6572 7469 6e69 5573 Next: bertiniUs │ │ │ │ -000122e0: 6572 486f 6d6f 746f 7079 2c20 5072 6576 erHomotopy, Prev │ │ │ │ -000122f0: 3a20 6265 7274 696e 6954 7261 636b 486f : bertiniTrackHo │ │ │ │ -00012300: 6d6f 746f 7079 2c20 5570 3a20 546f 700a motopy, Up: Top. │ │ │ │ -00012310: 0a62 6572 7469 6e69 5472 6163 6b48 6f6d .bertiniTrackHom │ │ │ │ -00012320: 6f74 6f70 7928 2e2e 2e2c 5665 7262 6f73 otopy(...,Verbos │ │ │ │ -00012330: 653d 3e2e 2e2e 2920 2d2d 204f 7074 696f e=>...) -- Optio │ │ │ │ -00012340: 6e20 746f 2073 696c 656e 6365 2061 6464 n to silence add │ │ │ │ -00012350: 6974 696f 6e61 6c20 6f75 7470 7574 0a2a itional output.* │ │ │ │ +000121f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012200: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00012210: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00012220: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00012230: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00012240: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +00012250: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00012260: 6163 6b61 6765 732f 4265 7274 696e 692e ackages/Bertini. │ │ │ │ +00012270: 6d32 3a0a 3238 3435 3a30 2e0a 1f0a 4669 m2:.2845:0....Fi │ │ │ │ +00012280: 6c65 3a20 4265 7274 696e 692e 696e 666f le: Bertini.info │ │ │ │ +00012290: 2c20 4e6f 6465 3a20 6265 7274 696e 6954 , Node: bertiniT │ │ │ │ +000122a0: 7261 636b 486f 6d6f 746f 7079 5f6c 705f rackHomotopy_lp_ │ │ │ │ +000122b0: 7064 5f70 645f 7064 5f63 6d56 6572 626f pd_pd_pd_cmVerbo │ │ │ │ +000122c0: 7365 3d3e 5f70 645f 7064 5f70 645f 7270 se=>_pd_pd_pd_rp │ │ │ │ +000122d0: 2c20 4e65 7874 3a20 6265 7274 696e 6955 , Next: bertiniU │ │ │ │ +000122e0: 7365 7248 6f6d 6f74 6f70 792c 2050 7265 serHomotopy, Pre │ │ │ │ +000122f0: 763a 2062 6572 7469 6e69 5472 6163 6b48 v: bertiniTrackH │ │ │ │ +00012300: 6f6d 6f74 6f70 792c 2055 703a 2054 6f70 omotopy, Up: Top │ │ │ │ +00012310: 0a0a 6265 7274 696e 6954 7261 636b 486f ..bertiniTrackHo │ │ │ │ +00012320: 6d6f 746f 7079 282e 2e2e 2c56 6572 626f motopy(...,Verbo │ │ │ │ +00012330: 7365 3d3e 2e2e 2e29 202d 2d20 4f70 7469 se=>...) -- Opti │ │ │ │ +00012340: 6f6e 2074 6f20 7369 6c65 6e63 6520 6164 on to silence ad │ │ │ │ +00012350: 6469 7469 6f6e 616c 206f 7574 7075 740a ditional output. │ │ │ │ 00012360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00012370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00012380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00012390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000123a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -000123b0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -000123c0: 2020 6265 7274 696e 6954 7261 636b 486f bertiniTrackHo │ │ │ │ -000123d0: 6d6f 746f 7079 5665 7262 6f73 6528 2e2e motopyVerbose(.. │ │ │ │ -000123e0: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65 .,Verbose=>Boole │ │ │ │ -000123f0: 616e 290a 2020 2020 2020 2020 6265 7274 an). bert │ │ │ │ -00012400: 696e 6955 7365 7248 6f6d 6f74 6f70 7956 iniUserHomotopyV │ │ │ │ -00012410: 6572 626f 7365 282e 2e2e 2c56 6572 626f erbose(...,Verbo │ │ │ │ -00012420: 7365 3d3e 426f 6f6c 6561 6e29 0a20 2020 se=>Boolean). │ │ │ │ -00012430: 2020 2020 2062 6572 7469 6e69 506f 7344 bertiniPosD │ │ │ │ -00012440: 696d 536f 6c76 6528 2e2e 2e2c 5665 7262 imSolve(...,Verb │ │ │ │ -00012450: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020 ose=>Boolean). │ │ │ │ -00012460: 2020 2020 2020 6265 7274 696e 6952 6566 bertiniRef │ │ │ │ -00012470: 696e 6553 6f6c 7328 2e2e 2e2c 5665 7262 ineSols(...,Verb │ │ │ │ -00012480: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020 ose=>Boolean). │ │ │ │ -00012490: 2020 2020 2020 6265 7274 696e 6953 616d bertiniSam │ │ │ │ -000124a0: 706c 6528 2e2e 2e2c 5665 7262 6f73 653d ple(...,Verbose= │ │ │ │ -000124b0: 3e42 6f6f 6c65 616e 290a 2020 2020 2020 >Boolean). │ │ │ │ -000124c0: 2020 6265 7274 696e 695a 6572 6f44 696d bertiniZeroDim │ │ │ │ -000124d0: 536f 6c76 6528 2e2e 2e2c 5665 7262 6f73 Solve(...,Verbos │ │ │ │ -000124e0: 653d 3e42 6f6f 6c65 616e 290a 2020 2020 e=>Boolean). │ │ │ │ -000124f0: 2020 2020 6265 7274 696e 6950 6172 616d bertiniParam │ │ │ │ -00012500: 6574 6572 486f 6d6f 746f 7079 282e 2e2e eterHomotopy(... │ │ │ │ -00012510: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561 ,Verbose=>Boolea │ │ │ │ -00012520: 6e29 0a20 2020 2020 2020 206d 616b 6542 n). makeB │ │ │ │ -00012530: 2749 6e70 7574 4669 6c65 282e 2e2e 2c56 'InputFile(...,V │ │ │ │ -00012540: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29 erbose=>Boolean) │ │ │ │ -00012550: 0a20 2020 2020 2020 206d 616b 654d 656d . makeMem │ │ │ │ -00012560: 6265 7273 6869 7046 696c 6528 2e2e 2e2c bershipFile(..., │ │ │ │ -00012570: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e Verbose=>Boolean │ │ │ │ -00012580: 290a 2020 2020 2020 2020 6227 5048 4761 ). b'PHGa │ │ │ │ -00012590: 6c6f 6973 4772 6f75 7028 2e2e 2e2c 5665 loisGroup(...,Ve │ │ │ │ -000125a0: 7262 6f73 653d 3e42 6f6f 6c65 616e 290a rbose=>Boolean). │ │ │ │ -000125b0: 2020 2020 2020 2020 6227 5048 4d6f 6e6f b'PHMono │ │ │ │ -000125c0: 6472 6f6d 7943 6f6c 6c65 6374 282e 2e2e dromyCollect(... │ │ │ │ -000125d0: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561 ,Verbose=>Boolea │ │ │ │ -000125e0: 6e29 0a20 2020 2020 2020 2069 6d70 6f72 n). impor │ │ │ │ -000125f0: 7449 6e63 6964 656e 6365 4d61 7472 6978 tIncidenceMatrix │ │ │ │ -00012600: 282e 2e2e 2c56 6572 626f 7365 3d3e 426f (...,Verbose=>Bo │ │ │ │ -00012610: 6f6c 6561 6e29 0a20 2020 2020 2020 2069 olean). i │ │ │ │ -00012620: 6d70 6f72 744d 6169 6e44 6174 6146 696c mportMainDataFil │ │ │ │ -00012630: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e42 e(...,Verbose=>B │ │ │ │ -00012640: 6f6f 6c65 616e 290a 2020 2020 2020 2020 oolean). │ │ │ │ -00012650: 696d 706f 7274 536c 6963 6546 696c 6528 importSliceFile( │ │ │ │ -00012660: 2e2e 2e2c 5665 7262 6f73 653d 3e42 6f6f ...,Verbose=>Boo │ │ │ │ -00012670: 6c65 616e 290a 2020 2020 2020 2020 696d lean). im │ │ │ │ -00012680: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ -00012690: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e42 e(...,Verbose=>B │ │ │ │ -000126a0: 6f6f 6c65 616e 290a 2020 2020 2020 2020 oolean). │ │ │ │ -000126b0: 7275 6e42 6572 7469 6e69 282e 2e2e 2c56 runBertini(...,V │ │ │ │ -000126c0: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29 erbose=>Boolean) │ │ │ │ -000126d0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -000126e0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 2056 =========..Use V │ │ │ │ -000126f0: 6572 626f 7365 3d3e 6661 6c73 6520 746f erbose=>false to │ │ │ │ -00012700: 2073 696c 656e 6365 2061 6464 6974 696f silence additio │ │ │ │ -00012710: 6e61 6c20 6f75 7470 7574 2e0a 0a46 756e nal output...Fun │ │ │ │ -00012720: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -00012730: 6f6e 616c 2061 7267 756d 656e 7420 6e61 onal argument na │ │ │ │ -00012740: 6d65 6420 5665 7262 6f73 653a 0a3d 3d3d med Verbose:.=== │ │ │ │ +000123a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +000123b0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +000123c0: 2020 2062 6572 7469 6e69 5472 6163 6b48 bertiniTrackH │ │ │ │ +000123d0: 6f6d 6f74 6f70 7956 6572 626f 7365 282e omotopyVerbose(. │ │ │ │ +000123e0: 2e2e 2c56 6572 626f 7365 3d3e 426f 6f6c ..,Verbose=>Bool │ │ │ │ +000123f0: 6561 6e29 0a20 2020 2020 2020 2062 6572 ean). ber │ │ │ │ +00012400: 7469 6e69 5573 6572 486f 6d6f 746f 7079 tiniUserHomotopy │ │ │ │ +00012410: 5665 7262 6f73 6528 2e2e 2e2c 5665 7262 Verbose(...,Verb │ │ │ │ +00012420: 6f73 653d 3e42 6f6f 6c65 616e 290a 2020 ose=>Boolean). │ │ │ │ +00012430: 2020 2020 2020 6265 7274 696e 6950 6f73 bertiniPos │ │ │ │ +00012440: 4469 6d53 6f6c 7665 282e 2e2e 2c56 6572 DimSolve(...,Ver │ │ │ │ +00012450: 626f 7365 3d3e 426f 6f6c 6561 6e29 0a20 bose=>Boolean). │ │ │ │ +00012460: 2020 2020 2020 2062 6572 7469 6e69 5265 bertiniRe │ │ │ │ +00012470: 6669 6e65 536f 6c73 282e 2e2e 2c56 6572 fineSols(...,Ver │ │ │ │ +00012480: 626f 7365 3d3e 426f 6f6c 6561 6e29 0a20 bose=>Boolean). │ │ │ │ +00012490: 2020 2020 2020 2062 6572 7469 6e69 5361 bertiniSa │ │ │ │ +000124a0: 6d70 6c65 282e 2e2e 2c56 6572 626f 7365 mple(...,Verbose │ │ │ │ +000124b0: 3d3e 426f 6f6c 6561 6e29 0a20 2020 2020 =>Boolean). │ │ │ │ +000124c0: 2020 2062 6572 7469 6e69 5a65 726f 4469 bertiniZeroDi │ │ │ │ +000124d0: 6d53 6f6c 7665 282e 2e2e 2c56 6572 626f mSolve(...,Verbo │ │ │ │ +000124e0: 7365 3d3e 426f 6f6c 6561 6e29 0a20 2020 se=>Boolean). │ │ │ │ +000124f0: 2020 2020 2062 6572 7469 6e69 5061 7261 bertiniPara │ │ │ │ +00012500: 6d65 7465 7248 6f6d 6f74 6f70 7928 2e2e meterHomotopy(.. │ │ │ │ +00012510: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65 .,Verbose=>Boole │ │ │ │ +00012520: 616e 290a 2020 2020 2020 2020 6d61 6b65 an). make │ │ │ │ +00012530: 4227 496e 7075 7446 696c 6528 2e2e 2e2c B'InputFile(..., │ │ │ │ +00012540: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e Verbose=>Boolean │ │ │ │ +00012550: 290a 2020 2020 2020 2020 6d61 6b65 4d65 ). makeMe │ │ │ │ +00012560: 6d62 6572 7368 6970 4669 6c65 282e 2e2e mbershipFile(... │ │ │ │ +00012570: 2c56 6572 626f 7365 3d3e 426f 6f6c 6561 ,Verbose=>Boolea │ │ │ │ +00012580: 6e29 0a20 2020 2020 2020 2062 2750 4847 n). b'PHG │ │ │ │ +00012590: 616c 6f69 7347 726f 7570 282e 2e2e 2c56 aloisGroup(...,V │ │ │ │ +000125a0: 6572 626f 7365 3d3e 426f 6f6c 6561 6e29 erbose=>Boolean) │ │ │ │ +000125b0: 0a20 2020 2020 2020 2062 2750 484d 6f6e . b'PHMon │ │ │ │ +000125c0: 6f64 726f 6d79 436f 6c6c 6563 7428 2e2e odromyCollect(.. │ │ │ │ +000125d0: 2e2c 5665 7262 6f73 653d 3e42 6f6f 6c65 .,Verbose=>Boole │ │ │ │ +000125e0: 616e 290a 2020 2020 2020 2020 696d 706f an). impo │ │ │ │ +000125f0: 7274 496e 6369 6465 6e63 654d 6174 7269 rtIncidenceMatri │ │ │ │ +00012600: 7828 2e2e 2e2c 5665 7262 6f73 653d 3e42 x(...,Verbose=>B │ │ │ │ +00012610: 6f6f 6c65 616e 290a 2020 2020 2020 2020 oolean). │ │ │ │ +00012620: 696d 706f 7274 4d61 696e 4461 7461 4669 importMainDataFi │ │ │ │ +00012630: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e le(...,Verbose=> │ │ │ │ +00012640: 426f 6f6c 6561 6e29 0a20 2020 2020 2020 Boolean). │ │ │ │ +00012650: 2069 6d70 6f72 7453 6c69 6365 4669 6c65 importSliceFile │ │ │ │ +00012660: 282e 2e2e 2c56 6572 626f 7365 3d3e 426f (...,Verbose=>Bo │ │ │ │ +00012670: 6f6c 6561 6e29 0a20 2020 2020 2020 2069 olean). i │ │ │ │ +00012680: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669 mportSolutionsFi │ │ │ │ +00012690: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e le(...,Verbose=> │ │ │ │ +000126a0: 426f 6f6c 6561 6e29 0a20 2020 2020 2020 Boolean). │ │ │ │ +000126b0: 2072 756e 4265 7274 696e 6928 2e2e 2e2c runBertini(..., │ │ │ │ +000126c0: 5665 7262 6f73 653d 3e42 6f6f 6c65 616e Verbose=>Boolean │ │ │ │ +000126d0: 290a 0a44 6573 6372 6970 7469 6f6e 0a3d )..Description.= │ │ │ │ +000126e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5573 6520 ==========..Use │ │ │ │ +000126f0: 5665 7262 6f73 653d 3e66 616c 7365 2074 Verbose=>false t │ │ │ │ +00012700: 6f20 7369 6c65 6e63 6520 6164 6469 7469 o silence additi │ │ │ │ +00012710: 6f6e 616c 206f 7574 7075 742e 0a0a 4675 onal output...Fu │ │ │ │ +00012720: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00012730: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +00012740: 616d 6564 2056 6572 626f 7365 3a0a 3d3d amed Verbose:.== │ │ │ │ 00012750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00012760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00012770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -00012780: 2a20 2262 6572 7469 6e69 436f 6d70 6f6e * "bertiniCompon │ │ │ │ -00012790: 656e 744d 656d 6265 7254 6573 7428 2e2e entMemberTest(.. │ │ │ │ -000127a0: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922 .,Verbose=>...)" │ │ │ │ -000127b0: 0a20 202a 2022 6265 7274 696e 6950 6172 . * "bertiniPar │ │ │ │ -000127c0: 616d 6574 6572 486f 6d6f 746f 7079 282e ameterHomotopy(. │ │ │ │ -000127d0: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29 ..,Verbose=>...) │ │ │ │ -000127e0: 220a 2020 2a20 2262 6572 7469 6e69 506f ". * "bertiniPo │ │ │ │ -000127f0: 7344 696d 536f 6c76 6528 2e2e 2e2c 5665 sDimSolve(...,Ve │ │ │ │ -00012800: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a rbose=>...)". * │ │ │ │ -00012810: 2022 6265 7274 696e 6952 6566 696e 6553 "bertiniRefineS │ │ │ │ -00012820: 6f6c 7328 2e2e 2e2c 5665 7262 6f73 653d ols(...,Verbose= │ │ │ │ -00012830: 3e2e 2e2e 2922 0a20 202a 2022 6265 7274 >...)". * "bert │ │ │ │ -00012840: 696e 6953 616d 706c 6528 2e2e 2e2c 5665 iniSample(...,Ve │ │ │ │ -00012850: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a rbose=>...)". * │ │ │ │ -00012860: 202a 6e6f 7465 2062 6572 7469 6e69 5472 *note bertiniTr │ │ │ │ -00012870: 6163 6b48 6f6d 6f74 6f70 7928 2e2e 2e2c ackHomotopy(..., │ │ │ │ -00012880: 5665 7262 6f73 653d 3e2e 2e2e 293a 0a20 Verbose=>...):. │ │ │ │ -00012890: 2020 2062 6572 7469 6e69 5472 6163 6b48 bertiniTrackH │ │ │ │ -000128a0: 6f6d 6f74 6f70 795f 6c70 5f70 645f 7064 omotopy_lp_pd_pd │ │ │ │ -000128b0: 5f70 645f 636d 5665 7262 6f73 653d 3e5f _pd_cmVerbose=>_ │ │ │ │ -000128c0: 7064 5f70 645f 7064 5f72 702c 202d 2d20 pd_pd_pd_rp, -- │ │ │ │ -000128d0: 4f70 7469 6f6e 2074 6f0a 2020 2020 7369 Option to. si │ │ │ │ -000128e0: 6c65 6e63 6520 6164 6469 7469 6f6e 616c lence additional │ │ │ │ -000128f0: 206f 7574 7075 740a 2020 2a20 2262 6572 output. * "ber │ │ │ │ -00012900: 7469 6e69 5573 6572 486f 6d6f 746f 7079 tiniUserHomotopy │ │ │ │ -00012910: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e (...,Verbose=>.. │ │ │ │ -00012920: 2e29 220a 2020 2a20 2262 6572 7469 6e69 .)". * "bertini │ │ │ │ -00012930: 5a65 726f 4469 6d53 6f6c 7665 282e 2e2e ZeroDimSolve(... │ │ │ │ -00012940: 2c56 6572 626f 7365 3d3e 2e2e 2e29 220a ,Verbose=>...)". │ │ │ │ -00012950: 2020 2a20 2269 6d70 6f72 7449 6e63 6964 * "importIncid │ │ │ │ -00012960: 656e 6365 4d61 7472 6978 282e 2e2e 2c56 enceMatrix(...,V │ │ │ │ -00012970: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020 erbose=>...)". │ │ │ │ -00012980: 2a20 2269 6d70 6f72 744d 6169 6e44 6174 * "importMainDat │ │ │ │ -00012990: 6146 696c 6528 2e2e 2e2c 5665 7262 6f73 aFile(...,Verbos │ │ │ │ -000129a0: 653d 3e2e 2e2e 2922 0a20 202a 2022 696d e=>...)". * "im │ │ │ │ -000129b0: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ -000129c0: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e2e e(...,Verbose=>. │ │ │ │ -000129d0: 2e2e 2922 0a20 202a 2022 6d61 6b65 4227 ..)". * "makeB' │ │ │ │ -000129e0: 496e 7075 7446 696c 6528 2e2e 2e2c 5665 InputFile(...,Ve │ │ │ │ -000129f0: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a rbose=>...)". * │ │ │ │ -00012a00: 2022 6d61 6b65 4d65 6d62 6572 7368 6970 "makeMembership │ │ │ │ -00012a10: 4669 6c65 282e 2e2e 2c56 6572 626f 7365 File(...,Verbose │ │ │ │ -00012a20: 3d3e 2e2e 2e29 220a 2020 2a20 226d 616b =>...)". * "mak │ │ │ │ -00012a30: 6553 616d 706c 6553 6f6c 7574 696f 6e73 eSampleSolutions │ │ │ │ -00012a40: 4669 6c65 282e 2e2e 2c56 6572 626f 7365 File(...,Verbose │ │ │ │ -00012a50: 3d3e 2e2e 2e29 220a 2020 2a20 2272 756e =>...)". * "run │ │ │ │ -00012a60: 4265 7274 696e 6928 2e2e 2e2c 5665 7262 Bertini(...,Verb │ │ │ │ -00012a70: 6f73 653d 3e2e 2e2e 2922 0a20 202a 2022 ose=>...)". * " │ │ │ │ -00012a80: 6368 6563 6b28 2e2e 2e2c 5665 7262 6f73 check(...,Verbos │ │ │ │ -00012a90: 653d 3e2e 2e2e 2922 202d 2d20 7365 6520 e=>...)" -- see │ │ │ │ -00012aa0: 2a6e 6f74 6520 6368 6563 6b3a 2028 4d61 *note check: (Ma │ │ │ │ -00012ab0: 6361 756c 6179 3244 6f63 2963 6865 636b caulay2Doc)check │ │ │ │ -00012ac0: 2c20 2d2d 0a20 2020 2070 6572 666f 726d , --. perform │ │ │ │ -00012ad0: 2074 6573 7473 206f 6620 6120 7061 636b tests of a pack │ │ │ │ -00012ae0: 6167 650a 2020 2a20 2263 6f70 7944 6972 age. * "copyDir │ │ │ │ -00012af0: 6563 746f 7279 282e 2e2e 2c56 6572 626f ectory(...,Verbo │ │ │ │ -00012b00: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565 se=>...)" -- see │ │ │ │ -00012b10: 202a 6e6f 7465 0a20 2020 2063 6f70 7944 *note. copyD │ │ │ │ -00012b20: 6972 6563 746f 7279 2853 7472 696e 672c irectory(String, │ │ │ │ -00012b30: 5374 7269 6e67 293a 0a20 2020 2028 4d61 String):. (Ma │ │ │ │ -00012b40: 6361 756c 6179 3244 6f63 2963 6f70 7944 caulay2Doc)copyD │ │ │ │ -00012b50: 6972 6563 746f 7279 5f6c 7053 7472 696e irectory_lpStrin │ │ │ │ -00012b60: 675f 636d 5374 7269 6e67 5f72 702c 0a20 g_cmString_rp,. │ │ │ │ -00012b70: 202a 2022 636f 7079 4669 6c65 282e 2e2e * "copyFile(... │ │ │ │ -00012b80: 2c56 6572 626f 7365 3d3e 2e2e 2e29 2220 ,Verbose=>...)" │ │ │ │ -00012b90: 2d2d 2073 6565 202a 6e6f 7465 2063 6f70 -- see *note cop │ │ │ │ -00012ba0: 7946 696c 6528 5374 7269 6e67 2c53 7472 yFile(String,Str │ │ │ │ -00012bb0: 696e 6729 3a0a 2020 2020 284d 6163 6175 ing):. (Macau │ │ │ │ -00012bc0: 6c61 7932 446f 6329 636f 7079 4669 6c65 lay2Doc)copyFile │ │ │ │ -00012bd0: 5f6c 7053 7472 696e 675f 636d 5374 7269 _lpString_cmStri │ │ │ │ -00012be0: 6e67 5f72 702c 0a20 202a 2022 6669 6e64 ng_rp,. * "find │ │ │ │ -00012bf0: 5072 6f67 7261 6d28 2e2e 2e2c 5665 7262 Program(...,Verb │ │ │ │ -00012c00: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365 ose=>...)" -- se │ │ │ │ -00012c10: 6520 2a6e 6f74 6520 6669 6e64 5072 6f67 e *note findProg │ │ │ │ -00012c20: 7261 6d3a 0a20 2020 2028 4d61 6361 756c ram:. (Macaul │ │ │ │ -00012c30: 6179 3244 6f63 2966 696e 6450 726f 6772 ay2Doc)findProgr │ │ │ │ -00012c40: 616d 2c20 2d2d 206c 6f61 6420 6578 7465 am, -- load exte │ │ │ │ -00012c50: 726e 616c 2070 726f 6772 616d 0a20 202a rnal program. * │ │ │ │ -00012c60: 2022 696e 7374 616c 6c50 6163 6b61 6765 "installPackage │ │ │ │ -00012c70: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e (...,Verbose=>.. │ │ │ │ -00012c80: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ -00012c90: 2069 6e73 7461 6c6c 5061 636b 6167 653a installPackage: │ │ │ │ -00012ca0: 0a20 2020 2028 4d61 6361 756c 6179 3244 . (Macaulay2D │ │ │ │ -00012cb0: 6f63 2969 6e73 7461 6c6c 5061 636b 6167 oc)installPackag │ │ │ │ -00012cc0: 652c 202d 2d20 6c6f 6164 2061 6e64 2069 e, -- load and i │ │ │ │ -00012cd0: 6e73 7461 6c6c 2061 2070 6163 6b61 6765 nstall a package │ │ │ │ -00012ce0: 2061 6e64 2069 7473 0a20 2020 2064 6f63 and its. doc │ │ │ │ -00012cf0: 756d 656e 7461 7469 6f6e 0a20 202a 2022 umentation. * " │ │ │ │ -00012d00: 6d6f 7665 4669 6c65 282e 2e2e 2c56 6572 moveFile(...,Ver │ │ │ │ -00012d10: 626f 7365 3d3e 2e2e 2e29 2220 2d2d 2073 bose=>...)" -- s │ │ │ │ -00012d20: 6565 202a 6e6f 7465 206d 6f76 6546 696c ee *note moveFil │ │ │ │ -00012d30: 6528 5374 7269 6e67 2c53 7472 696e 6729 e(String,String) │ │ │ │ -00012d40: 3a0a 2020 2020 284d 6163 6175 6c61 7932 :. (Macaulay2 │ │ │ │ -00012d50: 446f 6329 6d6f 7665 4669 6c65 5f6c 7053 Doc)moveFile_lpS │ │ │ │ -00012d60: 7472 696e 675f 636d 5374 7269 6e67 5f72 tring_cmString_r │ │ │ │ -00012d70: 702c 0a20 202a 2022 7275 6e50 726f 6772 p,. * "runProgr │ │ │ │ -00012d80: 616d 282e 2e2e 2c56 6572 626f 7365 3d3e am(...,Verbose=> │ │ │ │ -00012d90: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ -00012da0: 7465 2072 756e 5072 6f67 7261 6d3a 0a20 te runProgram:. │ │ │ │ -00012db0: 2020 2028 4d61 6361 756c 6179 3244 6f63 (Macaulay2Doc │ │ │ │ -00012dc0: 2972 756e 5072 6f67 7261 6d2c 202d 2d20 )runProgram, -- │ │ │ │ -00012dd0: 7275 6e20 616e 2065 7874 6572 6e61 6c20 run an external │ │ │ │ -00012de0: 7072 6f67 7261 6d0a 2020 2a20 2273 796d program. * "sym │ │ │ │ -00012df0: 6c69 6e6b 4469 7265 6374 6f72 7928 2e2e linkDirectory(.. │ │ │ │ -00012e00: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922 .,Verbose=>...)" │ │ │ │ -00012e10: 202d 2d20 7365 6520 2a6e 6f74 650a 2020 -- see *note. │ │ │ │ -00012e20: 2020 7379 6d6c 696e 6b44 6972 6563 746f symlinkDirecto │ │ │ │ -00012e30: 7279 2853 7472 696e 672c 5374 7269 6e67 ry(String,String │ │ │ │ -00012e40: 293a 0a20 2020 2028 4d61 6361 756c 6179 ):. (Macaulay │ │ │ │ -00012e50: 3244 6f63 2973 796d 6c69 6e6b 4469 7265 2Doc)symlinkDire │ │ │ │ -00012e60: 6374 6f72 795f 6c70 5374 7269 6e67 5f63 ctory_lpString_c │ │ │ │ -00012e70: 6d53 7472 696e 675f 7270 2c20 2d2d 206d mString_rp, -- m │ │ │ │ -00012e80: 616b 6520 7379 6d62 6f6c 6963 206c 696e ake symbolic lin │ │ │ │ -00012e90: 6b73 0a20 2020 2066 6f72 2061 6c6c 2066 ks. for all f │ │ │ │ -00012ea0: 696c 6573 2069 6e20 6120 6469 7265 6374 iles in a direct │ │ │ │ -00012eb0: 6f72 7920 7472 6565 0a0a 4675 7274 6865 ory tree..Furthe │ │ │ │ -00012ec0: 7220 696e 666f 726d 6174 696f 6e0a 3d3d r information.== │ │ │ │ +00012770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00012780: 202a 2022 6265 7274 696e 6943 6f6d 706f * "bertiniCompo │ │ │ │ +00012790: 6e65 6e74 4d65 6d62 6572 5465 7374 282e nentMemberTest(. │ │ │ │ +000127a0: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29 ..,Verbose=>...) │ │ │ │ +000127b0: 220a 2020 2a20 2262 6572 7469 6e69 5061 ". * "bertiniPa │ │ │ │ +000127c0: 7261 6d65 7465 7248 6f6d 6f74 6f70 7928 rameterHomotopy( │ │ │ │ +000127d0: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e ...,Verbose=>... │ │ │ │ +000127e0: 2922 0a20 202a 2022 6265 7274 696e 6950 )". * "bertiniP │ │ │ │ +000127f0: 6f73 4469 6d53 6f6c 7665 282e 2e2e 2c56 osDimSolve(...,V │ │ │ │ +00012800: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020 erbose=>...)". │ │ │ │ +00012810: 2a20 2262 6572 7469 6e69 5265 6669 6e65 * "bertiniRefine │ │ │ │ +00012820: 536f 6c73 282e 2e2e 2c56 6572 626f 7365 Sols(...,Verbose │ │ │ │ +00012830: 3d3e 2e2e 2e29 220a 2020 2a20 2262 6572 =>...)". * "ber │ │ │ │ +00012840: 7469 6e69 5361 6d70 6c65 282e 2e2e 2c56 tiniSample(...,V │ │ │ │ +00012850: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020 erbose=>...)". │ │ │ │ +00012860: 2a20 2a6e 6f74 6520 6265 7274 696e 6954 * *note bertiniT │ │ │ │ +00012870: 7261 636b 486f 6d6f 746f 7079 282e 2e2e rackHomotopy(... │ │ │ │ +00012880: 2c56 6572 626f 7365 3d3e 2e2e 2e29 3a0a ,Verbose=>...):. │ │ │ │ +00012890: 2020 2020 6265 7274 696e 6954 7261 636b bertiniTrack │ │ │ │ +000128a0: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70 Homotopy_lp_pd_p │ │ │ │ +000128b0: 645f 7064 5f63 6d56 6572 626f 7365 3d3e d_pd_cmVerbose=> │ │ │ │ +000128c0: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d _pd_pd_pd_rp, -- │ │ │ │ +000128d0: 204f 7074 696f 6e20 746f 0a20 2020 2073 Option to. s │ │ │ │ +000128e0: 696c 656e 6365 2061 6464 6974 696f 6e61 ilence additiona │ │ │ │ +000128f0: 6c20 6f75 7470 7574 0a20 202a 2022 6265 l output. * "be │ │ │ │ +00012900: 7274 696e 6955 7365 7248 6f6d 6f74 6f70 rtiniUserHomotop │ │ │ │ +00012910: 7928 2e2e 2e2c 5665 7262 6f73 653d 3e2e y(...,Verbose=>. │ │ │ │ +00012920: 2e2e 2922 0a20 202a 2022 6265 7274 696e ..)". * "bertin │ │ │ │ +00012930: 695a 6572 6f44 696d 536f 6c76 6528 2e2e iZeroDimSolve(.. │ │ │ │ +00012940: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922 .,Verbose=>...)" │ │ │ │ +00012950: 0a20 202a 2022 696d 706f 7274 496e 6369 . * "importInci │ │ │ │ +00012960: 6465 6e63 654d 6174 7269 7828 2e2e 2e2c denceMatrix(..., │ │ │ │ +00012970: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20 Verbose=>...)". │ │ │ │ +00012980: 202a 2022 696d 706f 7274 4d61 696e 4461 * "importMainDa │ │ │ │ +00012990: 7461 4669 6c65 282e 2e2e 2c56 6572 626f taFile(...,Verbo │ │ │ │ +000129a0: 7365 3d3e 2e2e 2e29 220a 2020 2a20 2269 se=>...)". * "i │ │ │ │ +000129b0: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669 mportSolutionsFi │ │ │ │ +000129c0: 6c65 282e 2e2e 2c56 6572 626f 7365 3d3e le(...,Verbose=> │ │ │ │ +000129d0: 2e2e 2e29 220a 2020 2a20 226d 616b 6542 ...)". * "makeB │ │ │ │ +000129e0: 2749 6e70 7574 4669 6c65 282e 2e2e 2c56 'InputFile(...,V │ │ │ │ +000129f0: 6572 626f 7365 3d3e 2e2e 2e29 220a 2020 erbose=>...)". │ │ │ │ +00012a00: 2a20 226d 616b 654d 656d 6265 7273 6869 * "makeMembershi │ │ │ │ +00012a10: 7046 696c 6528 2e2e 2e2c 5665 7262 6f73 pFile(...,Verbos │ │ │ │ +00012a20: 653d 3e2e 2e2e 2922 0a20 202a 2022 6d61 e=>...)". * "ma │ │ │ │ +00012a30: 6b65 5361 6d70 6c65 536f 6c75 7469 6f6e keSampleSolution │ │ │ │ +00012a40: 7346 696c 6528 2e2e 2e2c 5665 7262 6f73 sFile(...,Verbos │ │ │ │ +00012a50: 653d 3e2e 2e2e 2922 0a20 202a 2022 7275 e=>...)". * "ru │ │ │ │ +00012a60: 6e42 6572 7469 6e69 282e 2e2e 2c56 6572 nBertini(...,Ver │ │ │ │ +00012a70: 626f 7365 3d3e 2e2e 2e29 220a 2020 2a20 bose=>...)". * │ │ │ │ +00012a80: 2263 6865 636b 282e 2e2e 2c56 6572 626f "check(...,Verbo │ │ │ │ +00012a90: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565 se=>...)" -- see │ │ │ │ +00012aa0: 202a 6e6f 7465 2063 6865 636b 3a20 284d *note check: (M │ │ │ │ +00012ab0: 6163 6175 6c61 7932 446f 6329 6368 6563 acaulay2Doc)chec │ │ │ │ +00012ac0: 6b2c 202d 2d0a 2020 2020 7065 7266 6f72 k, --. perfor │ │ │ │ +00012ad0: 6d20 7465 7374 7320 6f66 2061 2070 6163 m tests of a pac │ │ │ │ +00012ae0: 6b61 6765 0a20 202a 2022 636f 7079 4469 kage. * "copyDi │ │ │ │ +00012af0: 7265 6374 6f72 7928 2e2e 2e2c 5665 7262 rectory(...,Verb │ │ │ │ +00012b00: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365 ose=>...)" -- se │ │ │ │ +00012b10: 6520 2a6e 6f74 650a 2020 2020 636f 7079 e *note. copy │ │ │ │ +00012b20: 4469 7265 6374 6f72 7928 5374 7269 6e67 Directory(String │ │ │ │ +00012b30: 2c53 7472 696e 6729 3a0a 2020 2020 284d ,String):. (M │ │ │ │ +00012b40: 6163 6175 6c61 7932 446f 6329 636f 7079 acaulay2Doc)copy │ │ │ │ +00012b50: 4469 7265 6374 6f72 795f 6c70 5374 7269 Directory_lpStri │ │ │ │ +00012b60: 6e67 5f63 6d53 7472 696e 675f 7270 2c0a ng_cmString_rp,. │ │ │ │ +00012b70: 2020 2a20 2263 6f70 7946 696c 6528 2e2e * "copyFile(.. │ │ │ │ +00012b80: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922 .,Verbose=>...)" │ │ │ │ +00012b90: 202d 2d20 7365 6520 2a6e 6f74 6520 636f -- see *note co │ │ │ │ +00012ba0: 7079 4669 6c65 2853 7472 696e 672c 5374 pyFile(String,St │ │ │ │ +00012bb0: 7269 6e67 293a 0a20 2020 2028 4d61 6361 ring):. (Maca │ │ │ │ +00012bc0: 756c 6179 3244 6f63 2963 6f70 7946 696c ulay2Doc)copyFil │ │ │ │ +00012bd0: 655f 6c70 5374 7269 6e67 5f63 6d53 7472 e_lpString_cmStr │ │ │ │ +00012be0: 696e 675f 7270 2c0a 2020 2a20 2266 696e ing_rp,. * "fin │ │ │ │ +00012bf0: 6450 726f 6772 616d 282e 2e2e 2c56 6572 dProgram(...,Ver │ │ │ │ +00012c00: 626f 7365 3d3e 2e2e 2e29 2220 2d2d 2073 bose=>...)" -- s │ │ │ │ +00012c10: 6565 202a 6e6f 7465 2066 696e 6450 726f ee *note findPro │ │ │ │ +00012c20: 6772 616d 3a0a 2020 2020 284d 6163 6175 gram:. (Macau │ │ │ │ +00012c30: 6c61 7932 446f 6329 6669 6e64 5072 6f67 lay2Doc)findProg │ │ │ │ +00012c40: 7261 6d2c 202d 2d20 6c6f 6164 2065 7874 ram, -- load ext │ │ │ │ +00012c50: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020 ernal program. │ │ │ │ +00012c60: 2a20 2269 6e73 7461 6c6c 5061 636b 6167 * "installPackag │ │ │ │ +00012c70: 6528 2e2e 2e2c 5665 7262 6f73 653d 3e2e e(...,Verbose=>. │ │ │ │ +00012c80: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74 ..)" -- see *not │ │ │ │ +00012c90: 6520 696e 7374 616c 6c50 6163 6b61 6765 e installPackage │ │ │ │ +00012ca0: 3a0a 2020 2020 284d 6163 6175 6c61 7932 :. (Macaulay2 │ │ │ │ +00012cb0: 446f 6329 696e 7374 616c 6c50 6163 6b61 Doc)installPacka │ │ │ │ +00012cc0: 6765 2c20 2d2d 206c 6f61 6420 616e 6420 ge, -- load and │ │ │ │ +00012cd0: 696e 7374 616c 6c20 6120 7061 636b 6167 install a packag │ │ │ │ +00012ce0: 6520 616e 6420 6974 730a 2020 2020 646f e and its. do │ │ │ │ +00012cf0: 6375 6d65 6e74 6174 696f 6e0a 2020 2a20 cumentation. * │ │ │ │ +00012d00: 226d 6f76 6546 696c 6528 2e2e 2e2c 5665 "moveFile(...,Ve │ │ │ │ +00012d10: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20 rbose=>...)" -- │ │ │ │ +00012d20: 7365 6520 2a6e 6f74 6520 6d6f 7665 4669 see *note moveFi │ │ │ │ +00012d30: 6c65 2853 7472 696e 672c 5374 7269 6e67 le(String,String │ │ │ │ +00012d40: 293a 0a20 2020 2028 4d61 6361 756c 6179 ):. (Macaulay │ │ │ │ +00012d50: 3244 6f63 296d 6f76 6546 696c 655f 6c70 2Doc)moveFile_lp │ │ │ │ +00012d60: 5374 7269 6e67 5f63 6d53 7472 696e 675f String_cmString_ │ │ │ │ +00012d70: 7270 2c0a 2020 2a20 2272 756e 5072 6f67 rp,. * "runProg │ │ │ │ +00012d80: 7261 6d28 2e2e 2e2c 5665 7262 6f73 653d ram(...,Verbose= │ │ │ │ +00012d90: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +00012da0: 6f74 6520 7275 6e50 726f 6772 616d 3a0a ote runProgram:. │ │ │ │ +00012db0: 2020 2020 284d 6163 6175 6c61 7932 446f (Macaulay2Do │ │ │ │ +00012dc0: 6329 7275 6e50 726f 6772 616d 2c20 2d2d c)runProgram, -- │ │ │ │ +00012dd0: 2072 756e 2061 6e20 6578 7465 726e 616c run an external │ │ │ │ +00012de0: 2070 726f 6772 616d 0a20 202a 2022 7379 program. * "sy │ │ │ │ +00012df0: 6d6c 696e 6b44 6972 6563 746f 7279 282e mlinkDirectory(. │ │ │ │ +00012e00: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29 ..,Verbose=>...) │ │ │ │ +00012e10: 2220 2d2d 2073 6565 202a 6e6f 7465 0a20 " -- see *note. │ │ │ │ +00012e20: 2020 2073 796d 6c69 6e6b 4469 7265 6374 symlinkDirect │ │ │ │ +00012e30: 6f72 7928 5374 7269 6e67 2c53 7472 696e ory(String,Strin │ │ │ │ +00012e40: 6729 3a0a 2020 2020 284d 6163 6175 6c61 g):. (Macaula │ │ │ │ +00012e50: 7932 446f 6329 7379 6d6c 696e 6b44 6972 y2Doc)symlinkDir │ │ │ │ +00012e60: 6563 746f 7279 5f6c 7053 7472 696e 675f ectory_lpString_ │ │ │ │ +00012e70: 636d 5374 7269 6e67 5f72 702c 202d 2d20 cmString_rp, -- │ │ │ │ +00012e80: 6d61 6b65 2073 796d 626f 6c69 6320 6c69 make symbolic li │ │ │ │ +00012e90: 6e6b 730a 2020 2020 666f 7220 616c 6c20 nks. for all │ │ │ │ +00012ea0: 6669 6c65 7320 696e 2061 2064 6972 6563 files in a direc │ │ │ │ +00012eb0: 746f 7279 2074 7265 650a 0a46 7572 7468 tory tree..Furth │ │ │ │ +00012ec0: 6572 2069 6e66 6f72 6d61 7469 6f6e 0a3d er information.= │ │ │ │ 00012ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00012ee0: 3d0a 0a20 202a 2044 6566 6175 6c74 2076 =.. * Default v │ │ │ │ -00012ef0: 616c 7565 3a20 2a6e 6f74 6520 6661 6c73 alue: *note fals │ │ │ │ -00012f00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00012f10: 2966 616c 7365 2c0a 2020 2a20 4675 6e63 )false,. * Func │ │ │ │ -00012f20: 7469 6f6e 3a20 2a6e 6f74 6520 6265 7274 tion: *note bert │ │ │ │ -00012f30: 696e 6954 7261 636b 486f 6d6f 746f 7079 iniTrackHomotopy │ │ │ │ -00012f40: 3a20 6265 7274 696e 6954 7261 636b 486f : bertiniTrackHo │ │ │ │ -00012f50: 6d6f 746f 7079 2c20 2d2d 2061 206d 6169 motopy, -- a mai │ │ │ │ -00012f60: 6e0a 2020 2020 6d65 7468 6f64 2074 6f20 n. method to │ │ │ │ -00012f70: 7472 6163 6b20 7573 696e 6720 6120 7573 track using a us │ │ │ │ -00012f80: 6572 2d64 6566 696e 6564 2068 6f6d 6f74 er-defined homot │ │ │ │ -00012f90: 6f70 790a 2020 2a20 4f70 7469 6f6e 206b opy. * Option k │ │ │ │ -00012fa0: 6579 3a20 2a6e 6f74 6520 5665 7262 6f73 ey: *note Verbos │ │ │ │ -00012fb0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00012fc0: 2956 6572 626f 7365 2c20 2d2d 2061 6e20 )Verbose, -- an │ │ │ │ -00012fd0: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ -00012fe0: 740a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d t.-------------- │ │ │ │ +00012ee0: 3d3d 0a0a 2020 2a20 4465 6661 756c 7420 ==.. * Default │ │ │ │ +00012ef0: 7661 6c75 653a 202a 6e6f 7465 2066 616c value: *note fal │ │ │ │ +00012f00: 7365 3a20 284d 6163 6175 6c61 7932 446f se: (Macaulay2Do │ │ │ │ +00012f10: 6329 6661 6c73 652c 0a20 202a 2046 756e c)false,. * Fun │ │ │ │ +00012f20: 6374 696f 6e3a 202a 6e6f 7465 2062 6572 ction: *note ber │ │ │ │ +00012f30: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70 tiniTrackHomotop │ │ │ │ +00012f40: 793a 2062 6572 7469 6e69 5472 6163 6b48 y: bertiniTrackH │ │ │ │ +00012f50: 6f6d 6f74 6f70 792c 202d 2d20 6120 6d61 omotopy, -- a ma │ │ │ │ +00012f60: 696e 0a20 2020 206d 6574 686f 6420 746f in. method to │ │ │ │ +00012f70: 2074 7261 636b 2075 7369 6e67 2061 2075 track using a u │ │ │ │ +00012f80: 7365 722d 6465 6669 6e65 6420 686f 6d6f ser-defined homo │ │ │ │ +00012f90: 746f 7079 0a20 202a 204f 7074 696f 6e20 topy. * Option │ │ │ │ +00012fa0: 6b65 793a 202a 6e6f 7465 2056 6572 626f key: *note Verbo │ │ │ │ +00012fb0: 7365 3a20 284d 6163 6175 6c61 7932 446f se: (Macaulay2Do │ │ │ │ +00012fc0: 6329 5665 7262 6f73 652c 202d 2d20 616e c)Verbose, -- an │ │ │ │ +00012fd0: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ +00012fe0: 6e74 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d nt.------------- │ │ │ │ 00012ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013030: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00013040: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00013050: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00013060: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00013070: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ -00013080: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00013090: 7061 636b 6167 6573 2f42 6572 7469 6e69 packages/Bertini │ │ │ │ -000130a0: 2e6d 323a 0a33 3437 323a 302e 0a1f 0a46 .m2:.3472:0....F │ │ │ │ -000130b0: 696c 653a 2042 6572 7469 6e69 2e69 6e66 ile: Bertini.inf │ │ │ │ -000130c0: 6f2c 204e 6f64 653a 2062 6572 7469 6e69 o, Node: bertini │ │ │ │ -000130d0: 5573 6572 486f 6d6f 746f 7079 2c20 4e65 UserHomotopy, Ne │ │ │ │ -000130e0: 7874 3a20 6265 7274 696e 695a 6572 6f44 xt: bertiniZeroD │ │ │ │ -000130f0: 696d 536f 6c76 652c 2050 7265 763a 2062 imSolve, Prev: b │ │ │ │ -00013100: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74 ertiniTrackHomot │ │ │ │ -00013110: 6f70 795f 6c70 5f70 645f 7064 5f70 645f opy_lp_pd_pd_pd_ │ │ │ │ -00013120: 636d 5665 7262 6f73 653d 3e5f 7064 5f70 cmVerbose=>_pd_p │ │ │ │ -00013130: 645f 7064 5f72 702c 2055 703a 2054 6f70 d_pd_rp, Up: Top │ │ │ │ -00013140: 0a0a 6265 7274 696e 6955 7365 7248 6f6d ..bertiniUserHom │ │ │ │ -00013150: 6f74 6f70 7920 2d2d 2061 206d 6169 6e20 otopy -- a main │ │ │ │ -00013160: 6d65 7468 6f64 2074 6f20 7472 6163 6b20 method to track │ │ │ │ -00013170: 6120 7573 6572 2d64 6566 696e 6564 2068 a user-defined h │ │ │ │ -00013180: 6f6d 6f74 6f70 790a 2a2a 2a2a 2a2a 2a2a omotopy.******** │ │ │ │ +00013030: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00013040: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00013050: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00013060: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00013070: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +00013080: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00013090: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ +000130a0: 692e 6d32 3a0a 3334 3732 3a30 2e0a 1f0a i.m2:.3472:0.... │ │ │ │ +000130b0: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ +000130c0: 666f 2c20 4e6f 6465 3a20 6265 7274 696e fo, Node: bertin │ │ │ │ +000130d0: 6955 7365 7248 6f6d 6f74 6f70 792c 204e iUserHomotopy, N │ │ │ │ +000130e0: 6578 743a 2062 6572 7469 6e69 5a65 726f ext: bertiniZero │ │ │ │ +000130f0: 4469 6d53 6f6c 7665 2c20 5072 6576 3a20 DimSolve, Prev: │ │ │ │ +00013100: 6265 7274 696e 6954 7261 636b 486f 6d6f bertiniTrackHomo │ │ │ │ +00013110: 746f 7079 5f6c 705f 7064 5f70 645f 7064 topy_lp_pd_pd_pd │ │ │ │ +00013120: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f _cmVerbose=>_pd_ │ │ │ │ +00013130: 7064 5f70 645f 7270 2c20 5570 3a20 546f pd_pd_rp, Up: To │ │ │ │ +00013140: 700a 0a62 6572 7469 6e69 5573 6572 486f p..bertiniUserHo │ │ │ │ +00013150: 6d6f 746f 7079 202d 2d20 6120 6d61 696e motopy -- a main │ │ │ │ +00013160: 206d 6574 686f 6420 746f 2074 7261 636b method to track │ │ │ │ +00013170: 2061 2075 7365 722d 6465 6669 6e65 6420 a user-defined │ │ │ │ +00013180: 686f 6d6f 746f 7079 0a2a 2a2a 2a2a 2a2a homotopy.******* │ │ │ │ 00013190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000131a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000131b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000131c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -000131d0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -000131e0: 2020 2053 303d 6265 7274 696e 6955 7365 S0=bertiniUse │ │ │ │ -000131f0: 7248 6f6d 6f74 6f70 7928 742c 2050 2c20 rHomotopy(t, P, │ │ │ │ -00013200: 482c 2053 3129 0a20 202a 2049 6e70 7574 H, S1). * Input │ │ │ │ -00013210: 733a 0a20 2020 2020 202a 2074 2c20 6120 s:. * t, a │ │ │ │ -00013220: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ -00013230: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ -00013240: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ -00013250: 6120 7061 7468 2076 6172 6961 626c 650a a path variable. │ │ │ │ -00013260: 2020 2020 2020 2a20 502c 2061 202a 6e6f * P, a *no │ │ │ │ -00013270: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -00013280: 6179 3244 6f63 294c 6973 742c 2c20 6120 ay2Doc)List,, a │ │ │ │ -00013290: 6c69 7374 206f 6620 6f70 7469 6f6e 7320 list of options │ │ │ │ -000132a0: 7468 6174 2073 6574 2074 6865 0a20 2020 that set the. │ │ │ │ -000132b0: 2020 2020 2070 6172 616d 6574 6572 730a parameters. │ │ │ │ -000132c0: 2020 2020 2020 2a20 482c 2061 202a 6e6f * H, a *no │ │ │ │ -000132d0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -000132e0: 6179 3244 6f63 294c 6973 742c 2c20 6120 ay2Doc)List,, a │ │ │ │ -000132f0: 6c69 7374 206f 6620 706f 6c79 6e6f 6d69 list of polynomi │ │ │ │ -00013300: 616c 7320 7468 6174 2064 6566 696e 650a als that define. │ │ │ │ -00013310: 2020 2020 2020 2020 7468 6520 686f 6d6f the homo │ │ │ │ -00013320: 746f 7079 0a20 2020 2020 202a 2053 312c topy. * S1, │ │ │ │ -00013330: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ -00013340: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -00013350: 742c 2c20 6120 6c69 7374 206f 6620 736f t,, a list of so │ │ │ │ -00013360: 6c75 7469 6f6e 7320 746f 2074 6865 2073 lutions to the s │ │ │ │ -00013370: 7461 7274 0a20 2020 2020 2020 2073 7973 tart. sys │ │ │ │ -00013380: 7465 6d0a 2020 2a20 2a6e 6f74 6520 4f70 tem. * *note Op │ │ │ │ -00013390: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -000133a0: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -000133b0: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -000133c0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -000133d0: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74 s,:. * *not │ │ │ │ -000133e0: 6520 4166 6656 6172 6961 626c 6547 726f e AffVariableGro │ │ │ │ -000133f0: 7570 3a20 5661 7269 6162 6c65 2067 726f up: Variable gro │ │ │ │ -00013400: 7570 732c 203d 3e20 2e2e 2e2c 2064 6566 ups, => ..., def │ │ │ │ -00013410: 6175 6c74 2076 616c 7565 207b 7d2c 2061 ault value {}, a │ │ │ │ -00013420: 6e0a 2020 2020 2020 2020 6f70 7469 6f6e n. option │ │ │ │ -00013430: 2074 6f20 6772 6f75 7020 7661 7269 6162 to group variab │ │ │ │ -00013440: 6c65 7320 616e 6420 7573 6520 6d75 6c74 les and use mult │ │ │ │ -00013450: 6968 6f6d 6f67 656e 656f 7573 2068 6f6d ihomogeneous hom │ │ │ │ -00013460: 6f74 6f70 6965 730a 2020 2020 2020 2a20 otopies. * │ │ │ │ -00013470: 4227 436f 6e73 7461 6e74 7320 286d 6973 B'Constants (mis │ │ │ │ -00013480: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ -00013490: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ -000134a0: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20 ult value {}, . │ │ │ │ -000134b0: 2020 2020 202a 2042 2746 756e 6374 696f * B'Functio │ │ │ │ -000134c0: 6e73 2028 6d69 7373 696e 6720 646f 6375 ns (missing docu │ │ │ │ -000134d0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -000134e0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -000134f0: 207b 7d2c 200a 2020 2020 2020 2a20 4265 {}, . * Be │ │ │ │ -00013500: 7274 696e 6949 6e70 7574 436f 6e66 6967 rtiniInputConfig │ │ │ │ -00013510: 7572 6174 696f 6e20 286d 6973 7369 6e67 uration (missing │ │ │ │ -00013520: 2064 6f63 756d 656e 7461 7469 6f6e 2920 documentation) │ │ │ │ -00013530: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00013540: 7661 6c75 650a 2020 2020 2020 2020 7b7d value. {} │ │ │ │ -00013550: 2c0a 2020 2020 2020 2a20 486f 6d56 6172 ,. * HomVar │ │ │ │ -00013560: 6961 626c 6547 726f 7570 2028 6d69 7373 iableGroup (miss │ │ │ │ -00013570: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00013580: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00013590: 6c74 2076 616c 7565 207b 7d2c 200a 2020 lt value {}, . │ │ │ │ -000135a0: 2020 2020 2a20 4d32 5072 6563 6973 696f * M2Precisio │ │ │ │ -000135b0: 6e20 286d 6973 7369 6e67 2064 6f63 756d n (missing docum │ │ │ │ -000135c0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -000135d0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000135e0: 3533 2c20 0a20 2020 2020 202a 204f 7574 53, . * Out │ │ │ │ -000135f0: 7075 7453 7479 6c65 2028 6d69 7373 696e putStyle (missin │ │ │ │ -00013600: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ -00013610: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00013620: 2076 616c 7565 2022 4f75 7450 6f69 6e74 value "OutPoint │ │ │ │ -00013630: 7322 2c20 0a20 2020 2020 202a 2052 616e s", . * Ran │ │ │ │ -00013640: 646f 6d43 6f6d 706c 6578 2028 6d69 7373 domComplex (miss │ │ │ │ -00013650: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00013660: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00013670: 6c74 2076 616c 7565 207b 7d2c 200a 2020 lt value {}, . │ │ │ │ -00013680: 2020 2020 2a20 5261 6e64 6f6d 5265 616c * RandomReal │ │ │ │ -00013690: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -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 2d37 3333 3638 2d30 2f30 222c p/M2-73368-0/0", │ │ │ │ -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= │ │ │ │ -00013790: 3e5f 7064 5f70 645f 7064 5f72 700a 2020 >_pd_pd_pd_rp. │ │ │ │ -000137a0: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20 , => ..., │ │ │ │ -000137b0: 6465 6661 756c 7420 7661 6c75 6520 6661 default value fa │ │ │ │ -000137c0: 6c73 652c 204f 7074 696f 6e20 746f 2073 lse, Option to s │ │ │ │ -000137d0: 696c 656e 6365 2061 6464 6974 696f 6e61 ilence additiona │ │ │ │ -000137e0: 6c20 6f75 7470 7574 0a20 202a 204f 7574 l output. * Out │ │ │ │ -000137f0: 7075 7473 3a0a 2020 2020 2020 2a20 5330 puts:. * S0 │ │ │ │ -00013800: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00013810: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00013820: 7374 2c2c 2061 206c 6973 7420 6f66 2073 st,, a list of s │ │ │ │ -00013830: 6f6c 7574 696f 6e73 2074 6f20 7468 650a olutions to the. │ │ │ │ -00013840: 2020 2020 2020 2020 7461 7267 6574 2073 target s │ │ │ │ -00013850: 7973 7465 6d0a 0a44 6573 6372 6970 7469 ystem..Descripti │ │ │ │ -00013860: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00013870: 5468 6973 206d 6574 686f 6420 6361 6c6c This method call │ │ │ │ -00013880: 7320 4265 7274 696e 6920 746f 2074 7261 s Bertini to tra │ │ │ │ -00013890: 636b 2061 2075 7365 722d 6465 6669 6e65 ck a user-define │ │ │ │ -000138a0: 6420 686f 6d6f 746f 7079 2e20 2054 6865 d homotopy. The │ │ │ │ -000138b0: 2075 7365 7220 6e65 6564 7320 746f 0a73 user needs to.s │ │ │ │ -000138c0: 7065 6369 6679 2074 6865 2068 6f6d 6f74 pecify the homot │ │ │ │ -000138d0: 6f70 7920 482c 2074 6865 2070 6174 6820 opy H, the path │ │ │ │ -000138e0: 7661 7269 6162 6c65 2074 2c20 616e 6420 variable t, and │ │ │ │ -000138f0: 6120 6c69 7374 206f 6620 7374 6172 7420 a list of start │ │ │ │ -00013900: 736f 6c75 7469 6f6e 7320 5331 2e0a 4265 solutions S1..Be │ │ │ │ -00013910: 7274 696e 6920 2831 2920 7772 6974 6573 rtini (1) writes │ │ │ │ -00013920: 2074 6865 2068 6f6d 6f74 6f70 7920 616e the homotopy an │ │ │ │ -00013930: 6420 7374 6172 7420 736f 6c75 7469 6f6e d start solution │ │ │ │ -00013940: 7320 746f 2074 656d 706f 7261 7279 2066 s to temporary f │ │ │ │ -00013950: 696c 6573 2c20 2832 290a 696e 766f 6b65 iles, (2).invoke │ │ │ │ -00013960: 7320 4265 7274 696e 6927 7320 736f 6c76 s Bertini's solv │ │ │ │ -00013970: 6572 2077 6974 6820 636f 6e66 6967 7572 er with configur │ │ │ │ -00013980: 6174 696f 6e20 6b65 7977 6f72 6420 5573 ation keyword Us │ │ │ │ -00013990: 6572 486f 6d6f 746f 7079 203d 3e20 322c erHomotopy => 2, │ │ │ │ -000139a0: 2028 3329 0a73 746f 7265 7320 7468 6520 (3).stores the │ │ │ │ -000139b0: 6f75 7470 7574 206f 6620 4265 7274 696e output of Bertin │ │ │ │ -000139c0: 6920 696e 2061 2074 656d 706f 7261 7279 i in a temporary │ │ │ │ -000139d0: 2066 696c 652c 2061 6e64 2028 3429 2070 file, and (4) p │ │ │ │ -000139e0: 6172 7365 7320 6120 6d61 6368 696e 650a arses a machine. │ │ │ │ -000139f0: 7265 6164 6162 6c65 2066 696c 6520 746f readable file to │ │ │ │ -00013a00: 206f 7574 7075 7420 6120 6c69 7374 206f output a list o │ │ │ │ -00013a10: 6620 736f 6c75 7469 6f6e 732e 0a0a 2b2d f solutions...+- │ │ │ │ -00013a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000131c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +000131d0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +000131e0: 2020 2020 5330 3d62 6572 7469 6e69 5573 S0=bertiniUs │ │ │ │ +000131f0: 6572 486f 6d6f 746f 7079 2874 2c20 502c erHomotopy(t, P, │ │ │ │ +00013200: 2048 2c20 5331 290a 2020 2a20 496e 7075 H, S1). * Inpu │ │ │ │ +00013210: 7473 3a0a 2020 2020 2020 2a20 742c 2061 ts:. * t, a │ │ │ │ +00013220: 202a 6e6f 7465 2072 696e 6720 656c 656d *note ring elem │ │ │ │ +00013230: 656e 743a 2028 4d61 6361 756c 6179 3244 ent: (Macaulay2D │ │ │ │ +00013240: 6f63 2952 696e 6745 6c65 6d65 6e74 2c2c oc)RingElement,, │ │ │ │ +00013250: 2061 2070 6174 6820 7661 7269 6162 6c65 a path variable │ │ │ │ +00013260: 0a20 2020 2020 202a 2050 2c20 6120 2a6e . * P, a *n │ │ │ │ +00013270: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +00013280: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ +00013290: 206c 6973 7420 6f66 206f 7074 696f 6e73 list of options │ │ │ │ +000132a0: 2074 6861 7420 7365 7420 7468 650a 2020 that set the. │ │ │ │ +000132b0: 2020 2020 2020 7061 7261 6d65 7465 7273 parameters │ │ │ │ +000132c0: 0a20 2020 2020 202a 2048 2c20 6120 2a6e . * H, a *n │ │ │ │ +000132d0: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +000132e0: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ +000132f0: 206c 6973 7420 6f66 2070 6f6c 796e 6f6d list of polynom │ │ │ │ +00013300: 6961 6c73 2074 6861 7420 6465 6669 6e65 ials that define │ │ │ │ +00013310: 0a20 2020 2020 2020 2074 6865 2068 6f6d . the hom │ │ │ │ +00013320: 6f74 6f70 790a 2020 2020 2020 2a20 5331 otopy. * S1 │ │ │ │ +00013330: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +00013340: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +00013350: 7374 2c2c 2061 206c 6973 7420 6f66 2073 st,, a list of s │ │ │ │ +00013360: 6f6c 7574 696f 6e73 2074 6f20 7468 6520 olutions to the │ │ │ │ +00013370: 7374 6172 740a 2020 2020 2020 2020 7379 start. sy │ │ │ │ +00013380: 7374 656d 0a20 202a 202a 6e6f 7465 204f stem. * *note O │ │ │ │ +00013390: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +000133a0: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +000133b0: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +000133c0: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +000133d0: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f ts,:. * *no │ │ │ │ +000133e0: 7465 2041 6666 5661 7269 6162 6c65 4772 te AffVariableGr │ │ │ │ +000133f0: 6f75 703a 2056 6172 6961 626c 6520 6772 oup: Variable gr │ │ │ │ +00013400: 6f75 7073 2c20 3d3e 202e 2e2e 2c20 6465 oups, => ..., de │ │ │ │ +00013410: 6661 756c 7420 7661 6c75 6520 7b7d 2c20 fault value {}, │ │ │ │ +00013420: 616e 0a20 2020 2020 2020 206f 7074 696f an. optio │ │ │ │ +00013430: 6e20 746f 2067 726f 7570 2076 6172 6961 n to group varia │ │ │ │ +00013440: 626c 6573 2061 6e64 2075 7365 206d 756c bles and use mul │ │ │ │ +00013450: 7469 686f 6d6f 6765 6e65 6f75 7320 686f tihomogeneous ho │ │ │ │ +00013460: 6d6f 746f 7069 6573 0a20 2020 2020 202a motopies. * │ │ │ │ +00013470: 2042 2743 6f6e 7374 616e 7473 2028 6d69 B'Constants (mi │ │ │ │ +00013480: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +00013490: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +000134a0: 6175 6c74 2076 616c 7565 207b 7d2c 200a ault value {}, . │ │ │ │ +000134b0: 2020 2020 2020 2a20 4227 4675 6e63 7469 * B'Functi │ │ │ │ +000134c0: 6f6e 7320 286d 6973 7369 6e67 2064 6f63 ons (missing doc │ │ │ │ +000134d0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +000134e0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +000134f0: 6520 7b7d 2c20 0a20 2020 2020 202a 2042 e {}, . * B │ │ │ │ +00013500: 6572 7469 6e69 496e 7075 7443 6f6e 6669 ertiniInputConfi │ │ │ │ +00013510: 6775 7261 7469 6f6e 2028 6d69 7373 696e guration (missin │ │ │ │ +00013520: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +00013530: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00013540: 2076 616c 7565 0a20 2020 2020 2020 207b value. { │ │ │ │ +00013550: 7d2c 0a20 2020 2020 202a 2048 6f6d 5661 },. * HomVa │ │ │ │ +00013560: 7269 6162 6c65 4772 6f75 7020 286d 6973 riableGroup (mis │ │ │ │ +00013570: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ +00013580: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ +00013590: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20 ult value {}, . │ │ │ │ +000135a0: 2020 2020 202a 204d 3250 7265 6369 7369 * M2Precisi │ │ │ │ +000135b0: 6f6e 2028 6d69 7373 696e 6720 646f 6375 on (missing docu │ │ │ │ +000135c0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ +000135d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +000135e0: 2035 332c 200a 2020 2020 2020 2a20 4f75 53, . * Ou │ │ │ │ +000135f0: 7470 7574 5374 796c 6520 286d 6973 7369 tputStyle (missi │ │ │ │ +00013600: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00013610: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ +00013620: 7420 7661 6c75 6520 224f 7574 506f 696e t value "OutPoin │ │ │ │ +00013630: 7473 222c 200a 2020 2020 2020 2a20 5261 ts", . * Ra │ │ │ │ +00013640: 6e64 6f6d 436f 6d70 6c65 7820 286d 6973 ndomComplex (mis │ │ │ │ +00013650: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ +00013660: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ +00013670: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20 ult value {}, . │ │ │ │ +00013680: 2020 2020 202a 2052 616e 646f 6d52 6561 * RandomRea │ │ │ │ +00013690: 6c20 286d 6973 7369 6e67 2064 6f63 756d l (missing docum │ │ │ │ +000136a0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +000136b0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000136c0: 7b7d 2c20 0a20 2020 2020 202a 202a 6e6f {}, . * *no │ │ │ │ +000136d0: 7465 2054 6f70 4469 7265 6374 6f72 793a te TopDirectory: │ │ │ │ +000136e0: 2054 6f70 4469 7265 6374 6f72 792c 203d TopDirectory, = │ │ │ │ +000136f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00013700: 616c 7565 0a20 2020 2020 2020 2022 2f74 alue. "/t │ │ │ │ +00013710: 6d70 2f4d 322d 3132 3633 3834 2d30 2f30 mp/M2-126384-0/0 │ │ │ │ +00013720: 222c 204f 7074 696f 6e20 746f 2063 6861 ", Option to cha │ │ │ │ +00013730: 6e67 6520 6469 7265 6374 6f72 7920 666f nge directory fo │ │ │ │ +00013740: 7220 6669 6c65 2073 746f 7261 6765 2e0a r file storage.. │ │ │ │ +00013750: 2020 2020 2020 2a20 2a6e 6f74 6520 5665 * *note Ve │ │ │ │ +00013760: 7262 6f73 653a 2062 6572 7469 6e69 5472 rbose: bertiniTr │ │ │ │ +00013770: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70 ackHomotopy_lp_p │ │ │ │ +00013780: 645f 7064 5f70 645f 636d 5665 7262 6f73 d_pd_pd_cmVerbos │ │ │ │ +00013790: 653d 3e5f 7064 5f70 645f 7064 5f72 700a e=>_pd_pd_pd_rp. │ │ │ │ +000137a0: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e , => ... │ │ │ │ +000137b0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000137c0: 6661 6c73 652c 204f 7074 696f 6e20 746f false, Option to │ │ │ │ +000137d0: 2073 696c 656e 6365 2061 6464 6974 696f silence additio │ │ │ │ +000137e0: 6e61 6c20 6f75 7470 7574 0a20 202a 204f nal output. * O │ │ │ │ +000137f0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00013800: 5330 2c20 6120 2a6e 6f74 6520 6c69 7374 S0, a *note list │ │ │ │ +00013810: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00013820: 4c69 7374 2c2c 2061 206c 6973 7420 6f66 List,, a list of │ │ │ │ +00013830: 2073 6f6c 7574 696f 6e73 2074 6f20 7468 solutions to th │ │ │ │ +00013840: 650a 2020 2020 2020 2020 7461 7267 6574 e. target │ │ │ │ +00013850: 2073 7973 7465 6d0a 0a44 6573 6372 6970 system..Descrip │ │ │ │ +00013860: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00013870: 0a0a 5468 6973 206d 6574 686f 6420 6361 ..This method ca │ │ │ │ +00013880: 6c6c 7320 4265 7274 696e 6920 746f 2074 lls Bertini to t │ │ │ │ +00013890: 7261 636b 2061 2075 7365 722d 6465 6669 rack a user-defi │ │ │ │ +000138a0: 6e65 6420 686f 6d6f 746f 7079 2e20 2054 ned homotopy. T │ │ │ │ +000138b0: 6865 2075 7365 7220 6e65 6564 7320 746f he user needs to │ │ │ │ +000138c0: 0a73 7065 6369 6679 2074 6865 2068 6f6d .specify the hom │ │ │ │ +000138d0: 6f74 6f70 7920 482c 2074 6865 2070 6174 otopy H, the pat │ │ │ │ +000138e0: 6820 7661 7269 6162 6c65 2074 2c20 616e h variable t, an │ │ │ │ +000138f0: 6420 6120 6c69 7374 206f 6620 7374 6172 d a list of star │ │ │ │ +00013900: 7420 736f 6c75 7469 6f6e 7320 5331 2e0a t solutions S1.. │ │ │ │ +00013910: 4265 7274 696e 6920 2831 2920 7772 6974 Bertini (1) writ │ │ │ │ +00013920: 6573 2074 6865 2068 6f6d 6f74 6f70 7920 es the homotopy │ │ │ │ +00013930: 616e 6420 7374 6172 7420 736f 6c75 7469 and start soluti │ │ │ │ +00013940: 6f6e 7320 746f 2074 656d 706f 7261 7279 ons to temporary │ │ │ │ +00013950: 2066 696c 6573 2c20 2832 290a 696e 766f files, (2).invo │ │ │ │ +00013960: 6b65 7320 4265 7274 696e 6927 7320 736f kes Bertini's so │ │ │ │ +00013970: 6c76 6572 2077 6974 6820 636f 6e66 6967 lver with config │ │ │ │ +00013980: 7572 6174 696f 6e20 6b65 7977 6f72 6420 uration keyword │ │ │ │ +00013990: 5573 6572 486f 6d6f 746f 7079 203d 3e20 UserHomotopy => │ │ │ │ +000139a0: 322c 2028 3329 0a73 746f 7265 7320 7468 2, (3).stores th │ │ │ │ +000139b0: 6520 6f75 7470 7574 206f 6620 4265 7274 e output of Bert │ │ │ │ +000139c0: 696e 6920 696e 2061 2074 656d 706f 7261 ini in a tempora │ │ │ │ +000139d0: 7279 2066 696c 652c 2061 6e64 2028 3429 ry file, and (4) │ │ │ │ +000139e0: 2070 6172 7365 7320 6120 6d61 6368 696e parses a machin │ │ │ │ +000139f0: 650a 7265 6164 6162 6c65 2066 696c 6520 e.readable file │ │ │ │ +00013a00: 746f 206f 7574 7075 7420 6120 6c69 7374 to output a list │ │ │ │ +00013a10: 206f 6620 736f 6c75 7469 6f6e 732e 0a0a of solutions... │ │ │ │ +00013a20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013a60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00013a70: 5220 3d20 4343 5b78 2c61 2c74 5d3b 202d R = CC[x,a,t]; - │ │ │ │ -00013a80: 2d20 696e 636c 7564 6520 7468 6520 7061 - include the pa │ │ │ │ -00013a90: 7468 2076 6172 6961 626c 6520 696e 2074 th variable in t │ │ │ │ -00013aa0: 6865 2072 696e 6720 2020 2020 2020 2020 he ring │ │ │ │ -00013ab0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00013a70: 3a20 5220 3d20 4343 5b78 2c61 2c74 5d3b : R = CC[x,a,t]; │ │ │ │ +00013a80: 202d 2d20 696e 636c 7564 6520 7468 6520 -- include the │ │ │ │ +00013a90: 7061 7468 2076 6172 6961 626c 6520 696e path variable in │ │ │ │ +00013aa0: 2074 6865 2072 696e 6720 2020 2020 2020 the ring │ │ │ │ +00013ab0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00013ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b00: 2b0a 7c69 3220 3a20 4820 3d20 7b20 2878 +.|i2 : H = { (x │ │ │ │ -00013b10: 5e32 2d31 292a 6120 2b20 2878 5e32 2d32 ^2-1)*a + (x^2-2 │ │ │ │ -00013b20: 292a 2831 2d61 297d 3b20 2020 2020 2020 )*(1-a)}; │ │ │ │ +00013b00: 2d2d 2b0a 7c69 3220 3a20 4820 3d20 7b20 --+.|i2 : H = { │ │ │ │ +00013b10: 2878 5e32 2d31 292a 6120 2b20 2878 5e32 (x^2-1)*a + (x^2 │ │ │ │ +00013b20: 2d32 292a 2831 2d61 297d 3b20 2020 2020 -2)*(1-a)}; │ │ │ │ 00013b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013b50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00013ba0: 736f 6c31 203d 2070 6f69 6e74 207b 7b31 sol1 = point {{1 │ │ │ │ -00013bb0: 7d7d 3b20 2020 2020 2020 2020 2020 2020 }}; │ │ │ │ +00013b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +00013ba0: 3a20 736f 6c31 203d 2070 6f69 6e74 207b : sol1 = point { │ │ │ │ +00013bb0: 7b31 7d7d 3b20 2020 2020 2020 2020 2020 {1}}; │ │ │ │ 00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013be0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00013be0: 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c30: 2b0a 7c69 3420 3a20 736f 6c32 203d 2070 +.|i4 : sol2 = p │ │ │ │ -00013c40: 6f69 6e74 207b 7b2d 317d 7d3b 2020 2020 oint {{-1}}; │ │ │ │ +00013c30: 2d2d 2b0a 7c69 3420 3a20 736f 6c32 203d --+.|i4 : sol2 = │ │ │ │ +00013c40: 2070 6f69 6e74 207b 7b2d 317d 7d3b 2020 point {{-1}}; │ │ │ │ 00013c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013c70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00013c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013c70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013c80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013cc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -00013cd0: 5331 3d20 7b20 736f 6c31 2c20 736f 6c32 S1= { sol1, sol2 │ │ │ │ -00013ce0: 2020 7d3b 2d2d 736f 6c75 7469 6f6e 7320 };--solutions │ │ │ │ -00013cf0: 746f 2048 2077 6865 6e20 743d 3120 2020 to H when t=1 │ │ │ │ +00013cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +00013cd0: 3a20 5331 3d20 7b20 736f 6c31 2c20 736f : S1= { sol1, so │ │ │ │ +00013ce0: 6c32 2020 7d3b 2d2d 736f 6c75 7469 6f6e l2 };--solution │ │ │ │ +00013cf0: 7320 746f 2048 2077 6865 6e20 743d 3120 s to H when t=1 │ │ │ │ 00013d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013d10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00013d10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013d60: 2b0a 7c69 3620 3a20 5330 203d 2062 6572 +.|i6 : S0 = ber │ │ │ │ -00013d70: 7469 6e69 5573 6572 486f 6d6f 746f 7079 tiniUserHomotopy │ │ │ │ -00013d80: 2028 742c 7b61 3d3e 747d 2c20 482c 2053 (t,{a=>t}, H, S │ │ │ │ -00013d90: 3129 202d 2d73 6f6c 7574 696f 6e73 2074 1) --solutions t │ │ │ │ -00013da0: 6f20 4820 7768 656e 2074 3d30 7c0a 7c20 o H when t=0|.| │ │ │ │ -00013db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d60: 2d2d 2b0a 7c69 3620 3a20 5330 203d 2062 --+.|i6 : S0 = b │ │ │ │ +00013d70: 6572 7469 6e69 5573 6572 486f 6d6f 746f ertiniUserHomoto │ │ │ │ +00013d80: 7079 2028 742c 7b61 3d3e 747d 2c20 482c py (t,{a=>t}, H, │ │ │ │ +00013d90: 2053 3129 202d 2d73 6f6c 7574 696f 6e73 S1) --solutions │ │ │ │ +00013da0: 2074 6f20 4820 7768 656e 2074 3d30 7c0a to H when t=0|. │ │ │ │ +00013db0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00013dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013df0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -00013e00: 7b7b 312e 3431 3432 317d 2c20 7b2d 312e {{1.41421}, {-1. │ │ │ │ -00013e10: 3431 3432 317d 7d20 2020 2020 2020 2020 41421}} │ │ │ │ +00013df0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00013e00: 3d20 7b7b 312e 3431 3432 317d 2c20 7b2d = {{1.41421}, {- │ │ │ │ +00013e10: 312e 3431 3432 317d 7d20 2020 2020 2020 1.41421}} │ │ │ │ 00013e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013e40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00013e40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013e90: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020 |.|o6 : List │ │ │ │ +00013e90: 2020 7c0a 7c6f 3620 3a20 4c69 7374 2020 |.|o6 : List │ │ │ │ 00013ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013ed0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013ee0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ -00013f30: 7065 656b 2053 305f 3020 2020 2020 2020 peek S0_0 │ │ │ │ +00013f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ +00013f30: 3a20 7065 656b 2053 305f 3020 2020 2020 : peek S0_0 │ │ │ │ 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 2020 │ │ │ │ -00013f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00013f70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013fc0: 7c0a 7c6f 3720 3d20 506f 696e 747b 6361 |.|o7 = Point{ca │ │ │ │ -00013fd0: 6368 6520 3d3e 2043 6163 6865 5461 626c che => CacheTabl │ │ │ │ -00013fe0: 657b 2e2e 2e31 342e 2e2e 7d7d 2020 2020 e{...14...}} │ │ │ │ +00013fc0: 2020 7c0a 7c6f 3720 3d20 506f 696e 747b |.|o7 = Point{ │ │ │ │ +00013fd0: 6361 6368 6520 3d3e 2043 6163 6865 5461 cache => CacheTa │ │ │ │ +00013fe0: 626c 657b 2e2e 2e31 342e 2e2e 7d7d 2020 ble{...14...}} │ │ │ │ 00013ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014000: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014010: 2020 2020 2020 2020 2020 436f 6f72 6469 Coordi │ │ │ │ -00014020: 6e61 7465 7320 3d3e 207b 312e 3431 3432 nates => {1.4142 │ │ │ │ -00014030: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +00014000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00014010: 7c20 2020 2020 2020 2020 2020 436f 6f72 | Coor │ │ │ │ +00014020: 6469 6e61 7465 7320 3d3e 207b 312e 3431 dinates => {1.41 │ │ │ │ +00014030: 3432 317d 2020 2020 2020 2020 2020 2020 421} │ │ │ │ 00014040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014050: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00014050: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00014060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000140a0: 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d ----+.+--------- │ │ │ │ +000140a0: 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d ------+.+------- │ │ │ │ 000140b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000140c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000140d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000140e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000140f0: 2d2d 2d2d 2b0a 7c69 3820 3a20 523d 4343 ----+.|i8 : R=CC │ │ │ │ -00014100: 5b78 2c79 2c74 2c61 5d3b 202d 2d20 696e [x,y,t,a]; -- in │ │ │ │ -00014110: 636c 7564 6520 7468 6520 7061 7468 2076 clude the path v │ │ │ │ -00014120: 6172 6961 626c 6520 696e 2074 6865 2072 ariable in the r │ │ │ │ -00014130: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00014140: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000140f0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 523d ------+.|i8 : R= │ │ │ │ +00014100: 4343 5b78 2c79 2c74 2c61 5d3b 202d 2d20 CC[x,y,t,a]; -- │ │ │ │ +00014110: 696e 636c 7564 6520 7468 6520 7061 7468 include the path │ │ │ │ +00014120: 2076 6172 6961 626c 6520 696e 2074 6865 variable in the │ │ │ │ +00014130: 2072 696e 6720 2020 2020 2020 2020 2020 ring │ │ │ │ +00014140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00014150: 2d2d 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 2b0a 7c69 3920 3a20 6631 3d28 ----+.|i9 : f1=( │ │ │ │ -000141a0: 785e 322d 795e 3229 3b20 2020 2020 2020 x^2-y^2); │ │ │ │ +00014190: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 6631 ------+.|i9 : f1 │ │ │ │ +000141a0: 3d28 785e 322d 795e 3229 3b20 2020 2020 =(x^2-y^2); │ │ │ │ 000141b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000141e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000141e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000141f0: 2d2d 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 2b0a 7c69 3130 203a 2066 323d ----+.|i10 : f2= │ │ │ │ -00014240: 2832 2a78 5e32 2d33 2a78 2a79 2b35 2a79 (2*x^2-3*x*y+5*y │ │ │ │ -00014250: 5e32 293b 2020 2020 2020 2020 2020 2020 ^2); │ │ │ │ +00014230: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2066 ------+.|i10 : f │ │ │ │ +00014240: 323d 2832 2a78 5e32 2d33 2a78 2a79 2b35 2=(2*x^2-3*x*y+5 │ │ │ │ +00014250: 2a79 5e32 293b 2020 2020 2020 2020 2020 *y^2); │ │ │ │ 00014260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014280: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014280: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00014290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000142a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000142b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000142c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000142d0: 2d2d 2d2d 2b0a 7c69 3131 203a 2048 203d ----+.|i11 : H = │ │ │ │ -000142e0: 207b 2066 312a 6120 2b20 6632 2a28 312d { f1*a + f2*(1- │ │ │ │ -000142f0: 6129 7d3b 202d 2d48 2069 7320 6120 6c69 a)}; --H is a li │ │ │ │ -00014300: 7374 206f 6620 706f 6c79 6e6f 6d69 616c st of polynomial │ │ │ │ -00014310: 7320 696e 2078 2c79 2c74 2020 2020 2020 s in x,y,t │ │ │ │ -00014320: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000142d0: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2048 ------+.|i11 : H │ │ │ │ +000142e0: 203d 207b 2066 312a 6120 2b20 6632 2a28 = { f1*a + f2*( │ │ │ │ +000142f0: 312d 6129 7d3b 202d 2d48 2069 7320 6120 1-a)}; --H is a │ │ │ │ +00014300: 6c69 7374 206f 6620 706f 6c79 6e6f 6d69 list of polynomi │ │ │ │ +00014310: 616c 7320 696e 2078 2c79 2c74 2020 2020 als in x,y,t │ │ │ │ +00014320: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00014330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014370: 2d2d 2d2d 2b0a 7c69 3132 203a 2073 6f6c ----+.|i12 : sol │ │ │ │ -00014380: 313d 2020 2020 706f 696e 747b 7b31 2c31 1= point{{1,1 │ │ │ │ -00014390: 7d7d 2d2d 7b7b 782c 797d 7d20 636f 6f72 }}--{{x,y}} coor │ │ │ │ -000143a0: 6469 6e61 7465 7320 2020 2020 2020 2020 dinates │ │ │ │ +00014370: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2073 ------+.|i12 : s │ │ │ │ +00014380: 6f6c 313d 2020 2020 706f 696e 747b 7b31 ol1= point{{1 │ │ │ │ +00014390: 2c31 7d7d 2d2d 7b7b 782c 797d 7d20 636f ,1}}--{{x,y}} co │ │ │ │ +000143a0: 6f72 6469 6e61 7465 7320 2020 2020 2020 ordinates │ │ │ │ 000143b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000143c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000143c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000143d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000143e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000143f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014410: 2020 2020 7c0a 7c6f 3132 203d 2073 6f6c |.|o12 = sol │ │ │ │ -00014420: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00014410: 2020 2020 2020 7c0a 7c6f 3132 203d 2073 |.|o12 = s │ │ │ │ +00014420: 6f6c 3120 2020 2020 2020 2020 2020 2020 ol1 │ │ │ │ 00014430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014460: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014460: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144b0: 2020 2020 7c0a 7c6f 3132 203a 2050 6f69 |.|o12 : Poi │ │ │ │ -000144c0: 6e74 2020 2020 2020 2020 2020 2020 2020 nt │ │ │ │ +000144b0: 2020 2020 2020 7c0a 7c6f 3132 203a 2050 |.|o12 : P │ │ │ │ +000144c0: 6f69 6e74 2020 2020 2020 2020 2020 2020 oint │ │ │ │ 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 2020 │ │ │ │ -00014500: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014500: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00014510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014550: 2d2d 2d2d 2b0a 7c69 3133 203a 2073 6f6c ----+.|i13 : sol │ │ │ │ -00014560: 323d 2020 2020 706f 696e 747b 7b20 2d31 2= point{{ -1 │ │ │ │ -00014570: 2c31 7d7d 2020 2020 2020 2020 2020 2020 ,1}} │ │ │ │ +00014550: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2073 ------+.|i13 : s │ │ │ │ +00014560: 6f6c 323d 2020 2020 706f 696e 747b 7b20 ol2= point{{ │ │ │ │ +00014570: 2d31 2c31 7d7d 2020 2020 2020 2020 2020 -1,1}} │ │ │ │ 00014580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000145a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000145a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 2020 │ │ │ │ -000145f0: 2020 2020 7c0a 7c6f 3133 203d 2073 6f6c |.|o13 = sol │ │ │ │ -00014600: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000145f0: 2020 2020 2020 7c0a 7c6f 3133 203d 2073 |.|o13 = s │ │ │ │ +00014600: 6f6c 3220 2020 2020 2020 2020 2020 2020 ol2 │ │ │ │ 00014610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014640: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014640: 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ -00014690: 2020 2020 7c0a 7c6f 3133 203a 2050 6f69 |.|o13 : Poi │ │ │ │ -000146a0: 6e74 2020 2020 2020 2020 2020 2020 2020 nt │ │ │ │ +00014690: 2020 2020 2020 7c0a 7c6f 3133 203a 2050 |.|o13 : P │ │ │ │ +000146a0: 6f69 6e74 2020 2020 2020 2020 2020 2020 oint │ │ │ │ 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000146e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000146f0: 2d2d 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 2b0a 7c69 3134 203a 2053 313d ----+.|i14 : S1= │ │ │ │ -00014740: 7b73 6f6c 312c 736f 6c32 7d2d 2d73 6f6c {sol1,sol2}--sol │ │ │ │ -00014750: 7574 696f 6e73 2074 6f20 4820 7768 656e utions to H when │ │ │ │ -00014760: 2074 3d31 2020 2020 2020 2020 2020 2020 t=1 │ │ │ │ +00014730: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2053 ------+.|i14 : S │ │ │ │ +00014740: 313d 7b73 6f6c 312c 736f 6c32 7d2d 2d73 1={sol1,sol2}--s │ │ │ │ +00014750: 6f6c 7574 696f 6e73 2074 6f20 4820 7768 olutions to H wh │ │ │ │ +00014760: 656e 2074 3d31 2020 2020 2020 2020 2020 en t=1 │ │ │ │ 00014770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014780: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014780: 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ -000147d0: 2020 2020 7c0a 7c6f 3134 203d 207b 736f |.|o14 = {so │ │ │ │ -000147e0: 6c31 2c20 736f 6c32 7d20 2020 2020 2020 l1, sol2} │ │ │ │ +000147d0: 2020 2020 2020 7c0a 7c6f 3134 203d 207b |.|o14 = { │ │ │ │ +000147e0: 736f 6c31 2c20 736f 6c32 7d20 2020 2020 sol1, sol2} │ │ │ │ 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014820: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 2020 │ │ │ │ -00014870: 2020 2020 7c0a 7c6f 3134 203a 204c 6973 |.|o14 : Lis │ │ │ │ -00014880: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00014870: 2020 2020 2020 7c0a 7c6f 3134 203a 204c |.|o14 : L │ │ │ │ +00014880: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00014890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000148c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000148c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000148d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014910: 2d2d 2d2d 2b0a 7c69 3135 203a 2053 303d ----+.|i15 : S0= │ │ │ │ -00014920: 6265 7274 696e 6955 7365 7248 6f6d 6f74 bertiniUserHomot │ │ │ │ -00014930: 6f70 7928 742c 7b61 3d3e 747d 2c20 482c opy(t,{a=>t}, H, │ │ │ │ -00014940: 2053 312c 2048 6f6d 5661 7269 6162 6c65 S1, HomVariable │ │ │ │ -00014950: 4772 6f75 703d 3e7b 782c 797d 2920 2020 Group=>{x,y}) │ │ │ │ -00014960: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014910: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2053 ------+.|i15 : S │ │ │ │ +00014920: 303d 6265 7274 696e 6955 7365 7248 6f6d 0=bertiniUserHom │ │ │ │ +00014930: 6f74 6f70 7928 742c 7b61 3d3e 747d 2c20 otopy(t,{a=>t}, │ │ │ │ +00014940: 482c 2053 312c 2048 6f6d 5661 7269 6162 H, S1, HomVariab │ │ │ │ +00014950: 6c65 4772 6f75 703d 3e7b 782c 797d 2920 leGroup=>{x,y}) │ │ │ │ +00014960: 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ -000149b0: 2020 2020 7c0a 7c6f 3135 203d 207b 7b31 |.|o15 = {{1 │ │ │ │ -000149c0: 2c20 2e33 2b2e 3535 3637 3736 2a69 697d , .3+.556776*ii} │ │ │ │ -000149d0: 2c20 7b31 2c20 2e33 2d2e 3535 3637 3736 , {1, .3-.556776 │ │ │ │ -000149e0: 2a69 697d 7d20 2020 2020 2020 2020 2020 *ii}} │ │ │ │ +000149b0: 2020 2020 2020 7c0a 7c6f 3135 203d 207b |.|o15 = { │ │ │ │ +000149c0: 7b31 2c20 2e33 2b2e 3535 3637 3736 2a69 {1, .3+.556776*i │ │ │ │ +000149d0: 697d 2c20 7b31 2c20 2e33 2d2e 3535 3637 i}, {1, .3-.5567 │ │ │ │ +000149e0: 3736 2a69 697d 7d20 2020 2020 2020 2020 76*ii}} │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014a00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 2020 │ │ │ │ -00014a50: 2020 2020 7c0a 7c6f 3135 203a 204c 6973 |.|o15 : Lis │ │ │ │ -00014a60: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00014a50: 2020 2020 2020 7c0a 7c6f 3135 203a 204c |.|o15 : L │ │ │ │ +00014a60: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00014a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014aa0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ +00014aa0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014af0: 2d2d 2d2d 7c0a 7c2d 2d73 6f6c 7574 696f ----|.|--solutio │ │ │ │ -00014b00: 6e73 2074 6f20 4820 7768 656e 2074 3d30 ns to H when t=0 │ │ │ │ -00014b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014af0: 2d2d 2d2d 2d2d 7c0a 7c2d 2d73 6f6c 7574 ------|.|--solut │ │ │ │ +00014b00: 696f 6e73 2074 6f20 4820 7768 656e 2074 ions to H when t │ │ │ │ +00014b10: 3d30 2020 2020 2020 2020 2020 2020 2020 =0 │ │ │ │ 00014b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014b40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00014b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014b90: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -00014ba0: 7365 2062 6572 7469 6e69 5573 6572 486f se bertiniUserHo │ │ │ │ -00014bb0: 6d6f 746f 7079 3a0a 3d3d 3d3d 3d3d 3d3d motopy:.======== │ │ │ │ +00014b90: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ +00014ba0: 2075 7365 2062 6572 7469 6e69 5573 6572 use bertiniUser │ │ │ │ +00014bb0: 486f 6d6f 746f 7079 3a0a 3d3d 3d3d 3d3d Homotopy:.====== │ │ │ │ 00014bc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00014bd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2262 ========.. * "b │ │ │ │ -00014be0: 6572 7469 6e69 5573 6572 486f 6d6f 746f ertiniUserHomoto │ │ │ │ -00014bf0: 7079 2854 6869 6e67 2c4c 6973 742c 4c69 py(Thing,List,Li │ │ │ │ -00014c00: 7374 2c4c 6973 7429 220a 0a46 6f72 2074 st,List)"..For t │ │ │ │ -00014c10: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +00014bd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00014be0: 2262 6572 7469 6e69 5573 6572 486f 6d6f "bertiniUserHomo │ │ │ │ +00014bf0: 746f 7079 2854 6869 6e67 2c4c 6973 742c topy(Thing,List, │ │ │ │ +00014c00: 4c69 7374 2c4c 6973 7429 220a 0a46 6f72 List,List)"..For │ │ │ │ +00014c10: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ 00014c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00014c30: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00014c40: 7465 2062 6572 7469 6e69 5573 6572 486f te bertiniUserHo │ │ │ │ -00014c50: 6d6f 746f 7079 3a20 6265 7274 696e 6955 motopy: bertiniU │ │ │ │ -00014c60: 7365 7248 6f6d 6f74 6f70 792c 2069 7320 serHomotopy, is │ │ │ │ -00014c70: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ -00014c80: 756e 6374 696f 6e20 7769 7468 206f 7074 unction with opt │ │ │ │ -00014c90: 696f 6e73 3a20 284d 6163 6175 6c61 7932 ions: (Macaulay2 │ │ │ │ -00014ca0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -00014cb0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ -00014cc0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00014c30: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00014c40: 6e6f 7465 2062 6572 7469 6e69 5573 6572 note bertiniUser │ │ │ │ +00014c50: 486f 6d6f 746f 7079 3a20 6265 7274 696e Homotopy: bertin │ │ │ │ +00014c60: 6955 7365 7248 6f6d 6f74 6f70 792c 2069 iUserHomotopy, i │ │ │ │ +00014c70: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +00014c80: 0a66 756e 6374 696f 6e20 7769 7468 206f .function with o │ │ │ │ +00014c90: 7074 696f 6e73 3a20 284d 6163 6175 6c61 ptions: (Macaula │ │ │ │ +00014ca0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +00014cb0: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +00014cc0: 2e0a 0a2d 2d2d 2d2d 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: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00014d20: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00014d30: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00014d40: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00014d50: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ -00014d60: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00014d70: 6163 6b61 6765 732f 4265 7274 696e 692e ackages/Bertini. │ │ │ │ -00014d80: 6d32 3a0a 3238 3933 3a30 2e0a 1f0a 4669 m2:.2893:0....Fi │ │ │ │ -00014d90: 6c65 3a20 4265 7274 696e 692e 696e 666f le: Bertini.info │ │ │ │ -00014da0: 2c20 4e6f 6465 3a20 6265 7274 696e 695a , Node: bertiniZ │ │ │ │ -00014db0: 6572 6f44 696d 536f 6c76 652c 204e 6578 eroDimSolve, Nex │ │ │ │ -00014dc0: 743a 2043 6f70 7942 2746 696c 652c 2050 t: CopyB'File, P │ │ │ │ -00014dd0: 7265 763a 2062 6572 7469 6e69 5573 6572 rev: bertiniUser │ │ │ │ -00014de0: 486f 6d6f 746f 7079 2c20 5570 3a20 546f Homotopy, Up: To │ │ │ │ -00014df0: 700a 0a62 6572 7469 6e69 5a65 726f 4469 p..bertiniZeroDi │ │ │ │ -00014e00: 6d53 6f6c 7665 202d 2d20 6120 6d61 696e mSolve -- a main │ │ │ │ -00014e10: 206d 6574 686f 6420 746f 2073 6f6c 7665 method to solve │ │ │ │ -00014e20: 2061 207a 6572 6f2d 6469 6d65 6e73 696f a zero-dimensio │ │ │ │ -00014e30: 6e61 6c20 7379 7374 656d 206f 6620 6571 nal system of eq │ │ │ │ -00014e40: 7561 7469 6f6e 730a 2a2a 2a2a 2a2a 2a2a uations.******** │ │ │ │ +00014d10: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00014d20: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00014d30: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00014d40: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00014d50: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +00014d60: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00014d70: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ +00014d80: 692e 6d32 3a0a 3238 3933 3a30 2e0a 1f0a i.m2:.2893:0.... │ │ │ │ +00014d90: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ +00014da0: 666f 2c20 4e6f 6465 3a20 6265 7274 696e fo, Node: bertin │ │ │ │ +00014db0: 695a 6572 6f44 696d 536f 6c76 652c 204e iZeroDimSolve, N │ │ │ │ +00014dc0: 6578 743a 2043 6f70 7942 2746 696c 652c ext: CopyB'File, │ │ │ │ +00014dd0: 2050 7265 763a 2062 6572 7469 6e69 5573 Prev: bertiniUs │ │ │ │ +00014de0: 6572 486f 6d6f 746f 7079 2c20 5570 3a20 erHomotopy, Up: │ │ │ │ +00014df0: 546f 700a 0a62 6572 7469 6e69 5a65 726f Top..bertiniZero │ │ │ │ +00014e00: 4469 6d53 6f6c 7665 202d 2d20 6120 6d61 DimSolve -- a ma │ │ │ │ +00014e10: 696e 206d 6574 686f 6420 746f 2073 6f6c in method to sol │ │ │ │ +00014e20: 7665 2061 207a 6572 6f2d 6469 6d65 6e73 ve a zero-dimens │ │ │ │ +00014e30: 696f 6e61 6c20 7379 7374 656d 206f 6620 ional system of │ │ │ │ +00014e40: 6571 7561 7469 6f6e 730a 2a2a 2a2a 2a2a equations.****** │ │ │ │ 00014e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00014e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00014e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00014e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00014e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -00014ea0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00014eb0: 2020 5320 3d20 6265 7274 696e 695a 6572 S = bertiniZer │ │ │ │ -00014ec0: 6f44 696d 536f 6c76 6520 460a 2020 2020 oDimSolve F. │ │ │ │ -00014ed0: 2020 2020 5320 3d20 6265 7274 696e 695a S = bertiniZ │ │ │ │ -00014ee0: 6572 6f44 696d 536f 6c76 6520 490a 2020 eroDimSolve I. │ │ │ │ -00014ef0: 2020 2020 2020 5320 3d20 6265 7274 696e S = bertin │ │ │ │ -00014f00: 695a 6572 6f44 696d 536f 6c76 6528 492c iZeroDimSolve(I, │ │ │ │ -00014f10: 2055 7365 5265 6765 6e65 7261 7469 6f6e UseRegeneration │ │ │ │ -00014f20: 3d3e 3129 0a20 202a 2049 6e70 7574 733a =>1). * Inputs: │ │ │ │ -00014f30: 0a20 2020 2020 202a 2046 2c20 6120 2a6e . * F, a *n │ │ │ │ -00014f40: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ -00014f50: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ -00014f60: 206c 6973 7420 6f66 2072 696e 6720 656c list of ring el │ │ │ │ -00014f70: 656d 656e 7473 2028 7379 7374 656d 0a20 ements (system. │ │ │ │ -00014f80: 2020 2020 2020 206e 6565 6420 6e6f 7420 need not │ │ │ │ -00014f90: 6265 2073 7175 6172 6529 0a20 2020 2020 be square). │ │ │ │ -00014fa0: 202a 2049 2c20 616e 202a 6e6f 7465 2069 * I, an *note i │ │ │ │ -00014fb0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ -00014fc0: 446f 6329 4964 6561 6c2c 2c20 616e 2069 Doc)Ideal,, an i │ │ │ │ -00014fd0: 6465 616c 2064 6566 696e 696e 6720 6120 deal defining a │ │ │ │ -00014fe0: 7661 7269 6574 790a 2020 2a20 2a6e 6f74 variety. * *not │ │ │ │ -00014ff0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ -00015000: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ -00015010: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ -00015020: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -00015030: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -00015040: 2a6e 6f74 6520 4166 6656 6172 6961 626c *note AffVariabl │ │ │ │ -00015050: 6547 726f 7570 3a20 5661 7269 6162 6c65 eGroup: Variable │ │ │ │ -00015060: 2067 726f 7570 732c 203d 3e20 2e2e 2e2c groups, => ..., │ │ │ │ -00015070: 2064 6566 6175 6c74 2076 616c 7565 207b default value { │ │ │ │ -00015080: 7d2c 2061 6e0a 2020 2020 2020 2020 6f70 }, an. op │ │ │ │ -00015090: 7469 6f6e 2074 6f20 6772 6f75 7020 7661 tion to group va │ │ │ │ -000150a0: 7269 6162 6c65 7320 616e 6420 7573 6520 riables and use │ │ │ │ -000150b0: 6d75 6c74 6968 6f6d 6f67 656e 656f 7573 multihomogeneous │ │ │ │ -000150c0: 2068 6f6d 6f74 6f70 6965 730a 2020 2020 homotopies. │ │ │ │ -000150d0: 2020 2a20 2a6e 6f74 6520 4227 436f 6e73 * *note B'Cons │ │ │ │ -000150e0: 7461 6e74 733a 2042 2743 6f6e 7374 616e tants: B'Constan │ │ │ │ -000150f0: 7473 2c20 3d3e 202e 2e2e 2c20 6465 6661 ts, => ..., defa │ │ │ │ -00015100: 756c 7420 7661 6c75 6520 7b7d 2c20 616e ult value {}, an │ │ │ │ -00015110: 206f 7074 696f 6e20 746f 0a20 2020 2020 option to. │ │ │ │ -00015120: 2020 2064 6573 6967 6e61 7465 2074 6865 designate the │ │ │ │ -00015130: 2063 6f6e 7374 616e 7473 2066 6f72 2061 constants for a │ │ │ │ -00015140: 2042 6572 7469 6e69 2049 6e70 7574 2066 Bertini Input f │ │ │ │ -00015150: 696c 650a 2020 2020 2020 2a20 4227 4675 ile. * B'Fu │ │ │ │ -00015160: 6e63 7469 6f6e 7320 286d 6973 7369 6e67 nctions (missing │ │ │ │ -00015170: 2064 6f63 756d 656e 7461 7469 6f6e 2920 documentation) │ │ │ │ -00015180: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00015190: 7661 6c75 6520 7b7d 2c20 0a20 2020 2020 value {}, . │ │ │ │ -000151a0: 202a 2042 6572 7469 6e69 496e 7075 7443 * BertiniInputC │ │ │ │ -000151b0: 6f6e 6669 6775 7261 7469 6f6e 2028 6d69 onfiguration (mi │ │ │ │ -000151c0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -000151d0: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -000151e0: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -000151f0: 2020 207b 7d2c 0a20 2020 2020 202a 202a {},. * * │ │ │ │ -00015200: 6e6f 7465 2048 6f6d 5661 7269 6162 6c65 note HomVariable │ │ │ │ -00015210: 4772 6f75 703a 2056 6172 6961 626c 6520 Group: Variable │ │ │ │ -00015220: 6772 6f75 7073 2c20 3d3e 202e 2e2e 2c20 groups, => ..., │ │ │ │ -00015230: 6465 6661 756c 7420 7661 6c75 6520 7b7d default value {} │ │ │ │ -00015240: 2c20 616e 0a20 2020 2020 2020 206f 7074 , an. opt │ │ │ │ -00015250: 696f 6e20 746f 2067 726f 7570 2076 6172 ion to group var │ │ │ │ -00015260: 6961 626c 6573 2061 6e64 2075 7365 206d iables and use m │ │ │ │ -00015270: 756c 7469 686f 6d6f 6765 6e65 6f75 7320 ultihomogeneous │ │ │ │ -00015280: 686f 6d6f 746f 7069 6573 0a20 2020 2020 homotopies. │ │ │ │ -00015290: 202a 2049 7350 726f 6a65 6374 6976 6520 * IsProjective │ │ │ │ -000152a0: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -000152b0: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -000152c0: 6465 6661 756c 7420 7661 6c75 6520 2d31 default value -1 │ │ │ │ -000152d0: 2c20 0a20 2020 2020 202a 204d 3250 7265 , . * M2Pre │ │ │ │ -000152e0: 6369 7369 6f6e 2028 6d69 7373 696e 6720 cision (missing │ │ │ │ -000152f0: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ -00015300: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00015310: 616c 7565 2035 332c 200a 2020 2020 2020 alue 53, . │ │ │ │ -00015320: 2a20 4e61 6d65 4d61 696e 4461 7461 4669 * NameMainDataFi │ │ │ │ -00015330: 6c65 2028 6d69 7373 696e 6720 646f 6375 le (missing docu │ │ │ │ -00015340: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -00015350: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00015360: 0a20 2020 2020 2020 2022 6d61 696e 5f64 . "main_d │ │ │ │ -00015370: 6174 6122 2c0a 2020 2020 2020 2a20 4e61 ata",. * Na │ │ │ │ -00015380: 6d65 536f 6c75 7469 6f6e 7346 696c 6520 meSolutionsFile │ │ │ │ -00015390: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -000153a0: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -000153b0: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ -000153c0: 2020 2020 2020 2272 6177 5f73 6f6c 7574 "raw_solut │ │ │ │ -000153d0: 696f 6e73 222c 0a20 2020 2020 202a 204f ions",. * O │ │ │ │ -000153e0: 7574 7075 7453 7479 6c65 2028 6d69 7373 utputStyle (miss │ │ │ │ -000153f0: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00015400: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00015410: 6c74 2076 616c 7565 2022 4f75 7450 6f69 lt value "OutPoi │ │ │ │ -00015420: 6e74 7322 2c20 0a20 2020 2020 202a 202a nts", . * * │ │ │ │ -00015430: 6e6f 7465 2052 616e 646f 6d43 6f6d 706c note RandomCompl │ │ │ │ -00015440: 6578 3a20 4265 7274 696e 6920 696e 7075 ex: Bertini inpu │ │ │ │ -00015450: 7420 6669 6c65 2064 6563 6c61 7261 7469 t file declarati │ │ │ │ -00015460: 6f6e 735f 636f 2072 616e 646f 6d20 6e75 ons_co random nu │ │ │ │ -00015470: 6d62 6572 732c 0a20 2020 2020 2020 203d mbers,. = │ │ │ │ -00015480: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00015490: 616c 7565 207b 7d2c 2061 6e20 6f70 7469 alue {}, an opti │ │ │ │ -000154a0: 6f6e 2077 6869 6368 2064 6573 6967 6e61 on which designa │ │ │ │ -000154b0: 7465 730a 2020 2020 2020 2020 7379 6d62 tes. symb │ │ │ │ -000154c0: 6f6c 732f 7374 7269 6e67 732f 7661 7269 ols/strings/vari │ │ │ │ -000154d0: 6162 6c65 7320 7468 6174 2077 696c 6c20 ables that will │ │ │ │ -000154e0: 6265 2073 6574 2074 6f20 6265 2061 2072 be set to be a r │ │ │ │ -000154f0: 616e 646f 6d20 7265 616c 206e 756d 6265 andom real numbe │ │ │ │ -00015500: 720a 2020 2020 2020 2020 6f72 2072 616e r. or ran │ │ │ │ -00015510: 646f 6d20 636f 6d70 6c65 7820 6e75 6d62 dom complex numb │ │ │ │ -00015520: 6572 0a20 2020 2020 202a 202a 6e6f 7465 er. * *note │ │ │ │ -00015530: 2052 616e 646f 6d52 6561 6c3a 2042 6572 RandomReal: Ber │ │ │ │ -00015540: 7469 6e69 2069 6e70 7574 2066 696c 6520 tini input file │ │ │ │ -00015550: 6465 636c 6172 6174 696f 6e73 5f63 6f20 declarations_co │ │ │ │ -00015560: 7261 6e64 6f6d 206e 756d 6265 7273 2c20 random numbers, │ │ │ │ -00015570: 3d3e 0a20 2020 2020 2020 202e 2e2e 2c20 =>. ..., │ │ │ │ -00015580: 6465 6661 756c 7420 7661 6c75 6520 7b7d default value {} │ │ │ │ -00015590: 2c20 616e 206f 7074 696f 6e20 7768 6963 , an option whic │ │ │ │ -000155a0: 6820 6465 7369 676e 6174 6573 0a20 2020 h designates. │ │ │ │ -000155b0: 2020 2020 2073 796d 626f 6c73 2f73 7472 symbols/str │ │ │ │ -000155c0: 696e 6773 2f76 6172 6961 626c 6573 2074 ings/variables t │ │ │ │ -000155d0: 6861 7420 7769 6c6c 2062 6520 7365 7420 hat will be set │ │ │ │ -000155e0: 746f 2062 6520 6120 7261 6e64 6f6d 2072 to be a random r │ │ │ │ -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 2d37 3333 "/tmp/M2-733 │ │ │ │ -00015670: 3638 2d30 2f30 222c 204f 7074 696f 6e20 68-0/0", 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, │ │ │ │ -000156f0: 0a20 2020 2020 202a 202a 6e6f 7465 2056 . * *note V │ │ │ │ -00015700: 6572 626f 7365 3a20 6265 7274 696e 6954 erbose: bertiniT │ │ │ │ -00015710: 7261 636b 486f 6d6f 746f 7079 5f6c 705f rackHomotopy_lp_ │ │ │ │ -00015720: 7064 5f70 645f 7064 5f63 6d56 6572 626f pd_pd_pd_cmVerbo │ │ │ │ -00015730: 7365 3d3e 5f70 645f 7064 5f70 645f 7270 se=>_pd_pd_pd_rp │ │ │ │ -00015740: 0a20 2020 2020 2020 202c 203d 3e20 2e2e . , => .. │ │ │ │ -00015750: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00015760: 2066 616c 7365 2c20 4f70 7469 6f6e 2074 false, Option t │ │ │ │ -00015770: 6f20 7369 6c65 6e63 6520 6164 6469 7469 o silence additi │ │ │ │ -00015780: 6f6e 616c 206f 7574 7075 740a 2020 2a20 onal output. * │ │ │ │ -00015790: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -000157a0: 2053 2c20 6120 2a6e 6f74 6520 6c69 7374 S, a *note list │ │ │ │ -000157b0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000157c0: 4c69 7374 2c2c 2061 206c 6973 7420 6f66 List,, a list of │ │ │ │ -000157d0: 2070 6f69 6e74 7320 7468 6174 2061 7265 points that are │ │ │ │ -000157e0: 0a20 2020 2020 2020 2063 6f6e 7461 696e . contain │ │ │ │ -000157f0: 6564 2069 6e20 7468 6520 7661 7269 6574 ed in the variet │ │ │ │ -00015800: 7920 6f66 2046 0a0a 4465 7363 7269 7074 y of F..Descript │ │ │ │ -00015810: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00015820: 0a54 6869 7320 6d65 7468 6f64 2066 696e .This method fin │ │ │ │ -00015830: 6473 2069 736f 6c61 7465 6420 736f 6c75 ds isolated solu │ │ │ │ -00015840: 7469 6f6e 7320 746f 2074 6865 2073 7973 tions to the sys │ │ │ │ -00015850: 7465 6d20 4620 7669 6120 6e75 6d65 7269 tem F via numeri │ │ │ │ -00015860: 6361 6c20 706f 6c79 6e6f 6d69 616c 0a68 cal polynomial.h │ │ │ │ -00015870: 6f6d 6f74 6f70 7920 636f 6e74 696e 7561 omotopy continua │ │ │ │ -00015880: 7469 6f6e 2062 7920 2831 2920 6275 696c tion by (1) buil │ │ │ │ -00015890: 6469 6e67 2061 2042 6572 7469 6e69 2069 ding a Bertini i │ │ │ │ -000158a0: 6e70 7574 2066 696c 6520 6672 6f6d 2074 nput file from t │ │ │ │ -000158b0: 6865 2073 7973 7465 6d20 462c 0a28 3229 he system F,.(2) │ │ │ │ -000158c0: 2063 616c 6c69 6e67 2042 6572 7469 6e69 calling Bertini │ │ │ │ -000158d0: 206f 6e20 7468 6973 2069 6e70 7574 2066 on this input f │ │ │ │ -000158e0: 696c 652c 2028 3329 2072 6574 7572 6e69 ile, (3) returni │ │ │ │ -000158f0: 6e67 2073 6f6c 7574 696f 6e73 2066 726f ng solutions fro │ │ │ │ -00015900: 6d20 6120 6d61 6368 696e 650a 7265 6164 m a machine.read │ │ │ │ -00015910: 6162 6c65 2066 696c 6520 7468 6174 2069 able file that i │ │ │ │ -00015920: 7320 616e 206f 7574 7075 7420 6672 6f6d s an output from │ │ │ │ -00015930: 2042 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d Bertini...+---- │ │ │ │ +00014e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00014ea0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00014eb0: 2020 2020 5320 3d20 6265 7274 696e 695a S = bertiniZ │ │ │ │ +00014ec0: 6572 6f44 696d 536f 6c76 6520 460a 2020 eroDimSolve F. │ │ │ │ +00014ed0: 2020 2020 2020 5320 3d20 6265 7274 696e S = bertin │ │ │ │ +00014ee0: 695a 6572 6f44 696d 536f 6c76 6520 490a iZeroDimSolve I. │ │ │ │ +00014ef0: 2020 2020 2020 2020 5320 3d20 6265 7274 S = bert │ │ │ │ +00014f00: 696e 695a 6572 6f44 696d 536f 6c76 6528 iniZeroDimSolve( │ │ │ │ +00014f10: 492c 2055 7365 5265 6765 6e65 7261 7469 I, UseRegenerati │ │ │ │ +00014f20: 6f6e 3d3e 3129 0a20 202a 2049 6e70 7574 on=>1). * Input │ │ │ │ +00014f30: 733a 0a20 2020 2020 202a 2046 2c20 6120 s:. * F, a │ │ │ │ +00014f40: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00014f50: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00014f60: 2061 206c 6973 7420 6f66 2072 696e 6720 a list of ring │ │ │ │ +00014f70: 656c 656d 656e 7473 2028 7379 7374 656d elements (system │ │ │ │ +00014f80: 0a20 2020 2020 2020 206e 6565 6420 6e6f . need no │ │ │ │ +00014f90: 7420 6265 2073 7175 6172 6529 0a20 2020 t be square). │ │ │ │ +00014fa0: 2020 202a 2049 2c20 616e 202a 6e6f 7465 * I, an *note │ │ │ │ +00014fb0: 2069 6465 616c 3a20 284d 6163 6175 6c61 ideal: (Macaula │ │ │ │ +00014fc0: 7932 446f 6329 4964 6561 6c2c 2c20 616e y2Doc)Ideal,, an │ │ │ │ +00014fd0: 2069 6465 616c 2064 6566 696e 696e 6720 ideal defining │ │ │ │ +00014fe0: 6120 7661 7269 6574 790a 2020 2a20 2a6e a variety. * *n │ │ │ │ +00014ff0: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +00015000: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ +00015010: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ +00015020: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00015030: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ +00015040: 2a20 2a6e 6f74 6520 4166 6656 6172 6961 * *note AffVaria │ │ │ │ +00015050: 626c 6547 726f 7570 3a20 5661 7269 6162 bleGroup: Variab │ │ │ │ +00015060: 6c65 2067 726f 7570 732c 203d 3e20 2e2e le groups, => .. │ │ │ │ +00015070: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +00015080: 207b 7d2c 2061 6e0a 2020 2020 2020 2020 {}, an. │ │ │ │ +00015090: 6f70 7469 6f6e 2074 6f20 6772 6f75 7020 option to group │ │ │ │ +000150a0: 7661 7269 6162 6c65 7320 616e 6420 7573 variables and us │ │ │ │ +000150b0: 6520 6d75 6c74 6968 6f6d 6f67 656e 656f e multihomogeneo │ │ │ │ +000150c0: 7573 2068 6f6d 6f74 6f70 6965 730a 2020 us homotopies. │ │ │ │ +000150d0: 2020 2020 2a20 2a6e 6f74 6520 4227 436f * *note B'Co │ │ │ │ +000150e0: 6e73 7461 6e74 733a 2042 2743 6f6e 7374 nstants: B'Const │ │ │ │ +000150f0: 616e 7473 2c20 3d3e 202e 2e2e 2c20 6465 ants, => ..., de │ │ │ │ +00015100: 6661 756c 7420 7661 6c75 6520 7b7d 2c20 fault value {}, │ │ │ │ +00015110: 616e 206f 7074 696f 6e20 746f 0a20 2020 an option to. │ │ │ │ +00015120: 2020 2020 2064 6573 6967 6e61 7465 2074 designate t │ │ │ │ +00015130: 6865 2063 6f6e 7374 616e 7473 2066 6f72 he constants for │ │ │ │ +00015140: 2061 2042 6572 7469 6e69 2049 6e70 7574 a Bertini Input │ │ │ │ +00015150: 2066 696c 650a 2020 2020 2020 2a20 4227 file. * B' │ │ │ │ +00015160: 4675 6e63 7469 6f6e 7320 286d 6973 7369 Functions (missi │ │ │ │ +00015170: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00015180: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ +00015190: 7420 7661 6c75 6520 7b7d 2c20 0a20 2020 t value {}, . │ │ │ │ +000151a0: 2020 202a 2042 6572 7469 6e69 496e 7075 * BertiniInpu │ │ │ │ +000151b0: 7443 6f6e 6669 6775 7261 7469 6f6e 2028 tConfiguration ( │ │ │ │ +000151c0: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ +000151d0: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ +000151e0: 6566 6175 6c74 2076 616c 7565 0a20 2020 efault value. │ │ │ │ +000151f0: 2020 2020 207b 7d2c 0a20 2020 2020 202a {},. * │ │ │ │ +00015200: 202a 6e6f 7465 2048 6f6d 5661 7269 6162 *note HomVariab │ │ │ │ +00015210: 6c65 4772 6f75 703a 2056 6172 6961 626c leGroup: Variabl │ │ │ │ +00015220: 6520 6772 6f75 7073 2c20 3d3e 202e 2e2e e groups, => ... │ │ │ │ +00015230: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00015240: 7b7d 2c20 616e 0a20 2020 2020 2020 206f {}, an. o │ │ │ │ +00015250: 7074 696f 6e20 746f 2067 726f 7570 2076 ption to group v │ │ │ │ +00015260: 6172 6961 626c 6573 2061 6e64 2075 7365 ariables and use │ │ │ │ +00015270: 206d 756c 7469 686f 6d6f 6765 6e65 6f75 multihomogeneou │ │ │ │ +00015280: 7320 686f 6d6f 746f 7069 6573 0a20 2020 s homotopies. │ │ │ │ +00015290: 2020 202a 2049 7350 726f 6a65 6374 6976 * IsProjectiv │ │ │ │ +000152a0: 6520 286d 6973 7369 6e67 2064 6f63 756d e (missing docum │ │ │ │ +000152b0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +000152c0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000152d0: 2d31 2c20 0a20 2020 2020 202a 204d 3250 -1, . * M2P │ │ │ │ +000152e0: 7265 6369 7369 6f6e 2028 6d69 7373 696e recision (missin │ │ │ │ +000152f0: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +00015300: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00015310: 2076 616c 7565 2035 332c 200a 2020 2020 value 53, . │ │ │ │ +00015320: 2020 2a20 4e61 6d65 4d61 696e 4461 7461 * NameMainData │ │ │ │ +00015330: 4669 6c65 2028 6d69 7373 696e 6720 646f File (missing do │ │ │ │ +00015340: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +00015350: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +00015360: 7565 0a20 2020 2020 2020 2022 6d61 696e ue. "main │ │ │ │ +00015370: 5f64 6174 6122 2c0a 2020 2020 2020 2a20 _data",. * │ │ │ │ +00015380: 4e61 6d65 536f 6c75 7469 6f6e 7346 696c NameSolutionsFil │ │ │ │ +00015390: 6520 286d 6973 7369 6e67 2064 6f63 756d e (missing docum │ │ │ │ +000153a0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +000153b0: 2c20 6465 6661 756c 7420 7661 6c75 650a , default value. │ │ │ │ +000153c0: 2020 2020 2020 2020 2272 6177 5f73 6f6c "raw_sol │ │ │ │ +000153d0: 7574 696f 6e73 222c 0a20 2020 2020 202a utions",. * │ │ │ │ +000153e0: 204f 7574 7075 7453 7479 6c65 2028 6d69 OutputStyle (mi │ │ │ │ +000153f0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +00015400: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +00015410: 6175 6c74 2076 616c 7565 2022 4f75 7450 ault value "OutP │ │ │ │ +00015420: 6f69 6e74 7322 2c20 0a20 2020 2020 202a oints", . * │ │ │ │ +00015430: 202a 6e6f 7465 2052 616e 646f 6d43 6f6d *note RandomCom │ │ │ │ +00015440: 706c 6578 3a20 4265 7274 696e 6920 696e plex: Bertini in │ │ │ │ +00015450: 7075 7420 6669 6c65 2064 6563 6c61 7261 put file declara │ │ │ │ +00015460: 7469 6f6e 735f 636f 2072 616e 646f 6d20 tions_co random │ │ │ │ +00015470: 6e75 6d62 6572 732c 0a20 2020 2020 2020 numbers,. │ │ │ │ +00015480: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00015490: 2076 616c 7565 207b 7d2c 2061 6e20 6f70 value {}, an op │ │ │ │ +000154a0: 7469 6f6e 2077 6869 6368 2064 6573 6967 tion which desig │ │ │ │ +000154b0: 6e61 7465 730a 2020 2020 2020 2020 7379 nates. sy │ │ │ │ +000154c0: 6d62 6f6c 732f 7374 7269 6e67 732f 7661 mbols/strings/va │ │ │ │ +000154d0: 7269 6162 6c65 7320 7468 6174 2077 696c riables that wil │ │ │ │ +000154e0: 6c20 6265 2073 6574 2074 6f20 6265 2061 l be set to be a │ │ │ │ +000154f0: 2072 616e 646f 6d20 7265 616c 206e 756d random real num │ │ │ │ +00015500: 6265 720a 2020 2020 2020 2020 6f72 2072 ber. or r │ │ │ │ +00015510: 616e 646f 6d20 636f 6d70 6c65 7820 6e75 andom complex nu │ │ │ │ +00015520: 6d62 6572 0a20 2020 2020 202a 202a 6e6f mber. * *no │ │ │ │ +00015530: 7465 2052 616e 646f 6d52 6561 6c3a 2042 te RandomReal: B │ │ │ │ +00015540: 6572 7469 6e69 2069 6e70 7574 2066 696c ertini input fil │ │ │ │ +00015550: 6520 6465 636c 6172 6174 696f 6e73 5f63 e declarations_c │ │ │ │ +00015560: 6f20 7261 6e64 6f6d 206e 756d 6265 7273 o random numbers │ │ │ │ +00015570: 2c20 3d3e 0a20 2020 2020 2020 202e 2e2e , =>. ... │ │ │ │ +00015580: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00015590: 7b7d 2c20 616e 206f 7074 696f 6e20 7768 {}, an option wh │ │ │ │ +000155a0: 6963 6820 6465 7369 676e 6174 6573 0a20 ich designates. │ │ │ │ +000155b0: 2020 2020 2020 2073 796d 626f 6c73 2f73 symbols/s │ │ │ │ +000155c0: 7472 696e 6773 2f76 6172 6961 626c 6573 trings/variables │ │ │ │ +000155d0: 2074 6861 7420 7769 6c6c 2062 6520 7365 that will be se │ │ │ │ +000155e0: 7420 746f 2062 6520 6120 7261 6e64 6f6d t to be a random │ │ │ │ +000155f0: 2072 6561 6c20 6e75 6d62 6572 0a20 2020 real number. │ │ │ │ +00015600: 2020 2020 206f 7220 7261 6e64 6f6d 2063 or random c │ │ │ │ +00015610: 6f6d 706c 6578 206e 756d 6265 720a 2020 omplex number. │ │ │ │ +00015620: 2020 2020 2a20 2a6e 6f74 6520 546f 7044 * *note TopD │ │ │ │ +00015630: 6972 6563 746f 7279 3a20 546f 7044 6972 irectory: TopDir │ │ │ │ +00015640: 6563 746f 7279 2c20 3d3e 202e 2e2e 2c20 ectory, => ..., │ │ │ │ +00015650: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ +00015660: 2020 2020 2020 222f 746d 702f 4d32 2d31 "/tmp/M2-1 │ │ │ │ +00015670: 3236 3338 342d 302f 3022 2c20 4f70 7469 26384-0/0", Opti │ │ │ │ +00015680: 6f6e 2074 6f20 6368 616e 6765 2064 6972 on to change dir │ │ │ │ +00015690: 6563 746f 7279 2066 6f72 2066 696c 6520 ectory for file │ │ │ │ +000156a0: 7374 6f72 6167 652e 0a20 2020 2020 202a storage.. * │ │ │ │ +000156b0: 2055 7365 5265 6765 6e65 7261 7469 6f6e UseRegeneration │ │ │ │ +000156c0: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +000156d0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +000156e0: 2064 6566 6175 6c74 2076 616c 7565 202d default value - │ │ │ │ +000156f0: 312c 200a 2020 2020 2020 2a20 2a6e 6f74 1, . * *not │ │ │ │ +00015700: 6520 5665 7262 6f73 653a 2062 6572 7469 e Verbose: berti │ │ │ │ +00015710: 6e69 5472 6163 6b48 6f6d 6f74 6f70 795f niTrackHomotopy_ │ │ │ │ +00015720: 6c70 5f70 645f 7064 5f70 645f 636d 5665 lp_pd_pd_pd_cmVe │ │ │ │ +00015730: 7262 6f73 653d 3e5f 7064 5f70 645f 7064 rbose=>_pd_pd_pd │ │ │ │ +00015740: 5f72 700a 2020 2020 2020 2020 2c20 3d3e _rp. , => │ │ │ │ +00015750: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +00015760: 6c75 6520 6661 6c73 652c 204f 7074 696f lue false, Optio │ │ │ │ +00015770: 6e20 746f 2073 696c 656e 6365 2061 6464 n to silence add │ │ │ │ +00015780: 6974 696f 6e61 6c20 6f75 7470 7574 0a20 itional output. │ │ │ │ +00015790: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +000157a0: 2020 2a20 532c 2061 202a 6e6f 7465 206c * S, a *note l │ │ │ │ +000157b0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000157c0: 6f63 294c 6973 742c 2c20 6120 6c69 7374 oc)List,, a list │ │ │ │ +000157d0: 206f 6620 706f 696e 7473 2074 6861 7420 of points that │ │ │ │ +000157e0: 6172 650a 2020 2020 2020 2020 636f 6e74 are. cont │ │ │ │ +000157f0: 6169 6e65 6420 696e 2074 6865 2076 6172 ained in the var │ │ │ │ +00015800: 6965 7479 206f 6620 460a 0a44 6573 6372 iety of F..Descr │ │ │ │ +00015810: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00015820: 3d3d 0a0a 5468 6973 206d 6574 686f 6420 ==..This method │ │ │ │ +00015830: 6669 6e64 7320 6973 6f6c 6174 6564 2073 finds isolated s │ │ │ │ +00015840: 6f6c 7574 696f 6e73 2074 6f20 7468 6520 olutions to the │ │ │ │ +00015850: 7379 7374 656d 2046 2076 6961 206e 756d system F via num │ │ │ │ +00015860: 6572 6963 616c 2070 6f6c 796e 6f6d 6961 erical polynomia │ │ │ │ +00015870: 6c0a 686f 6d6f 746f 7079 2063 6f6e 7469 l.homotopy conti │ │ │ │ +00015880: 6e75 6174 696f 6e20 6279 2028 3129 2062 nuation by (1) b │ │ │ │ +00015890: 7569 6c64 696e 6720 6120 4265 7274 696e uilding a Bertin │ │ │ │ +000158a0: 6920 696e 7075 7420 6669 6c65 2066 726f i input file fro │ │ │ │ +000158b0: 6d20 7468 6520 7379 7374 656d 2046 2c0a m the system F,. │ │ │ │ +000158c0: 2832 2920 6361 6c6c 696e 6720 4265 7274 (2) calling Bert │ │ │ │ +000158d0: 696e 6920 6f6e 2074 6869 7320 696e 7075 ini on this inpu │ │ │ │ +000158e0: 7420 6669 6c65 2c20 2833 2920 7265 7475 t file, (3) retu │ │ │ │ +000158f0: 726e 696e 6720 736f 6c75 7469 6f6e 7320 rning solutions │ │ │ │ +00015900: 6672 6f6d 2061 206d 6163 6869 6e65 0a72 from a machine.r │ │ │ │ +00015910: 6561 6461 626c 6520 6669 6c65 2074 6861 eadable file tha │ │ │ │ +00015920: 7420 6973 2061 6e20 6f75 7470 7574 2066 t is an output f │ │ │ │ +00015930: 726f 6d20 4265 7274 696e 692e 0a0a 2b2d rom Bertini...+- │ │ │ │ 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 2d2b 0a7c -------------+.| │ │ │ │ -00015980: 6931 203a 2052 203d 2043 435b 782c 795d i1 : R = CC[x,y] │ │ │ │ -00015990: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00015970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015980: 2b0a 7c69 3120 3a20 5220 3d20 4343 5b78 +.|i1 : R = CC[x │ │ │ │ +00015990: 2c79 5d3b 2020 2020 2020 2020 2020 2020 ,y]; │ │ │ │ 000159a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000159b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000159c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000159c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 000159d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000159e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000159f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015a00: 2d2d 2d2d 2d2b 0a7c 6932 203a 2046 203d -----+.|i2 : F = │ │ │ │ -00015a10: 207b 785e 322d 312c 795e 322d 327d 3b20 {x^2-1,y^2-2}; │ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015a00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ +00015a10: 4620 3d20 7b78 5e32 2d31 2c79 5e32 2d32 F = {x^2-1,y^2-2 │ │ │ │ +00015a20: 7d3b 2020 2020 2020 2020 2020 2020 2020 }; │ │ │ │ 00015a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015a40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00015a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00015a90: 6933 203a 2053 203d 2062 6572 7469 6e69 i3 : S = bertini │ │ │ │ -00015aa0: 5a65 726f 4469 6d53 6f6c 7665 2046 2020 ZeroDimSolve F │ │ │ │ -00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015a90: 2b0a 7c69 3320 3a20 5320 3d20 6265 7274 +.|i3 : S = bert │ │ │ │ +00015aa0: 696e 695a 6572 6f44 696d 536f 6c76 6520 iniZeroDimSolve │ │ │ │ +00015ab0: 4620 2020 2020 2020 2020 2020 2020 2020 F │ │ │ │ 00015ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ad0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 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 0a7c 6f33 203d 207b 7b31 |.|o3 = {{1 │ │ │ │ -00015b20: 2c20 312e 3431 3432 317d 2c20 7b31 2c20 , 1.41421}, {1, │ │ │ │ -00015b30: 2d31 2e34 3134 3231 7d2c 207b 2d31 2c20 -1.41421}, {-1, │ │ │ │ -00015b40: 312e 3431 3432 317d 2c20 7b2d 312c 202d 1.41421}, {-1, - │ │ │ │ -00015b50: 312e 3431 3432 317d 7d7c 0a7c 2020 2020 1.41421}}|.| │ │ │ │ +00015b10: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +00015b20: 7b7b 312c 2031 2e34 3134 3231 7d2c 207b {{1, 1.41421}, { │ │ │ │ +00015b30: 312c 202d 312e 3431 3432 317d 2c20 7b2d 1, -1.41421}, {- │ │ │ │ +00015b40: 312c 2031 2e34 3134 3231 7d2c 207b 2d31 1, 1.41421}, {-1 │ │ │ │ +00015b50: 2c20 2d31 2e34 3134 3231 7d7d 7c0a 7c20 , -1.41421}}|.| │ │ │ │ 00015b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015b90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015ba0: 6f33 203a 204c 6973 7420 2020 2020 2020 o3 : List │ │ │ │ +00015b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ba0: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020 |.|o3 : List │ │ │ │ 00015bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015be0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00015be0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00015bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015c20: 2d2d 2d2d 2d2b 0a0a 4561 6368 2073 6f6c -----+..Each sol │ │ │ │ -00015c30: 7574 696f 6e20 6973 206f 6620 7479 7065 ution is of type │ │ │ │ -00015c40: 202a 6e6f 7465 2050 6f69 6e74 3a20 284e *note Point: (N │ │ │ │ -00015c50: 4147 7479 7065 7329 4162 7374 7261 6374 AGtypes)Abstract │ │ │ │ -00015c60: 506f 696e 742c 2e20 2041 6464 6974 696f Point,. Additio │ │ │ │ -00015c70: 6e61 6c0a 696e 666f 726d 6174 696f 6e20 nal.information │ │ │ │ -00015c80: 6162 6f75 7420 7468 6520 736f 6c75 7469 about the soluti │ │ │ │ -00015c90: 6f6e 2063 616e 2062 6520 6163 6365 7373 on can be access │ │ │ │ -00015ca0: 6564 2062 7920 7573 696e 6720 2a6e 6f74 ed by using *not │ │ │ │ -00015cb0: 6520 7065 656b 3a0a 284d 6163 6175 6c61 e peek:.(Macaula │ │ │ │ -00015cc0: 7932 446f 6329 7065 656b 2c2e 0a0a 2b2d y2Doc)peek,...+- │ │ │ │ -00015cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c20: 2d2d 2d2d 2d2d 2d2d 2b0a 0a45 6163 6820 --------+..Each │ │ │ │ +00015c30: 736f 6c75 7469 6f6e 2069 7320 6f66 2074 solution is of t │ │ │ │ +00015c40: 7970 6520 2a6e 6f74 6520 506f 696e 743a ype *note Point: │ │ │ │ +00015c50: 2028 4e41 4774 7970 6573 2941 6273 7472 (NAGtypes)Abstr │ │ │ │ +00015c60: 6163 7450 6f69 6e74 2c2e 2020 4164 6469 actPoint,. Addi │ │ │ │ +00015c70: 7469 6f6e 616c 0a69 6e66 6f72 6d61 7469 tional.informati │ │ │ │ +00015c80: 6f6e 2061 626f 7574 2074 6865 2073 6f6c on about the sol │ │ │ │ +00015c90: 7574 696f 6e20 6361 6e20 6265 2061 6363 ution can be acc │ │ │ │ +00015ca0: 6573 7365 6420 6279 2075 7369 6e67 202a essed by using * │ │ │ │ +00015cb0: 6e6f 7465 2070 6565 6b3a 0a28 4d61 6361 note peek:.(Maca │ │ │ │ +00015cc0: 756c 6179 3244 6f63 2970 6565 6b2c 2e0a ulay2Doc)peek,.. │ │ │ │ +00015cd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00015ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015cf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00015d00: 7065 656b 2053 5f30 2020 2020 2020 2020 peek S_0 │ │ │ │ +00015cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +00015d00: 203a 2070 6565 6b20 535f 3020 2020 2020 : peek S_0 │ │ │ │ 00015d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015d20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00015d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d50: 7c0a 7c6f 3420 3d20 506f 696e 747b 6361 |.|o4 = Point{ca │ │ │ │ -00015d60: 6368 6520 3d3e 2043 6163 6865 5461 626c che => CacheTabl │ │ │ │ -00015d70: 657b 2e2e 2e31 342e 2e2e 7d7d 7c0a 7c20 e{...14...}}|.| │ │ │ │ -00015d80: 2020 2020 2020 2020 2020 436f 6f72 6469 Coordi │ │ │ │ -00015d90: 6e61 7465 7320 3d3e 207b 312c 2031 2e34 nates => {1, 1.4 │ │ │ │ -00015da0: 3134 3231 7d20 2020 7c0a 2b2d 2d2d 2d2d 1421} |.+----- │ │ │ │ +00015d50: 2020 207c 0a7c 6f34 203d 2050 6f69 6e74 |.|o4 = Point │ │ │ │ +00015d60: 7b63 6163 6865 203d 3e20 4361 6368 6554 {cache => CacheT │ │ │ │ +00015d70: 6162 6c65 7b2e 2e2e 3134 2e2e 2e7d 7d7c able{...14...}}| │ │ │ │ +00015d80: 0a7c 2020 2020 2020 2020 2020 2043 6f6f .| Coo │ │ │ │ +00015d90: 7264 696e 6174 6573 203d 3e20 7b31 2c20 rdinates => {1, │ │ │ │ +00015da0: 312e 3431 3432 317d 2020 207c 0a2b 2d2d 1.41421} |.+-- │ │ │ │ 00015db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015dd0: 2d2d 2d2d 2b0a 0a42 6572 7469 6e69 2075 ----+..Bertini u │ │ │ │ -00015de0: 7365 7320 6120 6d75 6c74 6968 6f6d 6f67 ses a multihomog │ │ │ │ -00015df0: 656e 656f 7573 2068 6f6d 6f74 6f70 7920 eneous homotopy │ │ │ │ -00015e00: 6173 2061 2064 6566 6175 6c74 2c20 6275 as a default, bu │ │ │ │ -00015e10: 7420 7265 6765 6e65 7261 7469 6f6e 2063 t regeneration c │ │ │ │ -00015e20: 616e 2062 650a 6465 706c 6f79 6564 2077 an be.deployed w │ │ │ │ -00015e30: 6974 6820 7468 6520 6f70 7469 6f6e 2055 ith the option U │ │ │ │ -00015e40: 7365 5265 6765 6e65 7261 7469 6f6e 3d3e seRegeneration=> │ │ │ │ -00015e50: 3120 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 1 ...+---------- │ │ │ │ +00015dd0: 2d2d 2d2d 2d2d 2d2b 0a0a 4265 7274 696e -------+..Bertin │ │ │ │ +00015de0: 6920 7573 6573 2061 206d 756c 7469 686f i uses a multiho │ │ │ │ +00015df0: 6d6f 6765 6e65 6f75 7320 686f 6d6f 746f mogeneous homoto │ │ │ │ +00015e00: 7079 2061 7320 6120 6465 6661 756c 742c py as a default, │ │ │ │ +00015e10: 2062 7574 2072 6567 656e 6572 6174 696f but regeneratio │ │ │ │ +00015e20: 6e20 6361 6e20 6265 0a64 6570 6c6f 7965 n can be.deploye │ │ │ │ +00015e30: 6420 7769 7468 2074 6865 206f 7074 696f d with the optio │ │ │ │ +00015e40: 6e20 5573 6552 6567 656e 6572 6174 696f n UseRegeneratio │ │ │ │ +00015e50: 6e3d 3e31 202e 0a0a 2b2d 2d2d 2d2d 2d2d n=>1 ...+------- │ │ │ │ 00015e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015e80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -00015e90: 5220 3d20 4343 5b78 5d3b 2020 2020 2020 R = CC[x]; │ │ │ │ +00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +00015e90: 203a 2052 203d 2043 435b 785d 3b20 2020 : R = CC[x]; │ │ │ │ 00015ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015eb0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ef0: 2d2d 2b0a 7c69 3620 3a20 4620 3d20 7b78 --+.|i6 : F = {x │ │ │ │ -00015f00: 5e32 2a28 782d 3129 7d3b 2020 2020 2020 ^2*(x-1)}; │ │ │ │ +00015ef0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2046 203d -----+.|i6 : F = │ │ │ │ +00015f00: 207b 785e 322a 2878 2d31 297d 3b20 2020 {x^2*(x-1)}; │ │ │ │ 00015f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00015f20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00015f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00015f60: 3720 3a20 5320 3d20 6265 7274 696e 695a 7 : S = bertiniZ │ │ │ │ -00015f70: 6572 6f44 696d 536f 6c76 6520 4620 2020 eroDimSolve F │ │ │ │ +00015f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00015f60: 0a7c 6937 203a 2053 203d 2062 6572 7469 .|i7 : S = berti │ │ │ │ +00015f70: 6e69 5a65 726f 4469 6d53 6f6c 7665 2046 niZeroDimSolve F │ │ │ │ 00015f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00015fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015fc0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7b7b |.|o7 = {{ │ │ │ │ -00015fd0: 317d 2c20 7b2d 312e 3535 3538 3965 2d31 1}, {-1.55589e-1 │ │ │ │ -00015fe0: 352d 322e 3436 3035 3165 2d31 352a 6969 5-2.46051e-15*ii │ │ │ │ -00015ff0: 7d7d 2020 2020 2020 2020 207c 0a7c 2020 }} |.| │ │ │ │ -00016000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015fc0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +00015fd0: 207b 7b31 7d2c 207b 2d31 2e35 3535 3839 {{1}, {-1.55589 │ │ │ │ +00015fe0: 652d 3135 2d32 2e34 3630 3531 652d 3135 e-15-2.46051e-15 │ │ │ │ +00015ff0: 2a69 697d 7d20 2020 2020 2020 2020 7c0a *ii}} |. │ │ │ │ +00016000: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00016010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016030: 7c0a 7c6f 3720 3a20 4c69 7374 2020 2020 |.|o7 : List │ │ │ │ +00016030: 2020 207c 0a7c 6f37 203a 204c 6973 7420 |.|o7 : List │ │ │ │ 00016040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00016060: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00016070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016090: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ -000160a0: 3a20 4220 3d20 6265 7274 696e 695a 6572 : B = bertiniZer │ │ │ │ -000160b0: 6f44 696d 536f 6c76 6528 462c 5573 6552 oDimSolve(F,UseR │ │ │ │ -000160c0: 6567 656e 6572 6174 696f 6e3d 3e31 297c egeneration=>1)| │ │ │ │ -000160d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000160a0: 6938 203a 2042 203d 2062 6572 7469 6e69 i8 : B = bertini │ │ │ │ +000160b0: 5a65 726f 4469 6d53 6f6c 7665 2846 2c55 ZeroDimSolve(F,U │ │ │ │ +000160c0: 7365 5265 6765 6e65 7261 7469 6f6e 3d3e seRegeneration=> │ │ │ │ +000160d0: 3129 7c0a 7c20 2020 2020 2020 2020 2020 1)|.| │ │ │ │ 000160e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000160f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016100: 2020 2020 7c0a 7c6f 3820 3d20 7b7b 317d |.|o8 = {{1} │ │ │ │ -00016110: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00016100: 2020 2020 2020 207c 0a7c 6f38 203d 207b |.|o8 = { │ │ │ │ +00016110: 7b31 7d7d 2020 2020 2020 2020 2020 2020 {1}} │ │ │ │ 00016120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016130: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00016130: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00016170: 7c6f 3820 3a20 4c69 7374 2020 2020 2020 |o8 : List │ │ │ │ +00016160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016170: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ 00016180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000161a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000161a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a56 6172 6961 --------+..Varia │ │ │ │ -000161e0: 626c 6573 206d 7573 7420 6265 6769 6e20 bles must begin │ │ │ │ -000161f0: 7769 7468 2061 206c 6574 7465 7220 286c with a letter (l │ │ │ │ -00016200: 6f77 6572 6361 7365 206f 7220 6361 7069 owercase or capi │ │ │ │ -00016210: 7461 6c29 2061 6e64 2063 616e 206f 6e6c tal) and can onl │ │ │ │ -00016220: 7920 636f 6e74 6169 6e0a 6c65 7474 6572 y contain.letter │ │ │ │ -00016230: 732c 206e 756d 6265 7273 2c20 756e 6465 s, numbers, unde │ │ │ │ -00016240: 7273 636f 7265 732c 2061 6e64 2073 7175 rscores, and squ │ │ │ │ -00016250: 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Re │ │ │ │ -00016260: 6765 6e65 7261 7469 6f6e 2069 6e0a 6265 generation in.be │ │ │ │ -00016270: 7274 696e 695a 6572 6f44 696d 536f 6c76 rtiniZeroDimSolv │ │ │ │ -00016280: 6520 6f6e 6c79 2066 696e 6473 206e 6f6e e only finds non │ │ │ │ -00016290: 7369 6e67 756c 6172 2069 736f 6c61 7465 singular isolate │ │ │ │ -000162a0: 6420 706f 696e 7473 2e0a 0a57 6179 7320 d points...Ways │ │ │ │ -000162b0: 746f 2075 7365 2062 6572 7469 6e69 5a65 to use bertiniZe │ │ │ │ -000162c0: 726f 4469 6d53 6f6c 7665 3a0a 3d3d 3d3d roDimSolve:.==== │ │ │ │ +000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5661 -----------+..Va │ │ │ │ +000161e0: 7269 6162 6c65 7320 6d75 7374 2062 6567 riables must beg │ │ │ │ +000161f0: 696e 2077 6974 6820 6120 6c65 7474 6572 in with a letter │ │ │ │ +00016200: 2028 6c6f 7765 7263 6173 6520 6f72 2063 (lowercase or c │ │ │ │ +00016210: 6170 6974 616c 2920 616e 6420 6361 6e20 apital) and can │ │ │ │ +00016220: 6f6e 6c79 2063 6f6e 7461 696e 0a6c 6574 only contain.let │ │ │ │ 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6974 684f 7074 696f unctionWithOptio │ │ │ │ -000163e0: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +000162e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +000162f0: 0a20 202a 2022 6265 7274 696e 695a 6572 . * "bertiniZer │ │ │ │ +00016300: 6f44 696d 536f 6c76 6528 4964 6561 6c29 oDimSolve(Ideal) │ │ │ │ +00016310: 220a 2020 2a20 2262 6572 7469 6e69 5a65 ". * "bertiniZe │ │ │ │ +00016320: 726f 4469 6d53 6f6c 7665 284c 6973 7429 roDimSolve(List) │ │ │ │ +00016330: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +00016340: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00016350: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00016360: 6a65 6374 202a 6e6f 7465 2062 6572 7469 ject *note berti │ │ │ │ +00016370: 6e69 5a65 726f 4469 6d53 6f6c 7665 3a20 niZeroDimSolve: │ │ │ │ +00016380: 6265 7274 696e 695a 6572 6f44 696d 536f bertiniZeroDimSo │ │ │ │ +00016390: 6c76 652c 2069 7320 6120 2a6e 6f74 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symbol:.(Mac │ │ │ │ -00016740: 6175 6c61 7932 446f 6329 5379 6d62 6f6c aulay2Doc)Symbol │ │ │ │ -00016750: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +000166b0: 3d3d 3d0a 0a20 202a 2022 6d6f 7665 4227 ===.. * "moveB' │ │ │ │ +000166c0: 4669 6c65 282e 2e2e 2c43 6f70 7942 2746 File(...,CopyB'F │ │ │ │ +000166d0: 696c 653d 3e2e 2e2e 2922 0a0a 466f 7220 ile=>...)"..For │ │ │ │ +000166e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +000166f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00016700: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00016710: 6f74 6520 436f 7079 4227 4669 6c65 3a20 ote CopyB'File: │ │ │ │ +00016720: 436f 7079 4227 4669 6c65 2c20 6973 2061 CopyB'File, is a │ │ │ │ +00016730: 202a 6e6f 7465 2073 796d 626f 6c3a 0a28 *note symbol:.( │ │ │ │ +00016740: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ +00016750: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 00016760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000167a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -000167b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -000167c0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -000167d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -000167e0: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ -000167f0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00016800: 322f 7061 636b 6167 6573 2f42 6572 7469 2/packages/Berti │ │ │ │ -00016810: 6e69 2e6d 323a 0a33 3533 303a 302e 0a1f ni.m2:.3530:0... │ │ │ │ -00016820: 0a46 696c 653a 2042 6572 7469 6e69 2e69 .File: Bertini.i │ │ │ │ -00016830: 6e66 6f2c 204e 6f64 653a 2069 6d70 6f72 nfo, Node: impor │ │ │ │ -00016840: 7449 6e63 6964 656e 6365 4d61 7472 6978 tIncidenceMatrix │ │ │ │ -00016850: 2c20 4e65 7874 3a20 696d 706f 7274 4d61 , Next: importMa │ │ │ │ -00016860: 696e 4461 7461 4669 6c65 2c20 5072 6576 inDataFile, Prev │ │ │ │ -00016870: 3a20 436f 7079 4227 4669 6c65 2c20 5570 : CopyB'File, Up │ │ │ │ -00016880: 3a20 546f 700a 0a69 6d70 6f72 7449 6e63 : Top..importInc │ │ │ │ -00016890: 6964 656e 6365 4d61 7472 6978 202d 2d20 idenceMatrix -- │ │ │ │ -000168a0: 496d 706f 7274 7320 616e 2069 6e63 6964 Imports an incid │ │ │ │ -000168b0: 656e 6365 206d 6174 7269 7820 6669 6c65 ence matrix file │ │ │ │ -000168c0: 2061 6674 6572 2072 756e 6e69 6e67 206d after running m │ │ │ │ -000168d0: 616b 654d 656d 6265 7273 6869 7046 696c akeMembershipFil │ │ │ │ -000168e0: 652e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e..************* │ │ │ │ +000167a0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +000167b0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +000167c0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +000167d0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +000167e0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +000167f0: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ +00016800: 6c61 7932 2f70 6163 6b61 6765 732f 4265 lay2/packages/Be │ │ │ │ +00016810: 7274 696e 692e 6d32 3a0a 3335 3330 3a30 rtini.m2:.3530:0 │ │ │ │ +00016820: 2e0a 1f0a 4669 6c65 3a20 4265 7274 696e ....File: Bertin │ │ │ │ +00016830: 692e 696e 666f 2c20 4e6f 6465 3a20 696d i.info, Node: im │ │ │ │ +00016840: 706f 7274 496e 6369 6465 6e63 654d 6174 portIncidenceMat │ │ │ │ +00016850: 7269 782c 204e 6578 743a 2069 6d70 6f72 rix, Next: impor │ │ │ │ +00016860: 744d 6169 6e44 6174 6146 696c 652c 2050 tMainDataFile, P │ │ │ │ +00016870: 7265 763a 2043 6f70 7942 2746 696c 652c rev: CopyB'File, │ │ │ │ +00016880: 2055 703a 2054 6f70 0a0a 696d 706f 7274 Up: Top..import │ │ │ │ +00016890: 496e 6369 6465 6e63 654d 6174 7269 7820 IncidenceMatrix │ │ │ │ +000168a0: 2d2d 2049 6d70 6f72 7473 2061 6e20 696e -- Imports an in │ │ │ │ +000168b0: 6369 6465 6e63 6520 6d61 7472 6978 2066 cidence matrix f │ │ │ │ +000168c0: 696c 6520 6166 7465 7220 7275 6e6e 696e ile after runnin │ │ │ │ +000168d0: 6720 6d61 6b65 4d65 6d62 6572 7368 6970 g makeMembership │ │ │ │ +000168e0: 4669 6c65 2e0a 2a2a 2a2a 2a2a 2a2a 2a2a File..********** │ │ │ │ 000168f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00016930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00016940: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00016950: 2020 2020 696d 706f 7274 496e 6369 6465 importIncide │ │ │ │ -00016960: 6e63 654d 6174 7269 7828 7329 0a20 202a nceMatrix(s). * │ │ │ │ -00016970: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00016980: 2073 2c20 6120 2a6e 6f74 6520 7374 7269 s, a *note stri │ │ │ │ -00016990: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ -000169a0: 6329 5374 7269 6e67 2c2c 2054 6865 2064 c)String,, The d │ │ │ │ -000169b0: 6972 6563 746f 7279 2077 6865 7265 2074 irectory where t │ │ │ │ -000169c0: 6865 2066 696c 650a 2020 2020 2020 2020 he file. │ │ │ │ -000169d0: 6973 2073 746f 7265 642e 0a20 202a 202a is stored.. * * │ │ │ │ -000169e0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -000169f0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00016a00: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00016a10: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00016a20: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00016a30: 202a 204e 616d 6549 6e63 6964 656e 6365 * NameIncidence │ │ │ │ -00016a40: 4d61 7472 6978 4669 6c65 2028 6d69 7373 MatrixFile (miss │ │ │ │ -00016a50: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00016a60: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00016a70: 6c74 2076 616c 7565 0a20 2020 2020 2020 lt value. │ │ │ │ -00016a80: 2022 696e 6369 6465 6e63 655f 6d61 7472 "incidence_matr │ │ │ │ -00016a90: 6978 222c 0a20 2020 2020 202a 2053 746f ix",. * Sto │ │ │ │ -00016aa0: 7261 6765 466f 6c64 6572 2028 6d69 7373 rageFolder (miss │ │ │ │ -00016ab0: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00016ac0: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00016ad0: 6c74 2076 616c 7565 206e 756c 6c2c 200a lt value null, . │ │ │ │ -00016ae0: 2020 2020 2020 2a20 2a6e 6f74 6520 5665 * *note Ve │ │ │ │ -00016af0: 7262 6f73 653a 2062 6572 7469 6e69 5472 rbose: bertiniTr │ │ │ │ -00016b00: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70 ackHomotopy_lp_p │ │ │ │ -00016b10: 645f 7064 5f70 645f 636d 5665 7262 6f73 d_pd_pd_cmVerbos │ │ │ │ -00016b20: 653d 3e5f 7064 5f70 645f 7064 5f72 700a e=>_pd_pd_pd_rp. │ │ │ │ -00016b30: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e , => ... │ │ │ │ -00016b40: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -00016b50: 6661 6c73 652c 204f 7074 696f 6e20 746f false, Option to │ │ │ │ -00016b60: 2073 696c 656e 6365 2061 6464 6974 696f silence additio │ │ │ │ -00016b70: 6e61 6c20 6f75 7470 7574 0a0a 4465 7363 nal output..Desc │ │ │ │ -00016b80: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00016b90: 3d3d 3d0a 0a41 6674 6572 2072 756e 6e69 ===..After runni │ │ │ │ -00016ba0: 6e67 206d 616b 654d 656d 6265 7273 6869 ng makeMembershi │ │ │ │ -00016bb0: 7046 696c 6520 4265 7274 696e 6920 7072 pFile Bertini pr │ │ │ │ -00016bc0: 6f64 7563 6573 2061 6e20 696e 6369 6465 oduces an incide │ │ │ │ -00016bd0: 6e63 655f 6d61 7472 6978 2066 696c 652e nce_matrix file. │ │ │ │ -00016be0: 2054 6865 0a69 6e63 6964 656e 6365 5f6d The.incidence_m │ │ │ │ -00016bf0: 6174 7269 7820 7361 7973 2077 6869 6368 atrix says which │ │ │ │ -00016c00: 2070 6f69 6e74 7320 6265 6c6f 6e67 2074 points belong t │ │ │ │ -00016c10: 6f20 7768 6963 6820 636f 6d70 6f6e 656e o which componen │ │ │ │ -00016c20: 7473 2e20 4f75 7220 696e 6369 6465 6e63 ts. Our incidenc │ │ │ │ -00016c30: 650a 6d61 7472 6978 2069 7320 666c 6174 e.matrix is flat │ │ │ │ -00016c40: 7465 6e65 6420 746f 2061 206c 6973 742e tened to a list. │ │ │ │ -00016c50: 2054 6865 206e 756d 6265 7220 6f66 2065 The number of e │ │ │ │ -00016c60: 6c65 6d65 6e74 7320 696e 2074 6865 494d lements in theIM │ │ │ │ -00016c70: 2069 7320 6571 7561 6c20 746f 2074 6865 is equal to the │ │ │ │ -00016c80: 0a6e 756d 6265 7220 6f66 2070 6f69 6e74 .number of point │ │ │ │ -00016c90: 7320 696e 2074 6865 2073 6f6c 7574 696f s in the solutio │ │ │ │ -00016ca0: 6e73 2066 696c 652e 2045 6163 6820 656c ns file. Each el │ │ │ │ -00016cb0: 656d 656e 7420 6f66 2074 6865 494d 2069 ement of theIM i │ │ │ │ -00016cc0: 7320 6120 6c69 7374 206f 660a 7365 7175 s a list of.sequ │ │ │ │ -00016cd0: 656e 6365 7320 6f66 2032 2065 6c65 6d65 ences of 2 eleme │ │ │ │ -00016ce0: 6e74 7320 2863 6f64 696d 2c63 6f6d 706f nts (codim,compo │ │ │ │ -00016cf0: 6e65 6e74 204e 756d 6265 7229 2e20 4e6f nent Number). No │ │ │ │ -00016d00: 7465 2074 6861 7420 7765 2066 6f6c 6c6f te that we follo │ │ │ │ -00016d10: 7720 7468 650a 4265 7274 696e 6920 636f w the.Bertini co │ │ │ │ -00016d20: 6e76 656e 7469 6f6e 2061 6e64 2073 7769 nvention and swi │ │ │ │ -00016d30: 7463 6820 6672 6f6d 2028 6469 6d65 6e73 tch from (dimens │ │ │ │ -00016d40: 696f 6e2c 636f 6d70 6f6e 656e 7420 6e75 ion,component nu │ │ │ │ -00016d50: 6d62 6572 2920 696e 6465 7869 6e67 2074 mber) indexing t │ │ │ │ -00016d60: 6f0a 2863 6f64 696d 656e 7369 6f6e 2c63 o.(codimension,c │ │ │ │ -00016d70: 6f6d 706f 6e65 6e74 206e 756d 6265 7229 omponent number) │ │ │ │ -00016d80: 2069 6e64 6578 696e 672e 0a0a 4966 2074 indexing...If t │ │ │ │ -00016d90: 6865 204e 616d 6549 6e63 6964 656e 6365 he NameIncidence │ │ │ │ -00016da0: 4d61 7472 6978 4669 6c65 206f 7074 696f MatrixFile optio │ │ │ │ -00016db0: 6e20 6973 2073 6574 2077 6865 6e20 7765 n is set when we │ │ │ │ -00016dc0: 2077 616e 7420 746f 2069 6d70 6f72 7420 want to import │ │ │ │ -00016dd0: 6669 6c65 7320 7769 7468 0a61 2064 6966 files with.a dif │ │ │ │ -00016de0: 6665 7265 6e74 206e 616d 652e 0a0a 2b2d ferent name...+- │ │ │ │ -00016df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00016940: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00016950: 2020 2020 2020 2069 6d70 6f72 7449 6e63 importInc │ │ │ │ +00016960: 6964 656e 6365 4d61 7472 6978 2873 290a idenceMatrix(s). │ │ │ │ +00016970: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00016980: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ +00016990: 7472 696e 673a 2028 4d61 6361 756c 6179 tring: (Macaulay │ │ │ │ +000169a0: 3244 6f63 2953 7472 696e 672c 2c20 5468 2Doc)String,, Th │ │ │ │ +000169b0: 6520 6469 7265 6374 6f72 7920 7768 6572 e directory wher │ │ │ │ +000169c0: 6520 7468 6520 6669 6c65 0a20 2020 2020 e the file. │ │ │ │ +000169d0: 2020 2069 7320 7374 6f72 6564 2e0a 2020 is stored.. │ │ │ │ +000169e0: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c * *note Optional │ │ │ │ +000169f0: 2069 6e70 7574 733a 2028 4d61 6361 756c inputs: (Macaul │ │ │ │ +00016a00: 6179 3244 6f63 2975 7369 6e67 2066 756e ay2Doc)using fun │ │ │ │ +00016a10: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ +00016a20: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020 onal inputs,:. │ │ │ │ +00016a30: 2020 2020 2a20 4e61 6d65 496e 6369 6465 * NameIncide │ │ │ │ +00016a40: 6e63 654d 6174 7269 7846 696c 6520 286d nceMatrixFile (m │ │ │ │ +00016a50: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +00016a60: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +00016a70: 6661 756c 7420 7661 6c75 650a 2020 2020 fault value. │ │ │ │ +00016a80: 2020 2020 2269 6e63 6964 656e 6365 5f6d "incidence_m │ │ │ │ +00016a90: 6174 7269 7822 2c0a 2020 2020 2020 2a20 atrix",. * │ │ │ │ +00016aa0: 5374 6f72 6167 6546 6f6c 6465 7220 286d StorageFolder (m │ │ │ │ +00016ab0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +00016ac0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +00016ad0: 6661 756c 7420 7661 6c75 6520 6e75 6c6c fault value null │ │ │ │ +00016ae0: 2c20 0a20 2020 2020 202a 202a 6e6f 7465 , . * *note │ │ │ │ +00016af0: 2056 6572 626f 7365 3a20 6265 7274 696e Verbose: bertin │ │ │ │ +00016b00: 6954 7261 636b 486f 6d6f 746f 7079 5f6c iTrackHomotopy_l │ │ │ │ +00016b10: 705f 7064 5f70 645f 7064 5f63 6d56 6572 p_pd_pd_pd_cmVer │ │ │ │ +00016b20: 626f 7365 3d3e 5f70 645f 7064 5f70 645f bose=>_pd_pd_pd_ │ │ │ │ +00016b30: 7270 0a20 2020 2020 2020 202c 203d 3e20 rp. , => │ │ │ │ +00016b40: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +00016b50: 7565 2066 616c 7365 2c20 4f70 7469 6f6e ue false, Option │ │ │ │ +00016b60: 2074 6f20 7369 6c65 6e63 6520 6164 6469 to silence addi │ │ │ │ +00016b70: 7469 6f6e 616c 206f 7574 7075 740a 0a44 tional output..D │ │ │ │ +00016b80: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00016b90: 3d3d 3d3d 3d3d 0a0a 4166 7465 7220 7275 ======..After ru │ │ │ │ +00016ba0: 6e6e 696e 6720 6d61 6b65 4d65 6d62 6572 nning makeMember │ │ │ │ +00016bb0: 7368 6970 4669 6c65 2042 6572 7469 6e69 shipFile Bertini │ │ │ │ +00016bc0: 2070 726f 6475 6365 7320 616e 2069 6e63 produces an inc │ │ │ │ +00016bd0: 6964 656e 6365 5f6d 6174 7269 7820 6669 idence_matrix fi │ │ │ │ +00016be0: 6c65 2e20 5468 650a 696e 6369 6465 6e63 le. The.incidenc │ │ │ │ +00016bf0: 655f 6d61 7472 6978 2073 6179 7320 7768 e_matrix says wh │ │ │ │ +00016c00: 6963 6820 706f 696e 7473 2062 656c 6f6e ich points belon │ │ │ │ +00016c10: 6720 746f 2077 6869 6368 2063 6f6d 706f g to which compo │ │ │ │ +00016c20: 6e65 6e74 732e 204f 7572 2069 6e63 6964 nents. Our incid │ │ │ │ +00016c30: 656e 6365 0a6d 6174 7269 7820 6973 2066 ence.matrix is f │ │ │ │ +00016c40: 6c61 7474 656e 6564 2074 6f20 6120 6c69 lattened to a li │ │ │ │ +00016c50: 7374 2e20 5468 6520 6e75 6d62 6572 206f st. The number o │ │ │ │ +00016c60: 6620 656c 656d 656e 7473 2069 6e20 7468 f elements in th │ │ │ │ +00016c70: 6549 4d20 6973 2065 7175 616c 2074 6f20 eIM is equal to │ │ │ │ +00016c80: 7468 650a 6e75 6d62 6572 206f 6620 706f the.number of po │ │ │ │ +00016c90: 696e 7473 2069 6e20 7468 6520 736f 6c75 ints in the solu │ │ │ │ +00016ca0: 7469 6f6e 7320 6669 6c65 2e20 4561 6368 tions file. Each │ │ │ │ +00016cb0: 2065 6c65 6d65 6e74 206f 6620 7468 6549 element of theI │ │ │ │ +00016cc0: 4d20 6973 2061 206c 6973 7420 6f66 0a73 M is a list of.s │ │ │ │ +00016cd0: 6571 7565 6e63 6573 206f 6620 3220 656c equences of 2 el │ │ │ │ +00016ce0: 656d 656e 7473 2028 636f 6469 6d2c 636f ements (codim,co │ │ │ │ +00016cf0: 6d70 6f6e 656e 7420 4e75 6d62 6572 292e mponent Number). │ │ │ │ +00016d00: 204e 6f74 6520 7468 6174 2077 6520 666f Note that we fo │ │ │ │ +00016d10: 6c6c 6f77 2074 6865 0a42 6572 7469 6e69 llow the.Bertini │ │ │ │ +00016d20: 2063 6f6e 7665 6e74 696f 6e20 616e 6420 convention and │ │ │ │ +00016d30: 7377 6974 6368 2066 726f 6d20 2864 696d switch from (dim │ │ │ │ +00016d40: 656e 7369 6f6e 2c63 6f6d 706f 6e65 6e74 ension,component │ │ │ │ +00016d50: 206e 756d 6265 7229 2069 6e64 6578 696e number) indexin │ │ │ │ +00016d60: 6720 746f 0a28 636f 6469 6d65 6e73 696f g to.(codimensio │ │ │ │ +00016d70: 6e2c 636f 6d70 6f6e 656e 7420 6e75 6d62 n,component numb │ │ │ │ +00016d80: 6572 2920 696e 6465 7869 6e67 2e0a 0a49 er) indexing...I │ │ │ │ +00016d90: 6620 7468 6520 4e61 6d65 496e 6369 6465 f the NameIncide │ │ │ │ +00016da0: 6e63 654d 6174 7269 7846 696c 6520 6f70 nceMatrixFile op │ │ │ │ +00016db0: 7469 6f6e 2069 7320 7365 7420 7768 656e tion is set when │ │ │ │ +00016dc0: 2077 6520 7761 6e74 2074 6f20 696d 706f we want to impo │ │ │ │ +00016dd0: 7274 2066 696c 6573 2077 6974 680a 6120 rt files with.a │ │ │ │ +00016de0: 6469 6666 6572 656e 7420 6e61 6d65 2e0a different name.. │ │ │ │ +00016df0: 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00016e40: 3120 3a20 6d61 6b65 4227 496e 7075 7446 1 : makeB'InputF │ │ │ │ -00016e50: 696c 6528 7374 6f72 6542 4d32 4669 6c65 ile(storeBM2File │ │ │ │ -00016e60: 732c 2020 2020 2020 2020 2020 2020 2020 s, │ │ │ │ +00016e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00016e40: 0a7c 6931 203a 206d 616b 6542 2749 6e70 .|i1 : makeB'Inp │ │ │ │ +00016e50: 7574 4669 6c65 2873 746f 7265 424d 3246 utFile(storeBM2F │ │ │ │ +00016e60: 696c 6573 2c20 2020 2020 2020 2020 2020 iles, │ │ │ │ 00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00016e90: 2020 2020 2020 2020 4265 7274 696e 6949 BertiniI │ │ │ │ -00016ea0: 6e70 7574 436f 6e66 6967 7572 6174 696f nputConfiguratio │ │ │ │ -00016eb0: 6e3d 3e7b 7b54 7261 636b 5479 7065 2c31 n=>{{TrackType,1 │ │ │ │ -00016ec0: 7d7d 2c20 2020 2020 2020 2020 2020 2020 }}, │ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016e90: 0a7c 2020 2020 2020 2020 2042 6572 7469 .| Berti │ │ │ │ +00016ea0: 6e69 496e 7075 7443 6f6e 6669 6775 7261 niInputConfigura │ │ │ │ +00016eb0: 7469 6f6e 3d3e 7b7b 5472 6163 6b54 7970 tion=>{{TrackTyp │ │ │ │ +00016ec0: 652c 317d 7d2c 2020 2020 2020 2020 2020 e,1}}, │ │ │ │ +00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016ee0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00016ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c41 ------------|.|A │ │ │ │ -00016f30: 6666 5661 7269 6162 6c65 4772 6f75 703d ffVariableGroup= │ │ │ │ -00016f40: 3e7b 782c 792c 7a7d 2c20 2020 2020 2020 >{x,y,z}, │ │ │ │ +00016f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00016f30: 0a7c 4166 6656 6172 6961 626c 6547 726f .|AffVariableGro │ │ │ │ +00016f40: 7570 3d3e 7b78 2c79 2c7a 7d2c 2020 2020 up=>{x,y,z}, │ │ │ │ 00016f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f70: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00016f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016f70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016f80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 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 2d2d 7c0a 7c42 ------------|.|B │ │ │ │ -00016fd0: 2750 6f6c 796e 6f6d 6961 6c73 3d3e 7b22 'Polynomials=>{" │ │ │ │ -00016fe0: 7a2a 2828 782b 792b 7a29 5e33 2d31 2922 z*((x+y+z)^3-1)" │ │ │ │ -00016ff0: 2c22 7a2a 2879 5e32 2d33 2b7a 2922 7d20 ,"z*(y^2-3+z)"} │ │ │ │ -00017000: 2020 2029 3b20 2020 2020 2020 2020 2020 ); │ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00017020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00016fd0: 0a7c 4227 506f 6c79 6e6f 6d69 616c 733d .|B'Polynomials= │ │ │ │ +00016fe0: 3e7b 227a 2a28 2878 2b79 2b7a 295e 332d >{"z*((x+y+z)^3- │ │ │ │ +00016ff0: 3129 222c 227a 2a28 795e 322d 332b 7a29 1)","z*(y^2-3+z) │ │ │ │ +00017000: 227d 2020 2020 293b 2020 2020 2020 2020 "} ); │ │ │ │ +00017010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017020: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00017030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00017070: 3220 3a20 7275 6e42 6572 7469 6e69 2873 2 : runBertini(s │ │ │ │ -00017080: 746f 7265 424d 3246 696c 6573 2920 2020 toreBM2Files) │ │ │ │ +00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00017070: 0a7c 6932 203a 2072 756e 4265 7274 696e .|i2 : runBertin │ │ │ │ +00017080: 6928 7374 6f72 6542 4d32 4669 6c65 7329 i(storeBM2Files) │ │ │ │ 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 7c0a 2b2d |.+- │ │ │ │ -000170c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000170c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000170d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000170e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000170f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00017110: 3320 3a20 6d61 6b65 5361 6d70 6c65 536f 3 : makeSampleSo │ │ │ │ -00017120: 6c75 7469 6f6e 7346 696c 6528 7374 6f72 lutionsFile(stor │ │ │ │ -00017130: 6542 4d32 4669 6c65 732c 322c 5370 6563 eBM2Files,2,Spec │ │ │ │ -00017140: 6966 7943 6f6d 706f 6e65 6e74 3d3e 7b31 ifyComponent=>{1 │ │ │ │ -00017150: 2c30 7d29 2020 2020 2020 2020 7c0a 2b2d ,0}) |.+- │ │ │ │ -00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00017110: 0a7c 6933 203a 206d 616b 6553 616d 706c .|i3 : makeSampl │ │ │ │ +00017120: 6553 6f6c 7574 696f 6e73 4669 6c65 2873 eSolutionsFile(s │ │ │ │ +00017130: 746f 7265 424d 3246 696c 6573 2c32 2c53 toreBM2Files,2,S │ │ │ │ +00017140: 7065 6369 6679 436f 6d70 6f6e 656e 743d pecifyComponent= │ │ │ │ +00017150: 3e7b 312c 307d 2920 2020 2020 2020 207c >{1,0}) | │ │ │ │ +00017160: 0a2b 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 2b0a 7c69 ------------+.|i │ │ │ │ -000171b0: 3420 3a20 6d61 6b65 4d65 6d62 6572 7368 4 : makeMembersh │ │ │ │ -000171c0: 6970 4669 6c65 2873 746f 7265 424d 3246 ipFile(storeBM2F │ │ │ │ -000171d0: 696c 6573 2c4e 616d 6553 6f6c 7574 696f iles,NameSolutio │ │ │ │ -000171e0: 6e73 4669 6c65 3d3e 2273 616d 706c 655f nsFile=>"sample_ │ │ │ │ -000171f0: 736f 6c75 7469 6f6e 735f 2020 7c0a 7c2d solutions_ |.|- │ │ │ │ -00017200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000171b0: 0a7c 6934 203a 206d 616b 654d 656d 6265 .|i4 : makeMembe │ │ │ │ +000171c0: 7273 6869 7046 696c 6528 7374 6f72 6542 rshipFile(storeB │ │ │ │ +000171d0: 4d32 4669 6c65 732c 4e61 6d65 536f 6c75 M2Files,NameSolu │ │ │ │ +000171e0: 7469 6f6e 7346 696c 653d 3e22 7361 6d70 tionsFile=>"samp │ │ │ │ +000171f0: 6c65 5f73 6f6c 7574 696f 6e73 5f20 207c le_solutions_ | │ │ │ │ +00017200: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00017210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c66 ------------|.|f │ │ │ │ -00017250: 696c 6522 2920 2020 2020 2020 2020 2020 ile") │ │ │ │ +00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00017250: 0a7c 6669 6c65 2229 2020 2020 2020 2020 .|file") │ │ │ │ 00017260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017290: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000172a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000172a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000172b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000172c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000172d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000172e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -000172f0: 3520 3a20 7468 6549 4d3d 696d 706f 7274 5 : theIM=import │ │ │ │ -00017300: 496e 6369 6465 6e63 654d 6174 7269 7828 IncidenceMatrix( │ │ │ │ -00017310: 7374 6f72 6542 4d32 4669 6c65 7329 2020 storeBM2Files) │ │ │ │ -00017320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017330: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00017340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000172e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000172f0: 0a7c 6935 203a 2074 6865 494d 3d69 6d70 .|i5 : theIM=imp │ │ │ │ +00017300: 6f72 7449 6e63 6964 656e 6365 4d61 7472 ortIncidenceMatr │ │ │ │ +00017310: 6978 2873 746f 7265 424d 3246 696c 6573 ix(storeBM2Files │ │ │ │ +00017320: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017340: 0a7c 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 2020 2020 2020 2020 │ │ │ │ -00017380: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00017390: 3520 3d20 7b7b 2832 2c20 3029 7d2c 207b 5 = {{(2, 0)}, { │ │ │ │ -000173a0: 2832 2c20 3029 7d7d 2020 2020 2020 2020 (2, 0)}} │ │ │ │ +00017380: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017390: 0a7c 6f35 203d 207b 7b28 322c 2030 297d .|o5 = {{(2, 0)} │ │ │ │ +000173a0: 2c20 7b28 322c 2030 297d 7d20 2020 2020 , {(2, 0)}} │ │ │ │ 000173b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000173c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000173d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000173e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000173d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000173e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000173f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00017430: 3520 3a20 4c69 7374 2020 2020 2020 2020 5 : List │ │ │ │ +00017420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017430: 0a7c 6f35 203a 204c 6973 7420 2020 2020 .|o5 : List │ │ │ │ 00017440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017470: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00017480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017470: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017480: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00017490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000174a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000174b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000174c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 ------------+..W │ │ │ │ -000174d0: 6179 7320 746f 2075 7365 2069 6d70 6f72 ays to use impor │ │ │ │ -000174e0: 7449 6e63 6964 656e 6365 4d61 7472 6978 tIncidenceMatrix │ │ │ │ -000174f0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +000174c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000174d0: 0a0a 5761 7973 2074 6f20 7573 6520 696d ..Ways to use im │ │ │ │ +000174e0: 706f 7274 496e 6369 6465 6e63 654d 6174 portIncidenceMat │ │ │ │ +000174f0: 7269 783a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d rix:.=========== │ │ │ │ 00017500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017510: 3d3d 3d3d 0a0a 2020 2a20 2269 6d70 6f72 ====.. * "impor │ │ │ │ -00017520: 7449 6e63 6964 656e 6365 4d61 7472 6978 tIncidenceMatrix │ │ │ │ -00017530: 2853 7472 696e 6729 220a 0a46 6f72 2074 (String)"..For t │ │ │ │ -00017540: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00017550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017560: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00017570: 7465 2069 6d70 6f72 7449 6e63 6964 656e te importInciden │ │ │ │ -00017580: 6365 4d61 7472 6978 3a20 696d 706f 7274 ceMatrix: import │ │ │ │ -00017590: 496e 6369 6465 6e63 654d 6174 7269 782c IncidenceMatrix, │ │ │ │ -000175a0: 2069 7320 6120 2a6e 6f74 650a 6d65 7468 is a *note.meth │ │ │ │ -000175b0: 6f64 2066 756e 6374 696f 6e20 7769 7468 od function with │ │ │ │ -000175c0: 206f 7074 696f 6e73 3a20 284d 6163 6175 options: (Macau │ │ │ │ -000175d0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -000175e0: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e nctionWithOption │ │ │ │ -000175f0: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d s,...----------- │ │ │ │ +00017510: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 696d =======.. * "im │ │ │ │ +00017520: 706f 7274 496e 6369 6465 6e63 654d 6174 portIncidenceMat │ │ │ │ +00017530: 7269 7828 5374 7269 6e67 2922 0a0a 466f rix(String)"..Fo │ │ │ │ +00017540: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00017550: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00017560: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00017570: 2a6e 6f74 6520 696d 706f 7274 496e 6369 *note importInci │ │ │ │ +00017580: 6465 6e63 654d 6174 7269 783a 2069 6d70 denceMatrix: imp │ │ │ │ +00017590: 6f72 7449 6e63 6964 656e 6365 4d61 7472 ortIncidenceMatr │ │ │ │ +000175a0: 6978 2c20 6973 2061 202a 6e6f 7465 0a6d ix, is a *note.m │ │ │ │ +000175b0: 6574 686f 6420 6675 6e63 7469 6f6e 2077 ethod function w │ │ │ │ +000175c0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61 ith options: (Ma │ │ │ │ +000175d0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +000175e0: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ +000175f0: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ 00017600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017640: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00017650: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00017660: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00017670: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00017680: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -00017690: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -000176a0: 7932 2f70 6163 6b61 6765 732f 4265 7274 y2/packages/Bert │ │ │ │ -000176b0: 696e 692e 6d32 3a0a 3332 3232 3a30 2e0a ini.m2:.3222:0.. │ │ │ │ -000176c0: 1f0a 4669 6c65 3a20 4265 7274 696e 692e ..File: Bertini. │ │ │ │ -000176d0: 696e 666f 2c20 4e6f 6465 3a20 696d 706f info, Node: impo │ │ │ │ -000176e0: 7274 4d61 696e 4461 7461 4669 6c65 2c20 rtMainDataFile, │ │ │ │ -000176f0: 4e65 7874 3a20 696d 706f 7274 5061 7261 Next: importPara │ │ │ │ -00017700: 6d65 7465 7246 696c 652c 2050 7265 763a meterFile, Prev: │ │ │ │ -00017710: 2069 6d70 6f72 7449 6e63 6964 656e 6365 importIncidence │ │ │ │ -00017720: 4d61 7472 6978 2c20 5570 3a20 546f 700a Matrix, Up: Top. │ │ │ │ -00017730: 0a69 6d70 6f72 744d 6169 6e44 6174 6146 .importMainDataF │ │ │ │ -00017740: 696c 6520 2d2d 2054 6869 7320 6675 6e63 ile -- This func │ │ │ │ -00017750: 7469 6f6e 2069 6d70 6f72 7473 2070 6f69 tion imports poi │ │ │ │ -00017760: 6e74 7320 6672 6f6d 2074 6865 206d 6169 nts from the mai │ │ │ │ -00017770: 6e20 6461 7461 2066 696c 6520 666f 726d n data file form │ │ │ │ -00017780: 2061 2042 6572 7469 6e69 2072 756e 2e0a a Bertini run.. │ │ │ │ -00017790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00017640: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00017650: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00017660: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00017670: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00017680: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00017690: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ +000176a0: 756c 6179 322f 7061 636b 6167 6573 2f42 ulay2/packages/B │ │ │ │ +000176b0: 6572 7469 6e69 2e6d 323a 0a33 3232 323a ertini.m2:.3222: │ │ │ │ +000176c0: 302e 0a1f 0a46 696c 653a 2042 6572 7469 0....File: Berti │ │ │ │ +000176d0: 6e69 2e69 6e66 6f2c 204e 6f64 653a 2069 ni.info, Node: i │ │ │ │ +000176e0: 6d70 6f72 744d 6169 6e44 6174 6146 696c mportMainDataFil │ │ │ │ +000176f0: 652c 204e 6578 743a 2069 6d70 6f72 7450 e, Next: importP │ │ │ │ +00017700: 6172 616d 6574 6572 4669 6c65 2c20 5072 arameterFile, Pr │ │ │ │ +00017710: 6576 3a20 696d 706f 7274 496e 6369 6465 ev: importIncide │ │ │ │ +00017720: 6e63 654d 6174 7269 782c 2055 703a 2054 nceMatrix, Up: T │ │ │ │ +00017730: 6f70 0a0a 696d 706f 7274 4d61 696e 4461 op..importMainDa │ │ │ │ +00017740: 7461 4669 6c65 202d 2d20 5468 6973 2066 taFile -- This f │ │ │ │ +00017750: 756e 6374 696f 6e20 696d 706f 7274 7320 unction imports │ │ │ │ +00017760: 706f 696e 7473 2066 726f 6d20 7468 6520 points from the │ │ │ │ +00017770: 6d61 696e 2064 6174 6120 6669 6c65 2066 main data file f │ │ │ │ +00017780: 6f72 6d20 6120 4265 7274 696e 6920 7275 orm a Bertini ru │ │ │ │ +00017790: 6e2e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n..************* │ │ │ │ 000177a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000177b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000177c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000177d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000177e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -000177f0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00017800: 2020 2020 696d 706f 7274 4d61 696e 4461 importMainDa │ │ │ │ -00017810: 7461 4669 6c65 2874 6865 4469 7229 0a20 taFile(theDir). │ │ │ │ -00017820: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00017830: 202a 2074 6865 4469 722c 2061 202a 6e6f * theDir, a *no │ │ │ │ -00017840: 7465 2073 7472 696e 673a 2028 4d61 6361 te string: (Maca │ │ │ │ -00017850: 756c 6179 3244 6f63 2953 7472 696e 672c ulay2Doc)String, │ │ │ │ -00017860: 2c20 5468 6520 6469 7265 6374 6f72 7920 , The directory │ │ │ │ -00017870: 7768 6572 6520 7468 650a 2020 2020 2020 where the. │ │ │ │ -00017880: 2020 6d61 696e 5f64 6174 6120 6669 6c65 main_data file │ │ │ │ -00017890: 2069 7320 6c6f 6361 7465 642e 0a20 202a is located.. * │ │ │ │ -000178a0: 202a 6e6f 7465 204f 7074 696f 6e61 6c20 *note Optional │ │ │ │ -000178b0: 696e 7075 7473 3a20 284d 6163 6175 6c61 inputs: (Macaula │ │ │ │ -000178c0: 7932 446f 6329 7573 696e 6720 6675 6e63 y2Doc)using func │ │ │ │ -000178d0: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -000178e0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ -000178f0: 2020 202a 204d 3250 7265 6369 7369 6f6e * M2Precision │ │ │ │ -00017900: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -00017910: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ -00017920: 2064 6566 6175 6c74 2076 616c 7565 2035 default value 5 │ │ │ │ -00017930: 332c 200a 2020 2020 2020 2a20 4e61 6d65 3, . * Name │ │ │ │ -00017940: 4d61 696e 4461 7461 4669 6c65 2028 6d69 MainDataFile (mi │ │ │ │ -00017950: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -00017960: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -00017970: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -00017980: 2020 2022 6d61 696e 5f64 6174 6122 2c0a "main_data",. │ │ │ │ -00017990: 2020 2020 2020 2a20 5061 7468 4c69 7374 * PathList │ │ │ │ -000179a0: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -000179b0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ -000179c0: 2064 6566 6175 6c74 2076 616c 7565 206e default value n │ │ │ │ -000179d0: 756c 6c2c 200a 2020 2020 2020 2a20 5370 ull, . * Sp │ │ │ │ -000179e0: 6563 6966 7944 696d 2028 6d69 7373 696e ecifyDim (missin │ │ │ │ -000179f0: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ -00017a00: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00017a10: 2076 616c 7565 2066 616c 7365 2c20 0a20 value false, . │ │ │ │ -00017a20: 2020 2020 202a 202a 6e6f 7465 2056 6572 * *note Ver │ │ │ │ -00017a30: 626f 7365 3a20 6265 7274 696e 6954 7261 bose: bertiniTra │ │ │ │ -00017a40: 636b 486f 6d6f 746f 7079 5f6c 705f 7064 ckHomotopy_lp_pd │ │ │ │ -00017a50: 5f70 645f 7064 5f63 6d56 6572 626f 7365 _pd_pd_cmVerbose │ │ │ │ -00017a60: 3d3e 5f70 645f 7064 5f70 645f 7270 0a20 =>_pd_pd_pd_rp. │ │ │ │ -00017a70: 2020 2020 2020 202c 203d 3e20 2e2e 2e2c , => ..., │ │ │ │ -00017a80: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ -00017a90: 616c 7365 2c20 4f70 7469 6f6e 2074 6f20 alse, Option to │ │ │ │ -00017aa0: 7369 6c65 6e63 6520 6164 6469 7469 6f6e silence addition │ │ │ │ -00017ab0: 616c 206f 7574 7075 740a 0a44 6573 6372 al output..Descr │ │ │ │ -00017ac0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00017ad0: 3d3d 0a0a 5468 6973 2066 756e 6374 696f ==..This functio │ │ │ │ -00017ae0: 6e20 646f 6573 206e 6f74 2069 6d70 6f72 n does not impor │ │ │ │ -00017af0: 7420 6120 6c69 7374 206f 6620 636f 6f72 t a list of coor │ │ │ │ -00017b00: 6469 6e61 7465 732e 2049 6e73 7465 6164 dinates. Instead │ │ │ │ -00017b10: 2069 7420 696d 706f 7274 7320 706f 696e it imports poin │ │ │ │ -00017b20: 7473 0a66 726f 6d20 6120 6d61 696e 5f64 ts.from a main_d │ │ │ │ -00017b30: 6174 6120 6669 6c65 2e20 5468 6573 6520 ata file. These │ │ │ │ -00017b40: 706f 696e 7473 2063 6f6e 7461 696e 2063 points contain c │ │ │ │ -00017b50: 6f6f 7264 696e 6174 6573 2c20 636f 6e64 oordinates, cond │ │ │ │ -00017b60: 6974 696f 6e20 6e75 6d62 6572 732c 2061 ition numbers, a │ │ │ │ -00017b70: 6e64 0a65 7463 2e20 5468 6520 696e 666f nd.etc. The info │ │ │ │ -00017b80: 726d 6174 696f 6e20 7468 6520 706f 696e rmation the poin │ │ │ │ -00017b90: 7473 2063 6f6e 7461 696e 2064 6570 656e ts contain depen │ │ │ │ -00017ba0: 6420 6f6e 2069 6620 7265 6765 6e65 7261 d on if regenera │ │ │ │ -00017bb0: 7469 6f6e 2077 6173 2075 7365 6420 616e tion was used an │ │ │ │ -00017bc0: 640a 6966 2061 2054 7261 636b 5479 7065 d.if a TrackType │ │ │ │ -00017bd0: 2030 206f 7220 3120 7761 7320 7573 6564 0 or 1 was used │ │ │ │ -00017be0: 2e20 5768 656e 2054 7261 636b 5479 7065 . When TrackType │ │ │ │ -00017bf0: 2031 2069 7320 7573 6564 2c20 554e 434c 1 is used, UNCL │ │ │ │ -00017c00: 4153 5349 4649 4544 2070 6f69 6e74 730a ASSIFIED points. │ │ │ │ -00017c10: 7769 6c6c 2068 6176 6520 636f 6d70 6f6e will have compon │ │ │ │ -00017c20: 656e 7420 6e75 6d62 6572 202d 312e 0a0a ent number -1... │ │ │ │ -00017c30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +000177e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000177f0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00017800: 2020 2020 2020 2069 6d70 6f72 744d 6169 importMai │ │ │ │ +00017810: 6e44 6174 6146 696c 6528 7468 6544 6972 nDataFile(theDir │ │ │ │ +00017820: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +00017830: 2020 2020 2a20 7468 6544 6972 2c20 6120 * theDir, a │ │ │ │ +00017840: 2a6e 6f74 6520 7374 7269 6e67 3a20 284d *note string: (M │ │ │ │ +00017850: 6163 6175 6c61 7932 446f 6329 5374 7269 acaulay2Doc)Stri │ │ │ │ +00017860: 6e67 2c2c 2054 6865 2064 6972 6563 746f ng,, The directo │ │ │ │ +00017870: 7279 2077 6865 7265 2074 6865 0a20 2020 ry where the. │ │ │ │ +00017880: 2020 2020 206d 6169 6e5f 6461 7461 2066 main_data f │ │ │ │ +00017890: 696c 6520 6973 206c 6f63 6174 6564 2e0a ile is located.. │ │ │ │ +000178a0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +000178b0: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ +000178c0: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ +000178d0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +000178e0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ +000178f0: 2020 2020 2020 2a20 4d32 5072 6563 6973 * M2Precis │ │ │ │ +00017900: 696f 6e20 286d 6973 7369 6e67 2064 6f63 ion (missing doc │ │ │ │ +00017910: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +00017920: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +00017930: 6520 3533 2c20 0a20 2020 2020 202a 204e e 53, . * N │ │ │ │ +00017940: 616d 654d 6169 6e44 6174 6146 696c 6520 ameMainDataFile │ │ │ │ +00017950: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +00017960: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ +00017970: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ +00017980: 2020 2020 2020 226d 6169 6e5f 6461 7461 "main_data │ │ │ │ +00017990: 222c 0a20 2020 2020 202a 2050 6174 684c ",. * PathL │ │ │ │ +000179a0: 6973 7420 286d 6973 7369 6e67 2064 6f63 ist (missing doc │ │ │ │ +000179b0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +000179c0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +000179d0: 6520 6e75 6c6c 2c20 0a20 2020 2020 202a e null, . * │ │ │ │ +000179e0: 2053 7065 6369 6679 4469 6d20 286d 6973 SpecifyDim (mis │ │ │ │ +000179f0: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ +00017a00: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ +00017a10: 756c 7420 7661 6c75 6520 6661 6c73 652c ult value false, │ │ │ │ +00017a20: 200a 2020 2020 2020 2a20 2a6e 6f74 6520 . * *note │ │ │ │ +00017a30: 5665 7262 6f73 653a 2062 6572 7469 6e69 Verbose: bertini │ │ │ │ +00017a40: 5472 6163 6b48 6f6d 6f74 6f70 795f 6c70 TrackHomotopy_lp │ │ │ │ +00017a50: 5f70 645f 7064 5f70 645f 636d 5665 7262 _pd_pd_pd_cmVerb │ │ │ │ +00017a60: 6f73 653d 3e5f 7064 5f70 645f 7064 5f72 ose=>_pd_pd_pd_r │ │ │ │ +00017a70: 700a 2020 2020 2020 2020 2c20 3d3e 202e p. , => . │ │ │ │ +00017a80: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +00017a90: 6520 6661 6c73 652c 204f 7074 696f 6e20 e false, Option │ │ │ │ +00017aa0: 746f 2073 696c 656e 6365 2061 6464 6974 to silence addit │ │ │ │ +00017ab0: 696f 6e61 6c20 6f75 7470 7574 0a0a 4465 ional output..De │ │ │ │ +00017ac0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00017ad0: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63 =====..This func │ │ │ │ +00017ae0: 7469 6f6e 2064 6f65 7320 6e6f 7420 696d tion does not im │ │ │ │ +00017af0: 706f 7274 2061 206c 6973 7420 6f66 2063 port a list of c │ │ │ │ +00017b00: 6f6f 7264 696e 6174 6573 2e20 496e 7374 oordinates. Inst │ │ │ │ +00017b10: 6561 6420 6974 2069 6d70 6f72 7473 2070 ead it imports p │ │ │ │ +00017b20: 6f69 6e74 730a 6672 6f6d 2061 206d 6169 oints.from a mai │ │ │ │ +00017b30: 6e5f 6461 7461 2066 696c 652e 2054 6865 n_data file. The │ │ │ │ +00017b40: 7365 2070 6f69 6e74 7320 636f 6e74 6169 se points contai │ │ │ │ +00017b50: 6e20 636f 6f72 6469 6e61 7465 732c 2063 n coordinates, c │ │ │ │ +00017b60: 6f6e 6469 7469 6f6e 206e 756d 6265 7273 ondition numbers │ │ │ │ +00017b70: 2c20 616e 640a 6574 632e 2054 6865 2069 , and.etc. The i │ │ │ │ +00017b80: 6e66 6f72 6d61 7469 6f6e 2074 6865 2070 nformation the p │ │ │ │ +00017b90: 6f69 6e74 7320 636f 6e74 6169 6e20 6465 oints contain de │ │ │ │ +00017ba0: 7065 6e64 206f 6e20 6966 2072 6567 656e pend on if regen │ │ │ │ +00017bb0: 6572 6174 696f 6e20 7761 7320 7573 6564 eration was used │ │ │ │ +00017bc0: 2061 6e64 0a69 6620 6120 5472 6163 6b54 and.if a TrackT │ │ │ │ +00017bd0: 7970 6520 3020 6f72 2031 2077 6173 2075 ype 0 or 1 was u │ │ │ │ +00017be0: 7365 642e 2057 6865 6e20 5472 6163 6b54 sed. When TrackT │ │ │ │ +00017bf0: 7970 6520 3120 6973 2075 7365 642c 2055 ype 1 is used, U │ │ │ │ +00017c00: 4e43 4c41 5353 4946 4945 4420 706f 696e NCLASSIFIED poin │ │ │ │ +00017c10: 7473 0a77 696c 6c20 6861 7665 2063 6f6d ts.will have com │ │ │ │ +00017c20: 706f 6e65 6e74 206e 756d 6265 7220 2d31 ponent number -1 │ │ │ │ +00017c30: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00017c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00017c80: 7c69 3120 3a20 6d61 6b65 4227 496e 7075 |i1 : makeB'Inpu │ │ │ │ -00017c90: 7446 696c 6528 7374 6f72 6542 4d32 4669 tFile(storeBM2Fi │ │ │ │ -00017ca0: 6c65 732c 2020 2020 2020 2020 2020 2020 les, │ │ │ │ +00017c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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│ │ +00018300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018310: 207c 0a7c 3120 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 00018320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 7c0a |. │ │ │ │ -00018360: 7c31 2020 2020 2020 2020 2020 2020 2020 |1 │ │ │ │ +00018350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018360: 207c 0a7c 3120 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 00018370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c0a |. │ │ │ │ -000183b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000183a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2020 2020 2020 7c0a |. │ │ │ │ -00018400: 7c6f 3420 3d20 7b7d 2020 2020 2020 2020 |o4 = {} │ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018400: 207c 0a7c 6f34 203d 207b 7d20 2020 2020 |.|o4 = {} │ │ │ │ 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 2020 7c0a |. │ │ │ │ -00018450: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018450: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 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 │ │ │ │ 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--------------|. │ │ │ │ -00018540: 7c63 616e 2063 686f 6f73 6520 7768 6963 |can choose whic │ │ │ │ -00018550: 6820 6469 6d65 6e73 696f 6e20 7765 2069 h dimension we i │ │ │ │ -00018560: 6d70 6f72 7420 706f 696e 7473 2066 726f mport points fro │ │ │ │ -00018570: 6d2e 2054 6865 7265 2061 7265 206e 6f20 m. There are no │ │ │ │ -00018580: 7769 746e 6573 7320 706f 696e 7473 7c0a witness points|. │ │ │ │ -00018590: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +00018530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018540: 2d7c 0a7c 6361 6e20 6368 6f6f 7365 2077 -|.|can choose w │ │ │ │ +00018550: 6869 6368 2064 696d 656e 7369 6f6e 2077 hich dimension w │ │ │ │ +00018560: 6520 696d 706f 7274 2070 6f69 6e74 7320 e import points │ │ │ │ +00018570: 6672 6f6d 2e20 5468 6572 6520 6172 6520 from. There are │ │ │ │ +00018580: 6e6f 2077 6974 6e65 7373 2070 6f69 6e74 no witness point │ │ │ │ +00018590: 737c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d s|.|------------ │ │ │ │ 000185a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000185e0: 7c69 6e20 6469 6d65 6e73 696f 6e20 312e |in dimension 1. │ │ │ │ -000185f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000185e0: 2d7c 0a7c 696e 2064 696d 656e 7369 6f6e -|.|in dimension │ │ │ │ +000185f0: 2031 2e20 2020 2020 2020 2020 2020 2020 1. │ │ │ │ 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 2020 7c0a |. │ │ │ │ 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------- │ │ │ │ 000188c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000188d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00018900: 7c20 2020 2020 2d2e 3330 3430 3534 2d2e | -.304054-. │ │ │ │ -00018910: 3935 3338 3934 2a69 697d 7d7d 2020 2020 953894*ii}}} │ │ │ │ +000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018900: 2d7c 0a7c 2020 2020 202d 2e33 3034 3035 -|.| -.30405 │ │ │ │ +00018910: 342d 2e39 3533 3839 342a 6969 7d7d 7d20 4-.953894*ii}}} │ │ │ │ 00018920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018940: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00018950: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00018940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2020 2020 2020 7c0a |. │ │ │ │ -000189a0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000189a0: 207c 0a7c 6f35 203a 204c 6973 7420 2020 |.|o5 : List │ │ │ │ 000189b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000189f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000189f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00018a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00018a40: 0a57 6179 7320 746f 2075 7365 2069 6d70 .Ways to use imp │ │ │ │ -00018a50: 6f72 744d 6169 6e44 6174 6146 696c 653a ortMainDataFile: │ │ │ │ -00018a60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00018a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018a40: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520 -+..Ways to use │ │ │ │ +00018a50: 696d 706f 7274 4d61 696e 4461 7461 4669 importMainDataFi │ │ │ │ +00018a60: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ 00018a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00018a80: 0a0a 2020 2a20 2269 6d70 6f72 744d 6169 .. * "importMai │ │ │ │ -00018a90: 6e44 6174 6146 696c 6528 5374 7269 6e67 nDataFile(String │ │ │ │ -00018aa0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -00018ab0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00018ac0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -00018ad0: 626a 6563 7420 2a6e 6f74 6520 696d 706f bject *note impo │ │ │ │ -00018ae0: 7274 4d61 696e 4461 7461 4669 6c65 3a20 rtMainDataFile: │ │ │ │ -00018af0: 696d 706f 7274 4d61 696e 4461 7461 4669 importMainDataFi │ │ │ │ -00018b00: 6c65 2c20 6973 2061 202a 6e6f 7465 206d le, is a *note m │ │ │ │ -00018b10: 6574 686f 640a 6675 6e63 7469 6f6e 2077 ethod.function w │ │ │ │ -00018b20: 6974 6820 6f70 7469 6f6e 733a 2028 4d61 ith options: (Ma │ │ │ │ -00018b30: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00018b40: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ -00018b50: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ +00018a80: 3d3d 3d0a 0a20 202a 2022 696d 706f 7274 ===.. * "import │ │ │ │ +00018a90: 4d61 696e 4461 7461 4669 6c65 2853 7472 MainDataFile(Str │ │ │ │ +00018aa0: 696e 6729 220a 0a46 6f72 2074 6865 2070 ing)"..For the p │ │ │ │ +00018ab0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00018ac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00018ad0: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ +00018ae0: 6d70 6f72 744d 6169 6e44 6174 6146 696c mportMainDataFil │ │ │ │ +00018af0: 653a 2069 6d70 6f72 744d 6169 6e44 6174 e: importMainDat │ │ │ │ +00018b00: 6146 696c 652c 2069 7320 6120 2a6e 6f74 aFile, is a *not │ │ │ │ +00018b10: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ +00018b20: 6e20 7769 7468 206f 7074 696f 6e73 3a20 n with options: │ │ │ │ +00018b30: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00018b40: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00018b50: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ 00018b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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ni.info, Node: i │ │ │ │ -00018c40: 6d70 6f72 7450 6172 616d 6574 6572 4669 mportParameterFi │ │ │ │ -00018c50: 6c65 2c20 4e65 7874 3a20 696d 706f 7274 le, Next: import │ │ │ │ -00018c60: 536f 6c75 7469 6f6e 7346 696c 652c 2050 SolutionsFile, P │ │ │ │ -00018c70: 7265 763a 2069 6d70 6f72 744d 6169 6e44 rev: importMainD │ │ │ │ -00018c80: 6174 6146 696c 652c 2055 703a 2054 6f70 ataFile, Up: Top │ │ │ │ -00018c90: 0a0a 696d 706f 7274 5061 7261 6d65 7465 ..importParamete │ │ │ │ -00018ca0: 7246 696c 6520 2d2d 2049 6d70 6f72 7473 rFile -- Imports │ │ │ │ -00018cb0: 2070 6172 616d 6574 6572 7320 6672 6f6d parameters from │ │ │ │ -00018cc0: 2061 2042 6572 7469 6e69 2070 6172 616d a Bertini param │ │ │ │ -00018cd0: 6574 6572 2066 696c 652e 0a2a 2a2a 2a2a eter file..***** │ │ │ │ +00018ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00018bb0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00018bc0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is 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+00018ca0: 6574 6572 4669 6c65 202d 2d20 496d 706f eterFile -- Impo │ │ │ │ +00018cb0: 7274 7320 7061 7261 6d65 7465 7273 2066 rts parameters f │ │ │ │ +00018cc0: 726f 6d20 6120 4265 7274 696e 6920 7061 rom a Bertini pa │ │ │ │ +00018cd0: 7261 6d65 7465 7220 6669 6c65 2e0a 2a2a rameter file..** │ │ │ │ 00018ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00018cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00018d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00018d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00018d20: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -00018d30: 0a20 2020 2020 2020 2069 6d70 6f72 7450 . importP │ │ │ │ -00018d40: 6172 616d 6574 6572 4669 6c65 2873 290a arameterFile(s). │ │ │ │ -00018d50: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00018d60: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ -00018d70: 7472 696e 673a 2028 4d61 6361 756c 6179 tring: (Macaulay │ │ │ │ -00018d80: 3244 6f63 2953 7472 696e 672c 2c20 5468 2Doc)String,, Th │ │ │ │ -00018d90: 6520 6469 7265 6374 6f72 7920 7768 6572 e directory wher │ │ │ │ -00018da0: 6520 7468 6520 6669 6c65 0a20 2020 2020 e the file. │ │ │ │ -00018db0: 2020 2069 7320 7374 6f72 6564 2e0a 2020 is stored.. │ │ │ │ -00018dc0: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c * *note Optional │ │ │ │ -00018dd0: 2069 6e70 7574 733a 2028 4d61 6361 756c inputs: (Macaul │ │ │ │ -00018de0: 6179 3244 6f63 2975 7369 6e67 2066 756e ay2Doc)using fun │ │ │ │ -00018df0: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -00018e00: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020 onal inputs,:. │ │ │ │ -00018e10: 2020 2020 2a20 4d32 5072 6563 6973 696f * M2Precisio │ │ │ │ -00018e20: 6e20 286d 6973 7369 6e67 2064 6f63 756d n (missing docum │ │ │ │ -00018e30: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -00018e40: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -00018e50: 3533 2c20 0a20 2020 2020 202a 204e 616d 53, . * Nam │ │ │ │ -00018e60: 6550 6172 616d 6574 6572 4669 6c65 2028 eParameterFile ( │ │ │ │ -00018e70: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -00018e80: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -00018e90: 6566 6175 6c74 2076 616c 7565 0a20 2020 efault value. │ │ │ │ -00018ea0: 2020 2020 2022 6669 6e61 6c5f 7061 7261 "final_para │ │ │ │ -00018eb0: 6d65 7465 7273 222c 0a20 2020 2020 202a meters",. * │ │ │ │ -00018ec0: 2053 746f 7261 6765 466f 6c64 6572 2028 StorageFolder ( │ │ │ │ -00018ed0: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -00018ee0: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -00018ef0: 6566 6175 6c74 2076 616c 7565 206e 756c efault value nul │ │ │ │ -00018f00: 6c2c 200a 0a44 6573 6372 6970 7469 6f6e l, ..Description │ │ │ │ -00018f10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4166 .===========..Af │ │ │ │ -00018f20: 7465 7220 4265 7274 696e 6920 646f 6573 ter Bertini does │ │ │ │ -00018f30: 2061 2070 6172 616d 6574 6572 2068 6f6d a parameter hom │ │ │ │ -00018f40: 6f74 6f70 7920 6d61 6e79 2066 696c 6573 otopy many files │ │ │ │ -00018f50: 2061 7265 2063 7265 6174 6564 2e20 5468 are created. Th │ │ │ │ -00018f60: 6973 2066 756e 6374 696f 6e0a 696d 706f is function.impo │ │ │ │ -00018f70: 7274 7320 7468 6520 7061 7261 6d65 7465 rts the paramete │ │ │ │ -00018f80: 7273 2066 726f 6d20 2074 6865 2022 6669 rs from the "fi │ │ │ │ -00018f90: 6e61 6c5f 7061 7261 6d65 7465 7273 2220 nal_parameters" │ │ │ │ -00018fa0: 6669 6c65 2061 7320 7468 6520 6465 6661 file as the defa │ │ │ │ -00018fb0: 756c 742e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ult...+--------- │ │ │ │ +00018d20: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ +00018d30: 653a 200a 2020 2020 2020 2020 696d 706f e: . impo │ │ │ │ +00018d40: 7274 5061 7261 6d65 7465 7246 696c 6528 rtParameterFile( │ │ │ │ +00018d50: 7329 0a20 202a 2049 6e70 7574 733a 0a20 s). * Inputs:. │ │ │ │ +00018d60: 2020 2020 202a 2073 2c20 6120 2a6e 6f74 * s, a *not │ │ │ │ +00018d70: 6520 7374 7269 6e67 3a20 284d 6163 6175 e string: (Macau │ │ │ │ +00018d80: 6c61 7932 446f 6329 5374 7269 6e67 2c2c lay2Doc)String,, │ │ │ │ +00018d90: 2054 6865 2064 6972 6563 746f 7279 2077 The directory w │ │ │ │ +00018da0: 6865 7265 2074 6865 2066 696c 650a 2020 here the file. │ │ │ │ +00018db0: 2020 2020 2020 6973 2073 746f 7265 642e is stored. │ │ │ │ +00018dc0: 0a20 202a 202a 6e6f 7465 204f 7074 696f . * *note Optio │ │ │ │ +00018dd0: 6e61 6c20 696e 7075 7473 3a20 284d 6163 nal inputs: (Mac │ │ │ │ +00018de0: 6175 6c61 7932 446f 6329 7573 696e 6720 aulay2Doc)using │ │ │ │ +00018df0: 6675 6e63 7469 6f6e 7320 7769 7468 206f functions with o │ │ │ │ +00018e00: 7074 696f 6e61 6c20 696e 7075 7473 2c3a ptional inputs,: │ │ │ │ +00018e10: 0a20 2020 2020 202a 204d 3250 7265 6369 . * M2Preci │ │ │ │ +00018e20: 7369 6f6e 2028 6d69 7373 696e 6720 646f sion (missing do │ │ │ │ +00018e30: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +00018e40: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +00018e50: 7565 2035 332c 200a 2020 2020 2020 2a20 ue 53, . * │ │ │ │ +00018e60: 4e61 6d65 5061 7261 6d65 7465 7246 696c NameParameterFil │ │ │ │ +00018e70: 6520 286d 6973 7369 6e67 2064 6f63 756d e (missing docum │ │ │ │ +00018e80: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +00018e90: 2c20 6465 6661 756c 7420 7661 6c75 650a , default value. │ │ │ │ +00018ea0: 2020 2020 2020 2020 2266 696e 616c 5f70 "final_p │ │ │ │ +00018eb0: 6172 616d 6574 6572 7322 2c0a 2020 2020 arameters",. │ │ │ │ +00018ec0: 2020 2a20 5374 6f72 6167 6546 6f6c 6465 * StorageFolde │ │ │ │ +00018ed0: 7220 286d 6973 7369 6e67 2064 6f63 756d r (missing docum │ │ │ │ +00018ee0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +00018ef0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00018f00: 6e75 6c6c 2c20 0a0a 4465 7363 7269 7074 null, ..Descript │ │ │ │ +00018f10: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00018f20: 0a41 6674 6572 2042 6572 7469 6e69 2064 .After Bertini d │ │ │ │ +00018f30: 6f65 7320 6120 7061 7261 6d65 7465 7220 oes a parameter │ │ │ │ +00018f40: 686f 6d6f 746f 7079 206d 616e 7920 6669 homotopy many fi │ │ │ │ +00018f50: 6c65 7320 6172 6520 6372 6561 7465 642e les are created. │ │ │ │ +00018f60: 2054 6869 7320 6675 6e63 7469 6f6e 0a69 This function.i │ │ │ │ +00018f70: 6d70 6f72 7473 2074 6865 2070 6172 616d mports the param │ │ │ │ +00018f80: 6574 6572 7320 6672 6f6d 2020 7468 6520 eters from the │ │ │ │ +00018f90: 2266 696e 616c 5f70 6172 616d 6574 6572 "final_parameter │ │ │ │ +00018fa0: 7322 2066 696c 6520 6173 2074 6865 2064 s" file as the d │ │ │ │ +00018fb0: 6566 6175 6c74 2e0a 0a2b 2d2d 2d2d 2d2d efault...+------ │ │ │ │ 00018fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019000: 2d2d 2d2d 2b0a 7c69 3120 3a20 7772 6974 ----+.|i1 : writ │ │ │ │ -00019010: 6550 6172 616d 6574 6572 4669 6c65 2873 eParameterFile(s │ │ │ │ -00019020: 746f 7265 424d 3246 696c 6573 2c7b 312c toreBM2Files,{1, │ │ │ │ -00019030: 327d 2c4e 616d 6550 6172 616d 6574 6572 2},NameParameter │ │ │ │ -00019040: 4669 6c65 3d3e 2266 696e 616c 5f20 2020 File=>"final_ │ │ │ │ -00019050: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ +00019000: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2077 -------+.|i1 : w │ │ │ │ +00019010: 7269 7465 5061 7261 6d65 7465 7246 696c riteParameterFil │ │ │ │ +00019020: 6528 7374 6f72 6542 4d32 4669 6c65 732c e(storeBM2Files, │ │ │ │ +00019030: 7b31 2c32 7d2c 4e61 6d65 5061 7261 6d65 {1,2},NameParame │ │ │ │ +00019040: 7465 7246 696c 653d 3e22 6669 6e61 6c5f terFile=>"final_ │ │ │ │ +00019050: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ 00019060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000190a0: 2d2d 2d2d 7c0a 7c70 6172 616d 6574 6572 ----|.|parameter │ │ │ │ -000190b0: 7322 293b 2020 2020 2020 2020 2020 2020 s"); │ │ │ │ +000190a0: 2d2d 2d2d 2d2d 2d7c 0a7c 7061 7261 6d65 -------|.|parame │ │ │ │ +000190b0: 7465 7273 2229 3b20 2020 2020 2020 2020 ters"); │ │ │ │ 000190c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000190d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000190e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000190f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000190f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00019100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019140: 2d2d 2d2d 2b0a 7c69 3220 3a20 696d 706f ----+.|i2 : impo │ │ │ │ -00019150: 7274 5061 7261 6d65 7465 7246 696c 6528 rtParameterFile( │ │ │ │ -00019160: 7374 6f72 6542 4d32 4669 6c65 7329 2020 storeBM2Files) │ │ │ │ -00019170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019140: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2069 -------+.|i2 : i │ │ │ │ +00019150: 6d70 6f72 7450 6172 616d 6574 6572 4669 mportParameterFi │ │ │ │ +00019160: 6c65 2873 746f 7265 424d 3246 696c 6573 le(storeBM2Files │ │ │ │ +00019170: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00019180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019190: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00019190: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000191a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000191b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000191c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000191d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000191e0: 2020 2020 7c0a 7c6f 3220 3d20 7b31 2c20 |.|o2 = {1, │ │ │ │ -000191f0: 327d 2020 2020 2020 2020 2020 2020 2020 2} │ │ │ │ +000191e0: 2020 2020 2020 207c 0a7c 6f32 203d 207b |.|o2 = { │ │ │ │ +000191f0: 312c 2032 7d20 2020 2020 2020 2020 2020 1, 2} │ │ │ │ 00019200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019230: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00019230: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00019240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019280: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374 |.|o2 : List │ │ │ │ -00019290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019280: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ +00019290: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 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 2020 2020 │ │ │ │ -000192d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000192d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019320: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -00019330: 7365 2069 6d70 6f72 7450 6172 616d 6574 se importParamet │ │ │ │ -00019340: 6572 4669 6c65 3a0a 3d3d 3d3d 3d3d 3d3d erFile:.======== │ │ │ │ +00019320: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +00019330: 6f20 7573 6520 696d 706f 7274 5061 7261 o use importPara │ │ │ │ +00019340: 6d65 7465 7246 696c 653a 0a3d 3d3d 3d3d meterFile:.===== │ │ │ │ 00019350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00019360: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 ========.. * "i │ │ │ │ -00019370: 6d70 6f72 7450 6172 616d 6574 6572 4669 mportParameterFi │ │ │ │ -00019380: 6c65 2853 7472 696e 6729 220a 0a46 6f72 le(String)"..For │ │ │ │ -00019390: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -000193a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000193b0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -000193c0: 6e6f 7465 2069 6d70 6f72 7450 6172 616d note importParam │ │ │ │ -000193d0: 6574 6572 4669 6c65 3a20 696d 706f 7274 eterFile: import │ │ │ │ -000193e0: 5061 7261 6d65 7465 7246 696c 652c 2069 ParameterFile, i │ │ │ │ -000193f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -00019400: 0a66 756e 6374 696f 6e20 7769 7468 206f .function with o │ │ │ │ -00019410: 7074 696f 6e73 3a20 284d 6163 6175 6c61 ptions: (Macaula │ │ │ │ -00019420: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00019430: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ -00019440: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +00019360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00019370: 2022 696d 706f 7274 5061 7261 6d65 7465 "importParamete │ │ │ │ +00019380: 7246 696c 6528 5374 7269 6e67 2922 0a0a rFile(String)".. │ │ │ │ +00019390: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +000193a0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +000193b0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +000193c0: 7420 2a6e 6f74 6520 696d 706f 7274 5061 t *note importPa │ │ │ │ +000193d0: 7261 6d65 7465 7246 696c 653a 2069 6d70 rameterFile: imp │ │ │ │ +000193e0: 6f72 7450 6172 616d 6574 6572 4669 6c65 ortParameterFile │ │ │ │ +000193f0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +00019400: 686f 640a 6675 6e63 7469 6f6e 2077 6974 hod.function wit │ │ │ │ +00019410: 6820 6f70 7469 6f6e 733a 2028 4d61 6361 h options: (Maca │ │ │ │ +00019420: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +00019430: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ +00019440: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ 00019450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019490: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -000194a0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -000194b0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -000194c0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -000194d0: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ -000194e0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -000194f0: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ -00019500: 692e 6d32 3a0a 3331 3633 3a30 2e0a 1f0a i.m2:.3163:0.... │ │ │ │ -00019510: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ -00019520: 666f 2c20 4e6f 6465 3a20 696d 706f 7274 fo, Node: import │ │ │ │ -00019530: 536f 6c75 7469 6f6e 7346 696c 652c 204e SolutionsFile, N │ │ │ │ -00019540: 6578 743a 2049 7350 726f 6a65 6374 6976 ext: IsProjectiv │ │ │ │ -00019550: 652c 2050 7265 763a 2069 6d70 6f72 7450 e, Prev: importP │ │ │ │ -00019560: 6172 616d 6574 6572 4669 6c65 2c20 5570 arameterFile, Up │ │ │ │ -00019570: 3a20 546f 700a 0a69 6d70 6f72 7453 6f6c : Top..importSol │ │ │ │ -00019580: 7574 696f 6e73 4669 6c65 202d 2d20 496d utionsFile -- Im │ │ │ │ -00019590: 706f 7274 7320 636f 6f72 6469 6e61 7465 ports coordinate │ │ │ │ -000195a0: 7320 6672 6f6d 2061 2042 6572 7469 6e69 s from a Bertini │ │ │ │ -000195b0: 2073 6f6c 7574 696f 6e20 6669 6c65 2e0a solution file.. │ │ │ │ -000195c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00019490: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +000194a0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +000194b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +000194c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +000194d0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +000194e0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +000194f0: 6179 322f 7061 636b 6167 6573 2f42 6572 ay2/packages/Ber │ │ │ │ +00019500: 7469 6e69 2e6d 323a 0a33 3136 333a 302e tini.m2:.3163:0. │ │ │ │ +00019510: 0a1f 0a46 696c 653a 2042 6572 7469 6e69 ...File: Bertini │ │ │ │ +00019520: 2e69 6e66 6f2c 204e 6f64 653a 2069 6d70 .info, Node: imp │ │ │ │ +00019530: 6f72 7453 6f6c 7574 696f 6e73 4669 6c65 ortSolutionsFile │ │ │ │ +00019540: 2c20 4e65 7874 3a20 4973 5072 6f6a 6563 , Next: IsProjec │ │ │ │ +00019550: 7469 7665 2c20 5072 6576 3a20 696d 706f tive, Prev: impo │ │ │ │ +00019560: 7274 5061 7261 6d65 7465 7246 696c 652c rtParameterFile, │ │ │ │ +00019570: 2055 703a 2054 6f70 0a0a 696d 706f 7274 Up: Top..import │ │ │ │ +00019580: 536f 6c75 7469 6f6e 7346 696c 6520 2d2d SolutionsFile -- │ │ │ │ +00019590: 2049 6d70 6f72 7473 2063 6f6f 7264 696e Imports coordin │ │ │ │ +000195a0: 6174 6573 2066 726f 6d20 6120 4265 7274 ates from a Bert │ │ │ │ +000195b0: 696e 6920 736f 6c75 7469 6f6e 2066 696c ini solution fil │ │ │ │ +000195c0: 652e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e..************* │ │ │ │ 000195d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000195e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000195f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00019600: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -00019610: 6167 653a 200a 2020 2020 2020 2020 696d age: . im │ │ │ │ -00019620: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ -00019630: 6528 7329 0a20 202a 2049 6e70 7574 733a e(s). * Inputs: │ │ │ │ -00019640: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ -00019650: 6f74 6520 7374 7269 6e67 3a20 284d 6163 ote string: (Mac │ │ │ │ -00019660: 6175 6c61 7932 446f 6329 5374 7269 6e67 aulay2Doc)String │ │ │ │ -00019670: 2c2c 2054 6865 2064 6972 6563 746f 7279 ,, The directory │ │ │ │ -00019680: 2077 6865 7265 2074 6865 2066 696c 650a where the file. │ │ │ │ -00019690: 2020 2020 2020 2020 6973 2073 746f 7265 is store │ │ │ │ -000196a0: 642e 0a20 202a 202a 6e6f 7465 204f 7074 d.. * *note Opt │ │ │ │ -000196b0: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ -000196c0: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ -000196d0: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ -000196e0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ -000196f0: 2c3a 0a20 2020 2020 202a 204d 3250 7265 ,:. * M2Pre │ │ │ │ -00019700: 6369 7369 6f6e 2028 6d69 7373 696e 6720 cision (missing │ │ │ │ -00019710: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ -00019720: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00019730: 616c 7565 2035 332c 200a 2020 2020 2020 alue 53, . │ │ │ │ -00019740: 2a20 4e61 6d65 536f 6c75 7469 6f6e 7346 * NameSolutionsF │ │ │ │ -00019750: 696c 6520 286d 6973 7369 6e67 2064 6f63 ile (missing doc │ │ │ │ -00019760: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -00019770: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -00019780: 650a 2020 2020 2020 2020 2272 6177 5f73 e. "raw_s │ │ │ │ -00019790: 6f6c 7574 696f 6e73 222c 0a20 2020 2020 olutions",. │ │ │ │ -000197a0: 202a 204f 7264 6572 5061 7468 7320 286d * OrderPaths (m │ │ │ │ -000197b0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ -000197c0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ -000197d0: 6661 756c 7420 7661 6c75 6520 6661 6c73 fault value fals │ │ │ │ -000197e0: 652c 200a 2020 2020 2020 2a20 5374 6f72 e, . * Stor │ │ │ │ -000197f0: 6167 6546 6f6c 6465 7220 286d 6973 7369 ageFolder (missi │ │ │ │ -00019800: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -00019810: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -00019820: 7420 7661 6c75 6520 6e75 6c6c 2c20 0a20 t value null, . │ │ │ │ -00019830: 2020 2020 202a 202a 6e6f 7465 2056 6572 * *note Ver │ │ │ │ -00019840: 626f 7365 3a20 6265 7274 696e 6954 7261 bose: bertiniTra │ │ │ │ -00019850: 636b 486f 6d6f 746f 7079 5f6c 705f 7064 ckHomotopy_lp_pd │ │ │ │ -00019860: 5f70 645f 7064 5f63 6d56 6572 626f 7365 _pd_pd_cmVerbose │ │ │ │ -00019870: 3d3e 5f70 645f 7064 5f70 645f 7270 0a20 =>_pd_pd_pd_rp. │ │ │ │ -00019880: 2020 2020 2020 202c 203d 3e20 2e2e 2e2c , => ..., │ │ │ │ -00019890: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ -000198a0: 616c 7365 2c20 4f70 7469 6f6e 2074 6f20 alse, Option to │ │ │ │ -000198b0: 7369 6c65 6e63 6520 6164 6469 7469 6f6e silence addition │ │ │ │ -000198c0: 616c 206f 7574 7075 740a 0a44 6573 6372 al output..Descr │ │ │ │ -000198d0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -000198e0: 3d3d 0a0a 4166 7465 7220 4265 7274 696e ==..After Bertin │ │ │ │ -000198f0: 6920 646f 6573 2061 2072 756e 206d 616e i does a run man │ │ │ │ -00019900: 7920 6669 6c65 7320 6172 6520 6372 6561 y files are crea │ │ │ │ -00019910: 7465 642e 2054 6869 7320 6675 6e63 7469 ted. This functi │ │ │ │ -00019920: 6f6e 2069 6d70 6f72 7473 2074 6865 0a63 on imports the.c │ │ │ │ -00019930: 6f6f 7264 696e 6174 6573 206f 6620 736f oordinates of so │ │ │ │ -00019940: 6c75 7469 6f6e 7320 6672 6f6d 2074 6865 lutions from the │ │ │ │ -00019950: 2073 696d 706c 6520 2272 6177 5f73 6f6c simple "raw_sol │ │ │ │ -00019960: 7574 696f 6e73 2220 6669 6c65 2e20 4279 utions" file. By │ │ │ │ -00019970: 2075 7369 6e67 2074 6865 0a6f 7074 696f using the.optio │ │ │ │ -00019980: 6e20 4e61 6d65 536f 6c75 7469 6f6e 7346 n NameSolutionsF │ │ │ │ -00019990: 696c 653d 3e22 7265 616c 5f66 696e 6974 ile=>"real_finit │ │ │ │ -000199a0: 655f 736f 6c75 7469 6f6e 7322 2077 6520 e_solutions" we │ │ │ │ -000199b0: 776f 756c 6420 696d 706f 7274 2073 6f6c would import sol │ │ │ │ -000199c0: 7574 696f 6e73 0a66 726f 6d20 7265 616c utions.from real │ │ │ │ -000199d0: 2066 696e 6974 6520 736f 6c75 7469 6f6e finite solution │ │ │ │ -000199e0: 732e 204f 7468 6572 2063 6f6d 6d6f 6e20 s. Other common │ │ │ │ -000199f0: 6669 6c65 206e 616d 6573 2061 7265 0a22 file names are." │ │ │ │ -00019a00: 6e6f 6e73 696e 6775 6c61 725f 736f 6c75 nonsingular_solu │ │ │ │ -00019a10: 7469 6f6e 7322 2c20 2266 696e 6974 655f tions", "finite_ │ │ │ │ -00019a20: 736f 6c75 7469 6f6e 7322 2c20 2269 6e66 solutions", "inf │ │ │ │ -00019a30: 696e 6974 655f 736f 6c75 7469 6f6e 7322 inite_solutions" │ │ │ │ -00019a40: 2c20 616e 640a 2273 696e 6775 6c61 725f , and."singular_ │ │ │ │ -00019a50: 736f 6c75 7469 6f6e 7322 2e0a 0a49 6620 solutions"...If │ │ │ │ -00019a60: 7468 6520 4e61 6d65 536f 6c75 7469 6f6e the NameSolution │ │ │ │ -00019a70: 7346 696c 6520 6f70 7469 6f6e 2069 7320 sFile option is │ │ │ │ -00019a80: 7365 7420 746f 2030 2074 6865 6e20 226e set to 0 then "n │ │ │ │ -00019a90: 6f6e 7369 6e67 756c 6172 5f73 6f6c 7574 onsingular_solut │ │ │ │ -00019aa0: 696f 6e73 2220 6973 0a69 6d70 6f72 7465 ions" is.importe │ │ │ │ -00019ab0: 642c 2069 7320 7365 7420 746f 2031 2074 d, is set to 1 t │ │ │ │ -00019ac0: 6865 6e20 2272 6561 6c5f 6669 6e69 7465 hen "real_finite │ │ │ │ -00019ad0: 5f73 6f6c 7574 696f 6e73 2220 6973 2069 _solutions" is i │ │ │ │ -00019ae0: 6d70 6f72 7465 642c 2069 7320 7365 7420 mported, is set │ │ │ │ -00019af0: 746f 2032 0a74 6865 6e20 2269 6e66 696e to 2.then "infin │ │ │ │ -00019b00: 6974 655f 736f 6c75 7469 6f6e 7322 2069 ite_solutions" i │ │ │ │ -00019b10: 7320 696d 706f 7274 6564 2c20 6973 2073 s imported, is s │ │ │ │ -00019b20: 6574 2074 6f20 3320 7468 656e 2022 6669 et to 3 then "fi │ │ │ │ -00019b30: 6e69 7465 5f73 6f6c 7574 696f 6e73 2220 nite_solutions" │ │ │ │ -00019b40: 6973 0a69 6d70 6f72 7465 642c 2069 7320 is.imported, is │ │ │ │ -00019b50: 7365 7420 746f 2034 2074 6865 6e20 2273 set to 4 then "s │ │ │ │ -00019b60: 7461 7274 2220 6973 2069 6d70 6f72 7465 tart" is importe │ │ │ │ -00019b70: 642c 2069 7320 7365 7420 746f 2035 2074 d, is set to 5 t │ │ │ │ -00019b80: 6865 6e0a 2272 6177 5f73 6f6c 7574 696f hen."raw_solutio │ │ │ │ -00019b90: 6e73 2220 6973 2069 6d70 6f72 7465 642e ns" is imported. │ │ │ │ -00019ba0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00019600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00019610: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00019620: 2069 6d70 6f72 7453 6f6c 7574 696f 6e73 importSolutions │ │ │ │ +00019630: 4669 6c65 2873 290a 2020 2a20 496e 7075 File(s). * Inpu │ │ │ │ +00019640: 7473 3a0a 2020 2020 2020 2a20 732c 2061 ts:. * s, a │ │ │ │ +00019650: 202a 6e6f 7465 2073 7472 696e 673a 2028 *note string: ( │ │ │ │ +00019660: 4d61 6361 756c 6179 3244 6f63 2953 7472 Macaulay2Doc)Str │ │ │ │ +00019670: 696e 672c 2c20 5468 6520 6469 7265 6374 ing,, The direct │ │ │ │ +00019680: 6f72 7920 7768 6572 6520 7468 6520 6669 ory where the fi │ │ │ │ +00019690: 6c65 0a20 2020 2020 2020 2069 7320 7374 le. is st │ │ │ │ +000196a0: 6f72 6564 2e0a 2020 2a20 2a6e 6f74 6520 ored.. * *note │ │ │ │ +000196b0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ +000196c0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ +000196d0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ +000196e0: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ +000196f0: 7574 732c 3a0a 2020 2020 2020 2a20 4d32 uts,:. * M2 │ │ │ │ +00019700: 5072 6563 6973 696f 6e20 286d 6973 7369 Precision (missi │ │ │ │ +00019710: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00019720: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ +00019730: 7420 7661 6c75 6520 3533 2c20 0a20 2020 t value 53, . │ │ │ │ +00019740: 2020 202a 204e 616d 6553 6f6c 7574 696f * NameSolutio │ │ │ │ +00019750: 6e73 4669 6c65 2028 6d69 7373 696e 6720 nsFile (missing │ │ │ │ +00019760: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ +00019770: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00019780: 616c 7565 0a20 2020 2020 2020 2022 7261 alue. "ra │ │ │ │ +00019790: 775f 736f 6c75 7469 6f6e 7322 2c0a 2020 w_solutions",. │ │ │ │ +000197a0: 2020 2020 2a20 4f72 6465 7250 6174 6873 * OrderPaths │ │ │ │ +000197b0: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +000197c0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +000197d0: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +000197e0: 616c 7365 2c20 0a20 2020 2020 202a 2053 alse, . * S │ │ │ │ +000197f0: 746f 7261 6765 466f 6c64 6572 2028 6d69 torageFolder (mi │ │ │ │ +00019800: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +00019810: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +00019820: 6175 6c74 2076 616c 7565 206e 756c 6c2c ault value null, │ │ │ │ +00019830: 200a 2020 2020 2020 2a20 2a6e 6f74 6520 . * *note │ │ │ │ +00019840: 5665 7262 6f73 653a 2062 6572 7469 6e69 Verbose: bertini │ │ │ │ +00019850: 5472 6163 6b48 6f6d 6f74 6f70 795f 6c70 TrackHomotopy_lp │ │ │ │ +00019860: 5f70 645f 7064 5f70 645f 636d 5665 7262 _pd_pd_pd_cmVerb │ │ │ │ +00019870: 6f73 653d 3e5f 7064 5f70 645f 7064 5f72 ose=>_pd_pd_pd_r │ │ │ │ +00019880: 700a 2020 2020 2020 2020 2c20 3d3e 202e p. , => . │ │ │ │ +00019890: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +000198a0: 6520 6661 6c73 652c 204f 7074 696f 6e20 e false, Option │ │ │ │ +000198b0: 746f 2073 696c 656e 6365 2061 6464 6974 to silence addit │ │ │ │ +000198c0: 696f 6e61 6c20 6f75 7470 7574 0a0a 4465 ional output..De │ │ │ │ +000198d0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +000198e0: 3d3d 3d3d 3d0a 0a41 6674 6572 2042 6572 =====..After Ber │ │ │ │ +000198f0: 7469 6e69 2064 6f65 7320 6120 7275 6e20 tini does a run │ │ │ │ +00019900: 6d61 6e79 2066 696c 6573 2061 7265 2063 many files are c │ │ │ │ +00019910: 7265 6174 6564 2e20 5468 6973 2066 756e reated. This fun │ │ │ │ +00019920: 6374 696f 6e20 696d 706f 7274 7320 7468 ction imports th │ │ │ │ +00019930: 650a 636f 6f72 6469 6e61 7465 7320 6f66 e.coordinates of │ │ │ │ +00019940: 2073 6f6c 7574 696f 6e73 2066 726f 6d20 solutions from │ │ │ │ +00019950: 7468 6520 7369 6d70 6c65 2022 7261 775f the simple "raw_ │ │ │ │ +00019960: 736f 6c75 7469 6f6e 7322 2066 696c 652e solutions" file. │ │ │ │ +00019970: 2042 7920 7573 696e 6720 7468 650a 6f70 By using the.op │ │ │ │ +00019980: 7469 6f6e 204e 616d 6553 6f6c 7574 696f tion NameSolutio │ │ │ │ +00019990: 6e73 4669 6c65 3d3e 2272 6561 6c5f 6669 nsFile=>"real_fi │ │ │ │ +000199a0: 6e69 7465 5f73 6f6c 7574 696f 6e73 2220 nite_solutions" │ │ │ │ +000199b0: 7765 2077 6f75 6c64 2069 6d70 6f72 7420 we would import │ │ │ │ +000199c0: 736f 6c75 7469 6f6e 730a 6672 6f6d 2072 solutions.from r │ │ │ │ +000199d0: 6561 6c20 6669 6e69 7465 2073 6f6c 7574 eal finite solut │ │ │ │ +000199e0: 696f 6e73 2e20 4f74 6865 7220 636f 6d6d ions. Other comm │ │ │ │ +000199f0: 6f6e 2066 696c 6520 6e61 6d65 7320 6172 on file names ar │ │ │ │ +00019a00: 650a 226e 6f6e 7369 6e67 756c 6172 5f73 e."nonsingular_s │ │ │ │ +00019a10: 6f6c 7574 696f 6e73 222c 2022 6669 6e69 olutions", "fini │ │ │ │ +00019a20: 7465 5f73 6f6c 7574 696f 6e73 222c 2022 te_solutions", " │ │ │ │ +00019a30: 696e 6669 6e69 7465 5f73 6f6c 7574 696f infinite_solutio │ │ │ │ +00019a40: 6e73 222c 2061 6e64 0a22 7369 6e67 756c ns", and."singul │ │ │ │ +00019a50: 6172 5f73 6f6c 7574 696f 6e73 222e 0a0a ar_solutions"... │ │ │ │ +00019a60: 4966 2074 6865 204e 616d 6553 6f6c 7574 If the NameSolut │ │ │ │ +00019a70: 696f 6e73 4669 6c65 206f 7074 696f 6e20 ionsFile option │ │ │ │ +00019a80: 6973 2073 6574 2074 6f20 3020 7468 656e is set to 0 then │ │ │ │ +00019a90: 2022 6e6f 6e73 696e 6775 6c61 725f 736f "nonsingular_so │ │ │ │ +00019aa0: 6c75 7469 6f6e 7322 2069 730a 696d 706f lutions" is.impo │ │ │ │ +00019ab0: 7274 6564 2c20 6973 2073 6574 2074 6f20 rted, is set to │ │ │ │ +00019ac0: 3120 7468 656e 2022 7265 616c 5f66 696e 1 then "real_fin │ │ │ │ +00019ad0: 6974 655f 736f 6c75 7469 6f6e 7322 2069 ite_solutions" i │ │ │ │ +00019ae0: 7320 696d 706f 7274 6564 2c20 6973 2073 s imported, is s │ │ │ │ +00019af0: 6574 2074 6f20 320a 7468 656e 2022 696e et to 2.then "in │ │ │ │ +00019b00: 6669 6e69 7465 5f73 6f6c 7574 696f 6e73 finite_solutions │ │ │ │ +00019b10: 2220 6973 2069 6d70 6f72 7465 642c 2069 " is imported, i │ │ │ │ +00019b20: 7320 7365 7420 746f 2033 2074 6865 6e20 s set to 3 then │ │ │ │ +00019b30: 2266 696e 6974 655f 736f 6c75 7469 6f6e "finite_solution │ │ │ │ +00019b40: 7322 2069 730a 696d 706f 7274 6564 2c20 s" is.imported, │ │ │ │ +00019b50: 6973 2073 6574 2074 6f20 3420 7468 656e is set to 4 then │ │ │ │ +00019b60: 2022 7374 6172 7422 2069 7320 696d 706f "start" is impo │ │ │ │ +00019b70: 7274 6564 2c20 6973 2073 6574 2074 6f20 rted, is set to │ │ │ │ +00019b80: 3520 7468 656e 0a22 7261 775f 736f 6c75 5 then."raw_solu │ │ │ │ +00019b90: 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7d7d 2020 5-.866025*ii}} │ │ │ │ -0001a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a5f0: 2d2d 2d7c 0a7c 2020 2020 202d 2e35 2b2e ---|.| -.5+. │ │ │ │ +0001a600: 3836 3630 3235 2a69 697d 2c20 7b2d 312c 866025*ii}, {-1, │ │ │ │ +0001a610: 202d 2e35 2d2e 3836 3630 3235 2a69 697d -.5-.866025*ii} │ │ │ │ +0001a620: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ 0001a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a640: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a640: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a690: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020 |.|o6 : List │ │ │ │ +0001a690: 2020 207c 0a7c 6f36 203a 204c 6973 7420 |.|o6 : List │ │ │ │ 0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001a6e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0001a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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: 2b0a 0a57 6179 7320 746f 2075 7365 2069 +..Ways to use i │ │ │ │ -0001a740: 6d70 6f72 7453 6f6c 7574 696f 6e73 4669 mportSolutionsFi │ │ │ │ -0001a750: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ +0001a730: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573 ---+..Ways to us │ │ │ │ +0001a740: 6520 696d 706f 7274 536f 6c75 7469 6f6e e importSolution │ │ │ │ +0001a750: 7346 696c 653a 0a3d 3d3d 3d3d 3d3d 3d3d sFile:.========= │ │ │ │ 0001a760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001a770: 3d3d 3d3d 0a0a 2020 2a20 2269 6d70 6f72 ====.. * "impor │ │ │ │ -0001a780: 7453 6f6c 7574 696f 6e73 4669 6c65 2853 tSolutionsFile(S │ │ │ │ -0001a790: 7472 696e 6729 220a 0a46 6f72 2074 6865 tring)"..For the │ │ │ │ -0001a7a0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -0001a7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0001a7c0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -0001a7d0: 2069 6d70 6f72 7453 6f6c 7574 696f 6e73 importSolutions │ │ │ │ -0001a7e0: 4669 6c65 3a20 696d 706f 7274 536f 6c75 File: importSolu │ │ │ │ -0001a7f0: 7469 6f6e 7346 696c 652c 2069 7320 6120 tionsFile, is a │ │ │ │ -0001a800: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ -0001a810: 6374 696f 6e20 7769 7468 206f 7074 696f ction with optio │ │ │ │ -0001a820: 6e73 3a20 284d 6163 6175 6c61 7932 446f ns: (Macaulay2Do │ │ │ │ -0001a830: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0001a840: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ -0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a770: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 696d =======.. * "im │ │ │ │ +0001a780: 706f 7274 536f 6c75 7469 6f6e 7346 696c portSolutionsFil │ │ │ │ +0001a790: 6528 5374 7269 6e67 2922 0a0a 466f 7220 e(String)"..For │ │ │ │ +0001a7a0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0001a7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001a7c0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0001a7d0: 6f74 6520 696d 706f 7274 536f 6c75 7469 ote importSoluti │ │ │ │ +0001a7e0: 6f6e 7346 696c 653a 2069 6d70 6f72 7453 onsFile: importS │ │ │ │ +0001a7f0: 6f6c 7574 696f 6e73 4669 6c65 2c20 6973 olutionsFile, is │ │ │ │ +0001a800: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ +0001a810: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ +0001a820: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ +0001a830: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0001a840: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +0001a850: 0a0a 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 ---------------- │ │ │ │ 0001a880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0001a8a0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0001a8b0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0001a8c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0001a8d0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0001a8e0: 6c61 7932 2d31 2e32 352e 3036 2b64 732f lay2-1.25.06+ds/ │ │ │ │ -0001a8f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -0001a900: 6b61 6765 732f 4265 7274 696e 692e 6d32 kages/Bertini.m2 │ │ │ │ -0001a910: 3a0a 3331 3431 3a30 2e0a 1f0a 4669 6c65 :.3141:0....File │ │ │ │ -0001a920: 3a20 4265 7274 696e 692e 696e 666f 2c20 : Bertini.info, │ │ │ │ -0001a930: 4e6f 6465 3a20 4973 5072 6f6a 6563 7469 Node: IsProjecti │ │ │ │ -0001a940: 7665 2c20 4e65 7874 3a20 4d61 696e 4461 ve, Next: MainDa │ │ │ │ -0001a950: 7461 4469 7265 6374 6f72 792c 2050 7265 taDirectory, Pre │ │ │ │ -0001a960: 763a 2069 6d70 6f72 7453 6f6c 7574 696f v: importSolutio │ │ │ │ -0001a970: 6e73 4669 6c65 2c20 5570 3a20 546f 700a nsFile, Up: Top. │ │ │ │ -0001a980: 0a49 7350 726f 6a65 6374 6976 6520 2d2d .IsProjective -- │ │ │ │ -0001a990: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ -0001a9a0: 6e74 2074 6f20 7370 6563 6966 7920 7768 nt to specify wh │ │ │ │ -0001a9b0: 6574 6865 7220 746f 2075 7365 2068 6f6d ether to use hom │ │ │ │ -0001a9c0: 6f67 656e 656f 7573 2063 6f6f 7264 696e ogeneous coordin │ │ │ │ -0001a9d0: 6174 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ates.*********** │ │ │ │ +0001a890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8a0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0001a8b0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0001a8c0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0001a8d0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0001a8e0: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ +0001a8f0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0001a900: 7061 636b 6167 6573 2f42 6572 7469 6e69 packages/Bertini │ │ │ │ +0001a910: 2e6d 323a 0a33 3134 313a 302e 0a1f 0a46 .m2:.3141:0....F │ │ │ │ +0001a920: 696c 653a 2042 6572 7469 6e69 2e69 6e66 ile: Bertini.inf │ │ │ │ +0001a930: 6f2c 204e 6f64 653a 2049 7350 726f 6a65 o, Node: IsProje │ │ │ │ +0001a940: 6374 6976 652c 204e 6578 743a 204d 6169 ctive, Next: Mai │ │ │ │ +0001a950: 6e44 6174 6144 6972 6563 746f 7279 2c20 nDataDirectory, │ │ │ │ +0001a960: 5072 6576 3a20 696d 706f 7274 536f 6c75 Prev: importSolu │ │ │ │ +0001a970: 7469 6f6e 7346 696c 652c 2055 703a 2054 tionsFile, Up: T │ │ │ │ +0001a980: 6f70 0a0a 4973 5072 6f6a 6563 7469 7665 op..IsProjective │ │ │ │ +0001a990: 202d 2d20 6f70 7469 6f6e 616c 2061 7267 -- optional arg │ │ │ │ +0001a9a0: 756d 656e 7420 746f 2073 7065 6369 6679 ument to specify │ │ │ │ +0001a9b0: 2077 6865 7468 6572 2074 6f20 7573 6520 whether to use │ │ │ │ +0001a9c0: 686f 6d6f 6765 6e65 6f75 7320 636f 6f72 homogeneous coor │ │ │ │ +0001a9d0: 6469 6e61 7465 730a 2a2a 2a2a 2a2a 2a2a dinates.******** │ │ │ │ 0001a9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001aa00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001aa10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001aa20: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 ********..Descri │ │ │ │ -0001aa30: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0001aa40: 3d0a 0a57 6865 6e20 7365 7420 746f 2031 =..When set to 1 │ │ │ │ -0001aa50: 2c20 7468 6973 206f 7074 696f 6e20 696e , this option in │ │ │ │ -0001aa60: 6469 6361 7465 7320 7468 6174 2074 6865 dicates that the │ │ │ │ -0001aa70: 2069 6e70 7574 2073 7973 7465 6d20 6973 input system is │ │ │ │ -0001aa80: 2068 6f6d 6f67 656e 697a 6564 2061 6e64 homogenized and │ │ │ │ -0001aa90: 0a74 6865 206f 7574 7075 7420 7368 6f75 .the output shou │ │ │ │ -0001aaa0: 6c64 2062 6520 6769 7665 6e20 696e 2070 ld be given in p │ │ │ │ -0001aab0: 726f 6a65 6374 6976 6520 7370 6163 652e rojective space. │ │ │ │ -0001aac0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +0001aa20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 ***********..Des │ │ │ │ +0001aa30: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0001aa40: 3d3d 3d3d 0a0a 5768 656e 2073 6574 2074 ====..When set t │ │ │ │ +0001aa50: 6f20 312c 2074 6869 7320 6f70 7469 6f6e o 1, this option │ │ │ │ +0001aa60: 2069 6e64 6963 6174 6573 2074 6861 7420 indicates that │ │ │ │ +0001aa70: 7468 6520 696e 7075 7420 7379 7374 656d the input system │ │ │ │ +0001aa80: 2069 7320 686f 6d6f 6765 6e69 7a65 6420 is homogenized │ │ │ │ +0001aa90: 616e 640a 7468 6520 6f75 7470 7574 2073 and.the output s │ │ │ │ +0001aaa0: 686f 756c 6420 6265 2067 6976 656e 2069 hould be given i │ │ │ │ +0001aab0: 6e20 7072 6f6a 6563 7469 7665 2073 7061 n projective spa │ │ │ │ +0001aac0: 6365 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ce...+---------- │ │ │ │ 0001aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ -0001ab00: 3d20 4343 5b78 2c79 2c7a 5d3b 2020 2020 = CC[x,y,z]; │ │ │ │ +0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +0001ab00: 2052 203d 2043 435b 782c 792c 7a5d 3b20 R = CC[x,y,z]; │ │ │ │ 0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ab20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ab30: 0a2b 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 2b0a 7c69 3220 3a20 6620 3d20 7b28 --+.|i2 : f = {( │ │ │ │ -0001ab70: 785e 322b 795e 322d 7a5e 3229 2a28 7a2d x^2+y^2-z^2)*(z- │ │ │ │ -0001ab80: 7829 2c28 785e 322b 795e 322d 7a5e 3229 x),(x^2+y^2-z^2) │ │ │ │ -0001ab90: 2a28 7a2b 7929 7d3b 7c0a 2b2d 2d2d 2d2d *(z+y)};|.+----- │ │ │ │ +0001ab60: 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 203d -----+.|i2 : f = │ │ │ │ +0001ab70: 207b 2878 5e32 2b79 5e32 2d7a 5e32 292a {(x^2+y^2-z^2)* │ │ │ │ +0001ab80: 287a 2d78 292c 2878 5e32 2b79 5e32 2d7a (z-x),(x^2+y^2-z │ │ │ │ +0001ab90: 5e32 292a 287a 2b79 297d 3b7c 0a2b 2d2d ^2)*(z+y)};|.+-- │ │ │ │ 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 2b0a --------------+. │ │ │ │ -0001abd0: 7c69 3320 3a20 6265 7274 696e 6950 6f73 |i3 : bertiniPos │ │ │ │ -0001abe0: 4469 6d53 6f6c 7665 2866 2c49 7350 726f DimSolve(f,IsPro │ │ │ │ -0001abf0: 6a65 6374 6976 653d 3e31 293b 2020 2020 jective=>1); │ │ │ │ -0001ac00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abd0: 2d2b 0a7c 6933 203a 2062 6572 7469 6e69 -+.|i3 : bertini │ │ │ │ +0001abe0: 506f 7344 696d 536f 6c76 6528 662c 4973 PosDimSolve(f,Is │ │ │ │ +0001abf0: 5072 6f6a 6563 7469 7665 3d3e 3129 3b20 Projective=>1); │ │ │ │ +0001ac00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 756e ----------+..Fun │ │ │ │ -0001ac40: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -0001ac50: 6f6e 616c 2061 7267 756d 656e 7420 6e61 onal argument na │ │ │ │ -0001ac60: 6d65 6420 4973 5072 6f6a 6563 7469 7665 med IsProjective │ │ │ │ -0001ac70: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0001ac40: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ +0001ac50: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ +0001ac60: 206e 616d 6564 2049 7350 726f 6a65 6374 named IsProject │ │ │ │ +0001ac70: 6976 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ive:.=========== │ │ │ │ 0001ac80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 0001ac90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001aca0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 6265 7274 ======.. * bert │ │ │ │ -0001acb0: 696e 695a 6572 6f44 696d 536f 6c76 6528 iniZeroDimSolve( │ │ │ │ -0001acc0: 2e2e 2e2c 4973 5072 6f6a 6563 7469 7665 ...,IsProjective │ │ │ │ -0001acd0: 3d3e 2e2e 2e29 2028 6d69 7373 696e 6720 =>...) (missing │ │ │ │ -0001ace0: 646f 6375 6d65 6e74 6174 696f 6e29 0a20 documentation). │ │ │ │ -0001acf0: 202a 2022 6265 7274 696e 6943 6f6d 706f * "bertiniCompo │ │ │ │ -0001ad00: 6e65 6e74 4d65 6d62 6572 5465 7374 282e nentMemberTest(. │ │ │ │ -0001ad10: 2e2e 2c49 7350 726f 6a65 6374 6976 653d ..,IsProjective= │ │ │ │ -0001ad20: 3e2e 2e2e 2922 0a20 202a 2022 6265 7274 >...)". * "bert │ │ │ │ -0001ad30: 696e 6950 6f73 4469 6d53 6f6c 7665 282e iniPosDimSolve(. │ │ │ │ -0001ad40: 2e2e 2c49 7350 726f 6a65 6374 6976 653d ..,IsProjective= │ │ │ │ -0001ad50: 3e2e 2e2e 2922 0a20 202a 2022 6265 7274 >...)". * "bert │ │ │ │ -0001ad60: 696e 6952 6566 696e 6553 6f6c 7328 2e2e iniRefineSols(.. │ │ │ │ -0001ad70: 2e2c 4973 5072 6f6a 6563 7469 7665 3d3e .,IsProjective=> │ │ │ │ -0001ad80: 2e2e 2e29 220a 2020 2a20 2262 6572 7469 ...)". * "berti │ │ │ │ -0001ad90: 6e69 5361 6d70 6c65 282e 2e2e 2c49 7350 niSample(...,IsP │ │ │ │ -0001ada0: 726f 6a65 6374 6976 653d 3e2e 2e2e 2922 rojective=>...)" │ │ │ │ -0001adb0: 0a20 202a 2022 6265 7274 696e 6954 7261 . * "bertiniTra │ │ │ │ -0001adc0: 636b 486f 6d6f 746f 7079 282e 2e2e 2c49 ckHomotopy(...,I │ │ │ │ -0001add0: 7350 726f 6a65 6374 6976 653d 3e2e 2e2e sProjective=>... │ │ │ │ -0001ade0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -0001adf0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0001ae00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0001ae10: 626a 6563 7420 2a6e 6f74 6520 4973 5072 bject *note IsPr │ │ │ │ -0001ae20: 6f6a 6563 7469 7665 3a20 4973 5072 6f6a ojective: IsProj │ │ │ │ -0001ae30: 6563 7469 7665 2c20 6973 2061 202a 6e6f ective, is a *no │ │ │ │ -0001ae40: 7465 2073 796d 626f 6c3a 0a28 4d61 6361 te symbol:.(Maca │ │ │ │ -0001ae50: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -0001ae60: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0001aca0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2062 =========.. * b │ │ │ │ +0001acb0: 6572 7469 6e69 5a65 726f 4469 6d53 6f6c ertiniZeroDimSol │ │ │ │ +0001acc0: 7665 282e 2e2e 2c49 7350 726f 6a65 6374 ve(...,IsProject │ │ │ │ +0001acd0: 6976 653d 3e2e 2e2e 2920 286d 6973 7369 ive=>...) (missi │ │ │ │ +0001ace0: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +0001acf0: 290a 2020 2a20 2262 6572 7469 6e69 436f ). * "bertiniCo │ │ │ │ +0001ad00: 6d70 6f6e 656e 744d 656d 6265 7254 6573 mponentMemberTes │ │ │ │ +0001ad10: 7428 2e2e 2e2c 4973 5072 6f6a 6563 7469 t(...,IsProjecti │ │ │ │ +0001ad20: 7665 3d3e 2e2e 2e29 220a 2020 2a20 2262 ve=>...)". * "b │ │ │ │ +0001ad30: 6572 7469 6e69 506f 7344 696d 536f 6c76 ertiniPosDimSolv │ │ │ │ +0001ad40: 6528 2e2e 2e2c 4973 5072 6f6a 6563 7469 e(...,IsProjecti │ │ │ │ +0001ad50: 7665 3d3e 2e2e 2e29 220a 2020 2a20 2262 ve=>...)". * "b │ │ │ │ +0001ad60: 6572 7469 6e69 5265 6669 6e65 536f 6c73 ertiniRefineSols │ │ │ │ +0001ad70: 282e 2e2e 2c49 7350 726f 6a65 6374 6976 (...,IsProjectiv │ │ │ │ +0001ad80: 653d 3e2e 2e2e 2922 0a20 202a 2022 6265 e=>...)". * "be │ │ │ │ +0001ad90: 7274 696e 6953 616d 706c 6528 2e2e 2e2c rtiniSample(..., │ │ │ │ +0001ada0: 4973 5072 6f6a 6563 7469 7665 3d3e 2e2e IsProjective=>.. │ │ │ │ +0001adb0: 2e29 220a 2020 2a20 2262 6572 7469 6e69 .)". * "bertini │ │ │ │ +0001adc0: 5472 6163 6b48 6f6d 6f74 6f70 7928 2e2e TrackHomotopy(.. │ │ │ │ +0001add0: 2e2c 4973 5072 6f6a 6563 7469 7665 3d3e .,IsProjective=> │ │ │ │ +0001ade0: 2e2e 2e29 220a 0a46 6f72 2074 6865 2070 ...)"..For the p │ │ │ │ +0001adf0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0001ae00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0001ae10: 6520 6f62 6a65 6374 202a 6e6f 7465 2049 e object *note I │ │ │ │ +0001ae20: 7350 726f 6a65 6374 6976 653a 2049 7350 sProjective: IsP │ │ │ │ +0001ae30: 726f 6a65 6374 6976 652c 2069 7320 6120 rojective, is a │ │ │ │ +0001ae40: 2a6e 6f74 6520 7379 6d62 6f6c 3a0a 284d *note symbol:.(M │ │ │ │ +0001ae50: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +0001ae60: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ 0001ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aeb0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0001aec0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0001aed0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0001aee0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0001aef0: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ -0001af00: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0001af10: 2f70 6163 6b61 6765 732f 4265 7274 696e /packages/Bertin │ │ │ │ -0001af20: 692e 6d32 3a0a 3332 3434 3a30 2e0a 1f0a i.m2:.3244:0.... │ │ │ │ -0001af30: 4669 6c65 3a20 4265 7274 696e 692e 696e File: Bertini.in │ │ │ │ -0001af40: 666f 2c20 4e6f 6465 3a20 4d61 696e 4461 fo, Node: MainDa │ │ │ │ -0001af50: 7461 4469 7265 6374 6f72 792c 204e 6578 taDirectory, Nex │ │ │ │ -0001af60: 743a 206d 616b 6542 2749 6e70 7574 4669 t: makeB'InputFi │ │ │ │ -0001af70: 6c65 2c20 5072 6576 3a20 4973 5072 6f6a le, Prev: IsProj │ │ │ │ -0001af80: 6563 7469 7665 2c20 5570 3a20 546f 700a ective, Up: Top. │ │ │ │ -0001af90: 0a4d 6169 6e44 6174 6144 6972 6563 746f .MainDataDirecto │ │ │ │ -0001afa0: 7279 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ry.************* │ │ │ │ -0001afb0: 2a2a 2a2a 0a0a 466f 7220 7468 6520 7072 ****..For the pr │ │ │ │ -0001afc0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -0001afd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -0001afe0: 206f 626a 6563 7420 4d61 696e 4461 7461 object MainData │ │ │ │ -0001aff0: 4469 7265 6374 6f72 7920 286d 6973 7369 Directory (missi │ │ │ │ -0001b000: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -0001b010: 2920 6973 2061 202a 6e6f 7465 2073 796d ) is a *note sym │ │ │ │ -0001b020: 626f 6c3a 0a28 4d61 6361 756c 6179 3244 bol:.(Macaulay2D │ │ │ │ -0001b030: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +0001aeb0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0001aec0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0001aed0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0001aee0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0001aef0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +0001af00: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0001af10: 6179 322f 7061 636b 6167 6573 2f42 6572 ay2/packages/Ber │ │ │ │ +0001af20: 7469 6e69 2e6d 323a 0a33 3234 343a 302e tini.m2:.3244:0. │ │ │ │ +0001af30: 0a1f 0a46 696c 653a 2042 6572 7469 6e69 ...File: Bertini │ │ │ │ +0001af40: 2e69 6e66 6f2c 204e 6f64 653a 204d 6169 .info, Node: Mai │ │ │ │ +0001af50: 6e44 6174 6144 6972 6563 746f 7279 2c20 nDataDirectory, │ │ │ │ +0001af60: 4e65 7874 3a20 6d61 6b65 4227 496e 7075 Next: makeB'Inpu │ │ │ │ +0001af70: 7446 696c 652c 2050 7265 763a 2049 7350 tFile, Prev: IsP │ │ │ │ +0001af80: 726f 6a65 6374 6976 652c 2055 703a 2054 rojective, Up: T │ │ │ │ +0001af90: 6f70 0a0a 4d61 696e 4461 7461 4469 7265 op..MainDataDire │ │ │ │ +0001afa0: 6374 6f72 790a 2a2a 2a2a 2a2a 2a2a 2a2a ctory.********** │ │ │ │ +0001afb0: 2a2a 2a2a 2a2a 2a0a 0a46 6f72 2074 6865 *******..For the │ │ │ │ +0001afc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +0001afd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0001afe0: 5468 6520 6f62 6a65 6374 204d 6169 6e44 The object MainD │ │ │ │ +0001aff0: 6174 6144 6972 6563 746f 7279 2028 6d69 ataDirectory (mi │ │ │ │ +0001b000: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +0001b010: 696f 6e29 2069 7320 6120 2a6e 6f74 6520 ion) is a *note │ │ │ │ +0001b020: 7379 6d62 6f6c 3a0a 284d 6163 6175 6c61 symbol:.(Macaula │ │ │ │ +0001b030: 7932 446f 6329 5379 6d62 6f6c 2c2e 0a0a y2Doc)Symbol,... │ │ │ │ 0001b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -0001b090: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -0001b0a0: 2064 6f63 756d 656e 7420 6973 2069 6e20 document is in │ │ │ │ -0001b0b0: 6375 7272 656e 7453 7472 696e 673a 313a currentString:1: │ │ │ │ -0001b0c0: 3338 322e 0a1f 0a46 696c 653a 2042 6572 382....File: Ber │ │ │ │ -0001b0d0: 7469 6e69 2e69 6e66 6f2c 204e 6f64 653a tini.info, Node: │ │ │ │ -0001b0e0: 206d 616b 6542 2749 6e70 7574 4669 6c65 makeB'InputFile │ │ │ │ -0001b0f0: 2c20 4e65 7874 3a20 6d61 6b65 4227 5365 , Next: makeB'Se │ │ │ │ -0001b100: 6374 696f 6e2c 2050 7265 763a 204d 6169 ction, Prev: Mai │ │ │ │ -0001b110: 6e44 6174 6144 6972 6563 746f 7279 2c20 nDataDirectory, │ │ │ │ -0001b120: 5570 3a20 546f 700a 0a6d 616b 6542 2749 Up: Top..makeB'I │ │ │ │ -0001b130: 6e70 7574 4669 6c65 202d 2d20 7772 6974 nputFile -- writ │ │ │ │ -0001b140: 6520 6120 4265 7274 696e 6920 696e 7075 e a Bertini inpu │ │ │ │ -0001b150: 7420 6669 6c65 2069 6e20 6120 6469 7265 t file in a dire │ │ │ │ -0001b160: 6374 6f72 790a 2a2a 2a2a 2a2a 2a2a 2a2a ctory.********** │ │ │ │ +0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0001b090: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0001b0a0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0001b0b0: 696e 2063 7572 7265 6e74 5374 7269 6e67 in currentString │ │ │ │ +0001b0c0: 3a31 3a33 3832 2e0a 1f0a 4669 6c65 3a20 :1:382....File: │ │ │ │ +0001b0d0: 4265 7274 696e 692e 696e 666f 2c20 4e6f Bertini.info, No │ │ │ │ +0001b0e0: 6465 3a20 6d61 6b65 4227 496e 7075 7446 de: makeB'InputF │ │ │ │ +0001b0f0: 696c 652c 204e 6578 743a 206d 616b 6542 ile, Next: makeB │ │ │ │ +0001b100: 2753 6563 7469 6f6e 2c20 5072 6576 3a20 'Section, Prev: │ │ │ │ +0001b110: 4d61 696e 4461 7461 4469 7265 6374 6f72 MainDataDirector │ │ │ │ +0001b120: 792c 2055 703a 2054 6f70 0a0a 6d61 6b65 y, Up: Top..make │ │ │ │ +0001b130: 4227 496e 7075 7446 696c 6520 2d2d 2077 B'InputFile -- w │ │ │ │ +0001b140: 7269 7465 2061 2042 6572 7469 6e69 2069 rite a Bertini i │ │ │ │ +0001b150: 6e70 7574 2066 696c 6520 696e 2061 2064 nput file in a d │ │ │ │ +0001b160: 6972 6563 746f 7279 0a2a 2a2a 2a2a 2a2a irectory.******* │ │ │ │ 0001b170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001b180: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001b190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001b1a0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -0001b1b0: 2020 2020 2020 2020 6d61 6b65 4227 496e makeB'In │ │ │ │ -0001b1c0: 7075 7446 696c 6528 7329 0a20 202a 2049 putFile(s). * I │ │ │ │ -0001b1d0: 6e70 7574 733a 0a20 2020 2020 202a 2073 nputs:. * s │ │ │ │ -0001b1e0: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67 , a *note string │ │ │ │ -0001b1f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001b200: 5374 7269 6e67 2c2c 2061 2064 6972 6563 String,, a direc │ │ │ │ -0001b210: 746f 7279 2077 6865 7265 2074 6865 2069 tory where the i │ │ │ │ -0001b220: 6e70 7574 0a20 2020 2020 2020 2066 696c nput. fil │ │ │ │ -0001b230: 6520 7769 6c6c 2062 6520 7772 6974 7465 e will be writte │ │ │ │ -0001b240: 6e0a 2020 2a20 2a6e 6f74 6520 4f70 7469 n. * *note Opti │ │ │ │ -0001b250: 6f6e 616c 2069 6e70 7574 733a 2028 4d61 onal inputs: (Ma │ │ │ │ -0001b260: 6361 756c 6179 3244 6f63 2975 7369 6e67 caulay2Doc)using │ │ │ │ -0001b270: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ -0001b280: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ -0001b290: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520 :. * *note │ │ │ │ -0001b2a0: 4166 6656 6172 6961 626c 6547 726f 7570 AffVariableGroup │ │ │ │ -0001b2b0: 3a20 5661 7269 6162 6c65 2067 726f 7570 : Variable group │ │ │ │ -0001b2c0: 732c 203d 3e20 2e2e 2e2c 2064 6566 6175 s, => ..., defau │ │ │ │ -0001b2d0: 6c74 2076 616c 7565 207b 7d2c 2061 6e0a lt value {}, an. │ │ │ │ -0001b2e0: 2020 2020 2020 2020 6f70 7469 6f6e 2074 option t │ │ │ │ -0001b2f0: 6f20 6772 6f75 7020 7661 7269 6162 6c65 o group variable │ │ │ │ -0001b300: 7320 616e 6420 7573 6520 6d75 6c74 6968 s and use multih │ │ │ │ -0001b310: 6f6d 6f67 656e 656f 7573 2068 6f6d 6f74 omogeneous homot │ │ │ │ -0001b320: 6f70 6965 730a 2020 2020 2020 2a20 2a6e opies. * *n │ │ │ │ -0001b330: 6f74 6520 4227 436f 6e73 7461 6e74 733a ote B'Constants: │ │ │ │ -0001b340: 2042 2743 6f6e 7374 616e 7473 2c20 3d3e B'Constants, => │ │ │ │ -0001b350: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0001b360: 6c75 6520 7b7d 2c20 616e 206f 7074 696f lue {}, an optio │ │ │ │ -0001b370: 6e20 746f 0a20 2020 2020 2020 2064 6573 n to. des │ │ │ │ -0001b380: 6967 6e61 7465 2074 6865 2063 6f6e 7374 ignate the const │ │ │ │ -0001b390: 616e 7473 2066 6f72 2061 2042 6572 7469 ants for a Berti │ │ │ │ -0001b3a0: 6e69 2049 6e70 7574 2066 696c 650a 2020 ni Input file. │ │ │ │ -0001b3b0: 2020 2020 2a20 4227 4675 6e63 7469 6f6e * B'Function │ │ │ │ -0001b3c0: 7320 286d 6973 7369 6e67 2064 6f63 756d s (missing docum │ │ │ │ -0001b3d0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -0001b3e0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -0001b3f0: 7b7d 2c20 0a20 2020 2020 202a 2042 2750 {}, . * B'P │ │ │ │ -0001b400: 6f6c 796e 6f6d 6961 6c73 2028 6d69 7373 olynomials (miss │ │ │ │ -0001b410: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -0001b420: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -0001b430: 6c74 2076 616c 7565 207b 7d2c 200a 2020 lt value {}, . │ │ │ │ -0001b440: 2020 2020 2a20 4265 7274 696e 6949 6e70 * BertiniInp │ │ │ │ -0001b450: 7574 436f 6e66 6967 7572 6174 696f 6e20 utConfiguration │ │ │ │ -0001b460: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -0001b470: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -0001b480: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ -0001b490: 2020 2020 2020 7b7d 2c0a 2020 2020 2020 {},. │ │ │ │ -0001b4a0: 2a20 2a6e 6f74 6520 486f 6d56 6172 6961 * *note HomVaria │ │ │ │ -0001b4b0: 626c 6547 726f 7570 3a20 5661 7269 6162 bleGroup: Variab │ │ │ │ -0001b4c0: 6c65 2067 726f 7570 732c 203d 3e20 2e2e le groups, => .. │ │ │ │ -0001b4d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -0001b4e0: 207b 7d2c 2061 6e0a 2020 2020 2020 2020 {}, an. │ │ │ │ -0001b4f0: 6f70 7469 6f6e 2074 6f20 6772 6f75 7020 option to group │ │ │ │ -0001b500: 7661 7269 6162 6c65 7320 616e 6420 7573 variables and us │ │ │ │ -0001b510: 6520 6d75 6c74 6968 6f6d 6f67 656e 656f e multihomogeneo │ │ │ │ -0001b520: 7573 2068 6f6d 6f74 6f70 6965 730a 2020 us homotopies. │ │ │ │ -0001b530: 2020 2020 2a20 4e61 6d65 4227 496e 7075 * NameB'Inpu │ │ │ │ -0001b540: 7446 696c 6520 286d 6973 7369 6e67 2064 tFile (missing d │ │ │ │ -0001b550: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ -0001b560: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0001b570: 6c75 6520 2269 6e70 7574 222c 200a 2020 lue "input", . │ │ │ │ -0001b580: 2020 2020 2a20 4e61 6d65 506f 6c79 6e6f * NamePolyno │ │ │ │ -0001b590: 6d69 616c 7320 286d 6973 7369 6e67 2064 mials (missing d │ │ │ │ -0001b5a0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ -0001b5b0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0001b5c0: 6c75 6520 7b7d 2c20 0a20 2020 2020 202a lue {}, . * │ │ │ │ -0001b5d0: 2050 6172 616d 6574 6572 4772 6f75 7020 ParameterGroup │ │ │ │ -0001b5e0: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -0001b5f0: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -0001b600: 6465 6661 756c 7420 7661 6c75 6520 7b7d default value {} │ │ │ │ -0001b610: 2c20 0a20 2020 2020 202a 2050 6174 6856 , . * PathV │ │ │ │ -0001b620: 6172 6961 626c 6520 286d 6973 7369 6e67 ariable (missing │ │ │ │ -0001b630: 2064 6f63 756d 656e 7461 7469 6f6e 2920 documentation) │ │ │ │ -0001b640: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -0001b650: 7661 6c75 6520 7b7d 2c20 0a20 2020 2020 value {}, . │ │ │ │ -0001b660: 202a 202a 6e6f 7465 2052 616e 646f 6d43 * *note RandomC │ │ │ │ -0001b670: 6f6d 706c 6578 3a20 4265 7274 696e 6920 omplex: Bertini │ │ │ │ -0001b680: 696e 7075 7420 6669 6c65 2064 6563 6c61 input file decla │ │ │ │ -0001b690: 7261 7469 6f6e 735f 636f 2072 616e 646f rations_co rando │ │ │ │ -0001b6a0: 6d20 6e75 6d62 6572 732c 0a20 2020 2020 m numbers,. │ │ │ │ -0001b6b0: 2020 203d 3e20 2e2e 2e2c 2064 6566 6175 => ..., defau │ │ │ │ -0001b6c0: 6c74 2076 616c 7565 207b 7d2c 2061 6e20 lt value {}, an │ │ │ │ -0001b6d0: 6f70 7469 6f6e 2077 6869 6368 2064 6573 option which des │ │ │ │ -0001b6e0: 6967 6e61 7465 730a 2020 2020 2020 2020 ignates. │ │ │ │ -0001b6f0: 7379 6d62 6f6c 732f 7374 7269 6e67 732f symbols/strings/ │ │ │ │ -0001b700: 7661 7269 6162 6c65 7320 7468 6174 2077 variables that w │ │ │ │ -0001b710: 696c 6c20 6265 2073 6574 2074 6f20 6265 ill be set to be │ │ │ │ -0001b720: 2061 2072 616e 646f 6d20 7265 616c 206e a random real n │ │ │ │ -0001b730: 756d 6265 720a 2020 2020 2020 2020 6f72 umber. or │ │ │ │ -0001b740: 2072 616e 646f 6d20 636f 6d70 6c65 7820 random complex │ │ │ │ -0001b750: 6e75 6d62 6572 0a20 2020 2020 202a 202a number. * * │ │ │ │ -0001b760: 6e6f 7465 2052 616e 646f 6d52 6561 6c3a note RandomReal: │ │ │ │ -0001b770: 2042 6572 7469 6e69 2069 6e70 7574 2066 Bertini input f │ │ │ │ -0001b780: 696c 6520 6465 636c 6172 6174 696f 6e73 ile declarations │ │ │ │ -0001b790: 5f63 6f20 7261 6e64 6f6d 206e 756d 6265 _co random numbe │ │ │ │ -0001b7a0: 7273 2c20 3d3e 0a20 2020 2020 2020 202e rs, =>. . │ │ │ │ -0001b7b0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -0001b7c0: 6520 7b7d 2c20 616e 206f 7074 696f 6e20 e {}, an option │ │ │ │ -0001b7d0: 7768 6963 6820 6465 7369 676e 6174 6573 which designates │ │ │ │ -0001b7e0: 0a20 2020 2020 2020 2073 796d 626f 6c73 . symbols │ │ │ │ -0001b7f0: 2f73 7472 696e 6773 2f76 6172 6961 626c /strings/variabl │ │ │ │ -0001b800: 6573 2074 6861 7420 7769 6c6c 2062 6520 es that will be │ │ │ │ -0001b810: 7365 7420 746f 2062 6520 6120 7261 6e64 set to be a rand │ │ │ │ -0001b820: 6f6d 2072 6561 6c20 6e75 6d62 6572 0a20 om real number. │ │ │ │ -0001b830: 2020 2020 2020 206f 7220 7261 6e64 6f6d or random │ │ │ │ -0001b840: 2063 6f6d 706c 6578 206e 756d 6265 720a complex number. │ │ │ │ -0001b850: 2020 2020 2020 2a20 5365 7450 6172 616d * SetParam │ │ │ │ -0001b860: 6574 6572 4772 6f75 7020 286d 6973 7369 eterGroup (missi │ │ │ │ -0001b870: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -0001b880: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -0001b890: 7420 7661 6c75 6520 7b7d 2c20 0a20 2020 t value {}, . │ │ │ │ -0001b8a0: 2020 202a 2053 746f 7261 6765 466f 6c64 * StorageFold │ │ │ │ -0001b8b0: 6572 2028 6d69 7373 696e 6720 646f 6375 er (missing docu │ │ │ │ -0001b8c0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -0001b8d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -0001b8e0: 206e 756c 6c2c 200a 2020 2020 2020 2a20 null, . * │ │ │ │ -0001b8f0: 5661 7269 6162 6c65 4c69 7374 2028 6d69 VariableList (mi │ │ │ │ -0001b900: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -0001b910: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -0001b920: 6175 6c74 2076 616c 7565 207b 7d2c 200a ault value {}, . │ │ │ │ -0001b930: 2020 2020 2020 2a20 2a6e 6f74 6520 5665 * *note Ve │ │ │ │ -0001b940: 7262 6f73 653a 2062 6572 7469 6e69 5472 rbose: bertiniTr │ │ │ │ -0001b950: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70 ackHomotopy_lp_p │ │ │ │ -0001b960: 645f 7064 5f70 645f 636d 5665 7262 6f73 d_pd_pd_cmVerbos │ │ │ │ -0001b970: 653d 3e5f 7064 5f70 645f 7064 5f72 700a e=>_pd_pd_pd_rp. │ │ │ │ -0001b980: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e , => ... │ │ │ │ -0001b990: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -0001b9a0: 6661 6c73 652c 204f 7074 696f 6e20 746f false, Option to │ │ │ │ -0001b9b0: 2073 696c 656e 6365 2061 6464 6974 696f silence additio │ │ │ │ -0001b9c0: 6e61 6c20 6f75 7470 7574 0a0a 4465 7363 nal output..Desc │ │ │ │ -0001b9d0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0001b9e0: 3d3d 3d0a 0a54 6869 7320 6675 6e63 7469 ===..This functi │ │ │ │ -0001b9f0: 6f6e 2077 7269 7465 7320 6120 4265 7274 on writes a Bert │ │ │ │ -0001ba00: 696e 6920 696e 7075 7420 6669 6c65 2e20 ini input file. │ │ │ │ -0001ba10: 5468 6520 7573 6572 2063 616e 2073 7065 The user can spe │ │ │ │ -0001ba20: 6369 6679 2043 4f4e 4649 4753 2066 6f72 cify CONFIGS for │ │ │ │ -0001ba30: 2074 6865 0a66 696c 6520 7573 696e 6720 the.file using │ │ │ │ -0001ba40: 7468 6520 4265 7274 696e 6949 6e70 7574 the BertiniInput │ │ │ │ -0001ba50: 436f 6e66 6967 7572 6174 696f 6e20 6f70 Configuration op │ │ │ │ -0001ba60: 7469 6f6e 2e20 5468 6520 7573 6572 2073 tion. The user s │ │ │ │ -0001ba70: 686f 756c 6420 7370 6563 6966 790a 7661 hould specify.va │ │ │ │ -0001ba80: 7269 6162 6c65 2067 726f 7570 7320 7769 riable groups wi │ │ │ │ -0001ba90: 7468 2074 6865 2041 6666 5661 7269 6162 th the AffVariab │ │ │ │ -0001baa0: 6c65 4772 6f75 7020 2861 6666 696e 6520 leGroup (affine │ │ │ │ -0001bab0: 7661 7269 6162 6c65 2067 726f 7570 2920 variable group) │ │ │ │ -0001bac0: 6f70 7469 6f6e 206f 720a 486f 6d56 6172 option or.HomVar │ │ │ │ -0001bad0: 6961 626c 6547 726f 7570 2028 686f 6d6f iableGroup (homo │ │ │ │ -0001bae0: 6765 6e65 6f75 7320 7661 7269 6162 6c65 geneous variable │ │ │ │ -0001baf0: 2067 726f 7570 2920 6f70 7469 6f6e 2e20 group) option. │ │ │ │ -0001bb00: 5468 6520 7573 6572 2073 686f 756c 6420 The user should │ │ │ │ -0001bb10: 7370 6563 6966 790a 7468 6520 706f 6c79 specify.the poly │ │ │ │ -0001bb20: 6e6f 6d69 616c 2073 7973 7465 6d20 7468 nomial system th │ │ │ │ -0001bb30: 6579 2077 616e 7420 746f 2073 6f6c 7665 ey want to solve │ │ │ │ -0001bb40: 2077 6974 6820 7468 6520 2042 2750 6f6c with the B'Pol │ │ │ │ -0001bb50: 796e 6f6d 6961 6c73 206f 7074 696f 6e20 ynomials option │ │ │ │ -0001bb60: 6f72 0a42 2746 756e 6374 696f 6e73 206f or.B'Functions o │ │ │ │ -0001bb70: 7074 696f 6e2e 2049 6620 4227 506f 6c79 ption. If B'Poly │ │ │ │ -0001bb80: 6e6f 6d69 616c 7320 6973 206e 6f74 2075 nomials is not u │ │ │ │ -0001bb90: 7365 6420 7468 656e 2074 6865 2075 7365 sed then the use │ │ │ │ -0001bba0: 7220 7368 6f75 6c64 2075 7365 2074 6865 r should use the │ │ │ │ -0001bbb0: 0a4e 616d 6550 6f6c 796e 6f6d 6961 6c73 .NamePolynomials │ │ │ │ -0001bbc0: 206f 7074 696f 6e2e 0a0a 2b2d 2d2d 2d2d option...+----- │ │ │ │ +0001b1a0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +0001b1b0: 3a20 0a20 2020 2020 2020 206d 616b 6542 : . makeB │ │ │ │ +0001b1c0: 2749 6e70 7574 4669 6c65 2873 290a 2020 'InputFile(s). │ │ │ │ +0001b1d0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +0001b1e0: 2a20 732c 2061 202a 6e6f 7465 2073 7472 * s, a *note str │ │ │ │ +0001b1f0: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +0001b200: 6f63 2953 7472 696e 672c 2c20 6120 6469 oc)String,, a di │ │ │ │ +0001b210: 7265 6374 6f72 7920 7768 6572 6520 7468 rectory where th │ │ │ │ +0001b220: 6520 696e 7075 740a 2020 2020 2020 2020 e input. │ │ │ │ +0001b230: 6669 6c65 2077 696c 6c20 6265 2077 7269 file will be wri │ │ │ │ +0001b240: 7474 656e 0a20 202a 202a 6e6f 7465 204f tten. * *note O │ │ │ │ +0001b250: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +0001b260: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +0001b270: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +0001b280: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +0001b290: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f ts,:. * *no │ │ │ │ +0001b2a0: 7465 2041 6666 5661 7269 6162 6c65 4772 te AffVariableGr │ │ │ │ +0001b2b0: 6f75 703a 2056 6172 6961 626c 6520 6772 oup: Variable gr │ │ │ │ +0001b2c0: 6f75 7073 2c20 3d3e 202e 2e2e 2c20 6465 oups, => ..., de │ │ │ │ +0001b2d0: 6661 756c 7420 7661 6c75 6520 7b7d 2c20 fault value {}, │ │ │ │ +0001b2e0: 616e 0a20 2020 2020 2020 206f 7074 696f an. optio │ │ │ │ +0001b2f0: 6e20 746f 2067 726f 7570 2076 6172 6961 n to group varia │ │ │ │ +0001b300: 626c 6573 2061 6e64 2075 7365 206d 756c bles and use mul │ │ │ │ +0001b310: 7469 686f 6d6f 6765 6e65 6f75 7320 686f tihomogeneous ho │ │ │ │ +0001b320: 6d6f 746f 7069 6573 0a20 2020 2020 202a motopies. * │ │ │ │ +0001b330: 202a 6e6f 7465 2042 2743 6f6e 7374 616e *note B'Constan │ │ │ │ +0001b340: 7473 3a20 4227 436f 6e73 7461 6e74 732c ts: B'Constants, │ │ │ │ +0001b350: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +0001b360: 2076 616c 7565 207b 7d2c 2061 6e20 6f70 value {}, an op │ │ │ │ +0001b370: 7469 6f6e 2074 6f0a 2020 2020 2020 2020 tion to. │ │ │ │ +0001b380: 6465 7369 676e 6174 6520 7468 6520 636f designate the co │ │ │ │ +0001b390: 6e73 7461 6e74 7320 666f 7220 6120 4265 nstants for a Be │ │ │ │ +0001b3a0: 7274 696e 6920 496e 7075 7420 6669 6c65 rtini Input file │ │ │ │ +0001b3b0: 0a20 2020 2020 202a 2042 2746 756e 6374 . * B'Funct │ │ │ │ +0001b3c0: 696f 6e73 2028 6d69 7373 696e 6720 646f ions (missing do │ │ │ │ +0001b3d0: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +0001b3e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +0001b3f0: 7565 207b 7d2c 200a 2020 2020 2020 2a20 ue {}, . * │ │ │ │ +0001b400: 4227 506f 6c79 6e6f 6d69 616c 7320 286d B'Polynomials (m │ │ │ │ +0001b410: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +0001b420: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +0001b430: 6661 756c 7420 7661 6c75 6520 7b7d 2c20 fault value {}, │ │ │ │ +0001b440: 0a20 2020 2020 202a 2042 6572 7469 6e69 . * Bertini │ │ │ │ +0001b450: 496e 7075 7443 6f6e 6669 6775 7261 7469 InputConfigurati │ │ │ │ +0001b460: 6f6e 2028 6d69 7373 696e 6720 646f 6375 on (missing docu │ │ │ │ +0001b470: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ +0001b480: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +0001b490: 0a20 2020 2020 2020 207b 7d2c 0a20 2020 . {},. │ │ │ │ +0001b4a0: 2020 202a 202a 6e6f 7465 2048 6f6d 5661 * *note HomVa │ │ │ │ +0001b4b0: 7269 6162 6c65 4772 6f75 703a 2056 6172 riableGroup: Var │ │ │ │ +0001b4c0: 6961 626c 6520 6772 6f75 7073 2c20 3d3e iable groups, => │ │ │ │ +0001b4d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +0001b4e0: 6c75 6520 7b7d 2c20 616e 0a20 2020 2020 lue {}, an. │ │ │ │ +0001b4f0: 2020 206f 7074 696f 6e20 746f 2067 726f option to gro │ │ │ │ +0001b500: 7570 2076 6172 6961 626c 6573 2061 6e64 up variables and │ │ │ │ +0001b510: 2075 7365 206d 756c 7469 686f 6d6f 6765 use multihomoge │ │ │ │ +0001b520: 6e65 6f75 7320 686f 6d6f 746f 7069 6573 neous homotopies │ │ │ │ +0001b530: 0a20 2020 2020 202a 204e 616d 6542 2749 . * NameB'I │ │ │ │ +0001b540: 6e70 7574 4669 6c65 2028 6d69 7373 696e nputFile (missin │ │ │ │ +0001b550: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +0001b560: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +0001b570: 2076 616c 7565 2022 696e 7075 7422 2c20 value "input", │ │ │ │ +0001b580: 0a20 2020 2020 202a 204e 616d 6550 6f6c . * NamePol │ │ │ │ +0001b590: 796e 6f6d 6961 6c73 2028 6d69 7373 696e ynomials (missin │ │ │ │ +0001b5a0: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +0001b5b0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +0001b5c0: 2076 616c 7565 207b 7d2c 200a 2020 2020 value {}, . │ │ │ │ +0001b5d0: 2020 2a20 5061 7261 6d65 7465 7247 726f * ParameterGro │ │ │ │ +0001b5e0: 7570 2028 6d69 7373 696e 6720 646f 6375 up (missing docu │ │ │ │ +0001b5f0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ +0001b600: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +0001b610: 207b 7d2c 200a 2020 2020 2020 2a20 5061 {}, . * Pa │ │ │ │ +0001b620: 7468 5661 7269 6162 6c65 2028 6d69 7373 thVariable (miss │ │ │ │ +0001b630: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +0001b640: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ +0001b650: 6c74 2076 616c 7565 207b 7d2c 200a 2020 lt value {}, . │ │ │ │ +0001b660: 2020 2020 2a20 2a6e 6f74 6520 5261 6e64 * *note Rand │ │ │ │ +0001b670: 6f6d 436f 6d70 6c65 783a 2042 6572 7469 omComplex: Berti │ │ │ │ +0001b680: 6e69 2069 6e70 7574 2066 696c 6520 6465 ni input file de │ │ │ │ +0001b690: 636c 6172 6174 696f 6e73 5f63 6f20 7261 clarations_co ra │ │ │ │ +0001b6a0: 6e64 6f6d 206e 756d 6265 7273 2c0a 2020 ndom numbers,. │ │ │ │ +0001b6b0: 2020 2020 2020 3d3e 202e 2e2e 2c20 6465 => ..., de │ │ │ │ +0001b6c0: 6661 756c 7420 7661 6c75 6520 7b7d 2c20 fault value {}, │ │ │ │ +0001b6d0: 616e 206f 7074 696f 6e20 7768 6963 6820 an option which │ │ │ │ +0001b6e0: 6465 7369 676e 6174 6573 0a20 2020 2020 designates. │ │ │ │ +0001b6f0: 2020 2073 796d 626f 6c73 2f73 7472 696e symbols/strin │ │ │ │ +0001b700: 6773 2f76 6172 6961 626c 6573 2074 6861 gs/variables tha │ │ │ │ +0001b710: 7420 7769 6c6c 2062 6520 7365 7420 746f t will be set to │ │ │ │ +0001b720: 2062 6520 6120 7261 6e64 6f6d 2072 6561 be a random rea │ │ │ │ +0001b730: 6c20 6e75 6d62 6572 0a20 2020 2020 2020 l number. │ │ │ │ +0001b740: 206f 7220 7261 6e64 6f6d 2063 6f6d 706c or random compl │ │ │ │ +0001b750: 6578 206e 756d 6265 720a 2020 2020 2020 ex number. │ │ │ │ +0001b760: 2a20 2a6e 6f74 6520 5261 6e64 6f6d 5265 * *note RandomRe │ │ │ │ +0001b770: 616c 3a20 4265 7274 696e 6920 696e 7075 al: Bertini inpu │ │ │ │ +0001b780: 7420 6669 6c65 2064 6563 6c61 7261 7469 t file declarati │ │ │ │ +0001b790: 6f6e 735f 636f 2072 616e 646f 6d20 6e75 ons_co random nu │ │ │ │ +0001b7a0: 6d62 6572 732c 203d 3e0a 2020 2020 2020 mbers, =>. │ │ │ │ +0001b7b0: 2020 2e2e 2e2c 2064 6566 6175 6c74 2076 ..., default v │ │ │ │ +0001b7c0: 616c 7565 207b 7d2c 2061 6e20 6f70 7469 alue {}, an opti │ │ │ │ +0001b7d0: 6f6e 2077 6869 6368 2064 6573 6967 6e61 on which designa │ │ │ │ +0001b7e0: 7465 730a 2020 2020 2020 2020 7379 6d62 tes. symb │ │ │ │ +0001b7f0: 6f6c 732f 7374 7269 6e67 732f 7661 7269 ols/strings/vari │ │ │ │ +0001b800: 6162 6c65 7320 7468 6174 2077 696c 6c20 ables that will │ │ │ │ +0001b810: 6265 2073 6574 2074 6f20 6265 2061 2072 be set to be a r │ │ │ │ +0001b820: 616e 646f 6d20 7265 616c 206e 756d 6265 andom real numbe │ │ │ │ +0001b830: 720a 2020 2020 2020 2020 6f72 2072 616e r. or ran │ │ │ │ +0001b840: 646f 6d20 636f 6d70 6c65 7820 6e75 6d62 dom complex numb │ │ │ │ +0001b850: 6572 0a20 2020 2020 202a 2053 6574 5061 er. * SetPa │ │ │ │ +0001b860: 7261 6d65 7465 7247 726f 7570 2028 6d69 rameterGroup (mi │ │ │ │ +0001b870: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +0001b880: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +0001b890: 6175 6c74 2076 616c 7565 207b 7d2c 200a ault value {}, . │ │ │ │ +0001b8a0: 2020 2020 2020 2a20 5374 6f72 6167 6546 * StorageF │ │ │ │ +0001b8b0: 6f6c 6465 7220 286d 6973 7369 6e67 2064 older (missing d │ │ │ │ +0001b8c0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ +0001b8d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +0001b8e0: 6c75 6520 6e75 6c6c 2c20 0a20 2020 2020 lue null, . │ │ │ │ +0001b8f0: 202a 2056 6172 6961 626c 654c 6973 7420 * VariableList │ │ │ │ +0001b900: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +0001b910: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ +0001b920: 6465 6661 756c 7420 7661 6c75 6520 7b7d default value {} │ │ │ │ +0001b930: 2c20 0a20 2020 2020 202a 202a 6e6f 7465 , . * *note │ │ │ │ +0001b940: 2056 6572 626f 7365 3a20 6265 7274 696e Verbose: bertin │ │ │ │ +0001b950: 6954 7261 636b 486f 6d6f 746f 7079 5f6c iTrackHomotopy_l │ │ │ │ +0001b960: 705f 7064 5f70 645f 7064 5f63 6d56 6572 p_pd_pd_pd_cmVer │ │ │ │ +0001b970: 626f 7365 3d3e 5f70 645f 7064 5f70 645f bose=>_pd_pd_pd_ │ │ │ │ +0001b980: 7270 0a20 2020 2020 2020 202c 203d 3e20 rp. , => │ │ │ │ +0001b990: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +0001b9a0: 7565 2066 616c 7365 2c20 4f70 7469 6f6e ue false, Option │ │ │ │ +0001b9b0: 2074 6f20 7369 6c65 6e63 6520 6164 6469 to silence addi │ │ │ │ +0001b9c0: 7469 6f6e 616c 206f 7574 7075 740a 0a44 tional output..D │ │ │ │ +0001b9d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0001b9e0: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e ======..This fun │ │ │ │ +0001b9f0: 6374 696f 6e20 7772 6974 6573 2061 2042 ction writes a B │ │ │ │ +0001ba00: 6572 7469 6e69 2069 6e70 7574 2066 696c ertini input fil │ │ │ │ +0001ba10: 652e 2054 6865 2075 7365 7220 6361 6e20 e. The user can │ │ │ │ +0001ba20: 7370 6563 6966 7920 434f 4e46 4947 5320 specify CONFIGS │ │ │ │ +0001ba30: 666f 7220 7468 650a 6669 6c65 2075 7369 for the.file usi │ │ │ │ +0001ba40: 6e67 2074 6865 2042 6572 7469 6e69 496e ng the BertiniIn │ │ │ │ +0001ba50: 7075 7443 6f6e 6669 6775 7261 7469 6f6e putConfiguration │ │ │ │ +0001ba60: 206f 7074 696f 6e2e 2054 6865 2075 7365 option. The use │ │ │ │ +0001ba70: 7220 7368 6f75 6c64 2073 7065 6369 6679 r should specify │ │ │ │ +0001ba80: 0a76 6172 6961 626c 6520 6772 6f75 7073 .variable groups │ │ │ │ +0001ba90: 2077 6974 6820 7468 6520 4166 6656 6172 with the AffVar │ │ │ │ +0001baa0: 6961 626c 6547 726f 7570 2028 6166 6669 iableGroup (affi │ │ │ │ +0001bab0: 6e65 2076 6172 6961 626c 6520 6772 6f75 ne variable grou │ │ │ │ +0001bac0: 7029 206f 7074 696f 6e20 6f72 0a48 6f6d p) option or.Hom │ │ │ │ +0001bad0: 5661 7269 6162 6c65 4772 6f75 7020 2868 VariableGroup (h │ │ │ │ +0001bae0: 6f6d 6f67 656e 656f 7573 2076 6172 6961 omogeneous varia │ │ │ │ +0001baf0: 626c 6520 6772 6f75 7029 206f 7074 696f ble group) optio │ │ │ │ +0001bb00: 6e2e 2054 6865 2075 7365 7220 7368 6f75 n. The user shou │ │ │ │ +0001bb10: 6c64 2073 7065 6369 6679 0a74 6865 2070 ld specify.the p │ │ │ │ +0001bb20: 6f6c 796e 6f6d 6961 6c20 7379 7374 656d olynomial system │ │ │ │ +0001bb30: 2074 6865 7920 7761 6e74 2074 6f20 736f they want to so │ │ │ │ +0001bb40: 6c76 6520 7769 7468 2074 6865 2020 4227 lve with the B' │ │ │ │ +0001bb50: 506f 6c79 6e6f 6d69 616c 7320 6f70 7469 Polynomials opti │ │ │ │ +0001bb60: 6f6e 206f 720a 4227 4675 6e63 7469 6f6e on or.B'Function │ │ │ │ +0001bb70: 7320 6f70 7469 6f6e 2e20 4966 2042 2750 s option. If B'P │ │ │ │ +0001bb80: 6f6c 796e 6f6d 6961 6c73 2069 7320 6e6f olynomials is no │ │ │ │ +0001bb90: 7420 7573 6564 2074 6865 6e20 7468 6520 t used then the │ │ │ │ +0001bba0: 7573 6572 2073 686f 756c 6420 7573 6520 user should use │ │ │ │ +0001bbb0: 7468 650a 4e61 6d65 506f 6c79 6e6f 6d69 the.NamePolynomi │ │ │ │ +0001bbc0: 616c 7320 6f70 7469 6f6e 2e0a 0a2b 2d2d als option...+-- │ │ │ │ 0001bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bc00: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 523d ------+.|i1 : R= │ │ │ │ -0001bc10: 5151 5b78 312c 7832 2c79 5d20 2020 2020 QQ[x1,x2,y] │ │ │ │ +0001bc00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +0001bc10: 2052 3d51 515b 7831 2c78 322c 795d 2020 R=QQ[x1,x2,y] │ │ │ │ 0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001bc40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001bc50: 2020 2020 2020 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 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ +0001bc80: 2020 2020 207c 0a7c 6f31 203d 2052 2020 |.|o1 = R │ │ │ │ 0001bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bcc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001bd00: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ -0001bd10: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0001bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bd00: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ +0001bd10: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ 0001bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bd40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -0001bd80: 3a20 7468 6544 6972 203d 2074 656d 706f : theDir = tempo │ │ │ │ -0001bd90: 7261 7279 4669 6c65 4e61 6d65 2829 3b20 raryFileName(); │ │ │ │ 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+0001be00: 206d 616b 6544 6972 6563 746f 7279 2074 makeDirectory t │ │ │ │ +0001be10: 6865 4469 7220 2020 2020 2020 2020 2020 heDir │ │ │ │ 0001be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001be30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001be70: 2d2d 2b0a 7c69 3420 3a20 6d61 6b65 4227 --+.|i4 : makeB' │ │ │ │ -0001be80: 496e 7075 7446 696c 6528 7468 6544 6972 InputFile(theDir │ │ │ │ -0001be90: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ +0001be70: 2d2d 2d2d 2d2b 0a7c 6934 203a 206d 616b -----+.|i4 : mak │ │ │ │ +0001be80: 6542 2749 6e70 7574 4669 6c65 2874 6865 eB'InputFile(the │ │ │ │ +0001be90: 4469 722c 2020 2020 2020 2020 2020 2020 Dir, │ │ │ │ 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7469 6e69 496e 7075 7443 BertiniInputC │ │ │ │ +0001bed0: 6f6e 6669 6775 7261 7469 6f6e 3d3e 7b4d onfiguration=>{M │ │ │ │ +0001bee0: 5054 7970 653d 3e32 7d2c 2020 2020 2020 PType=>2}, │ │ │ │ +0001bef0: 207c 0a7c 2020 2020 2020 2020 2041 6666 |.| Aff │ │ │ │ +0001bf00: 5661 7269 6162 6c65 4772 6f75 703d 3e7b VariableGroup=>{ │ │ │ │ +0001bf10: 7b78 312c 7832 7d2c 7b79 7d7d 2c20 2020 {x1,x2},{y}}, │ │ │ │ +0001bf20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bf30: 0a7c 2020 2020 2020 2020 2020 2020 2042 .| B │ │ │ │ +0001bf40: 2750 6f6c 796e 6f6d 6961 6c73 3d3e 7b79 'Polynomials=>{y │ │ │ │ +0001bf50: 2a28 7831 2b78 322b 3129 5e32 2b31 2c78 *(x1+x2+1)^2+1,x │ │ │ │ +0001bf60: 312d 7832 2b31 2c79 2d32 7d29 3b7c 0a2b 1-x2+1,y-2});|.+ │ │ │ │ 0001bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2b0a 2b2d 2d2d 2d2d --------+.+----- │ │ │ │ +0001bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d -----------+.+-- │ │ │ │ 0001bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001bfe0: 0a7c 6935 203a 2052 3d51 515b 7831 2c78 .|i5 : R=QQ[x1,x │ │ │ │ -0001bff0: 322c 792c 585d 2020 2020 2020 2020 2020 2,y,X] │ │ │ │ +0001bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bfe0: 2d2d 2b0a 7c69 3520 3a20 523d 5151 5b78 --+.|i5 : R=QQ[x │ │ │ │ +0001bff0: 312c 7832 2c79 2c58 5d20 2020 2020 2020 1,x2,y,X] │ │ │ │ 0001c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c010: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c010: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0001c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c040: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001c050: 6f35 203d 2052 2020 2020 2020 2020 2020 o5 = R │ │ │ │ +0001c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c050: 7c0a 7c6f 3520 3d20 5220 2020 2020 2020 |.|o5 = R │ │ │ │ 0001c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001c080: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -0001c0c0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0001c0d0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0001c0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001c0c0: 7c6f 3520 3a20 506f 6c79 6e6f 6d69 616c |o5 : Polynomial │ │ │ │ +0001c0d0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c0f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001c0f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0001c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c120: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -0001c130: 206d 616b 6542 2749 6e70 7574 4669 6c65 makeB'InputFile │ │ │ │ -0001c140: 2874 6865 4469 722c 2020 2020 2020 2020 (theDir, │ │ │ │ +0001c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001c130: 3620 3a20 6d61 6b65 4227 496e 7075 7446 6 : makeB'InputF │ │ │ │ +0001c140: 696c 6528 7468 6544 6972 2c20 2020 2020 ile(theDir, │ │ │ │ 0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c160: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c170: 4265 7274 696e 6949 6e70 7574 436f 6e66 BertiniInputConf │ │ │ │ -0001c180: 6967 7572 6174 696f 6e3d 3e7b 4d50 5479 iguration=>{MPTy │ │ │ │ -0001c190: 7065 3d3e 327d 2c7c 0a7c 2020 2020 2020 pe=>2},|.| │ │ │ │ -0001c1a0: 2020 2041 6666 5661 7269 6162 6c65 4772 AffVariableGr │ │ │ │ -0001c1b0: 6f75 703d 3e7b 7b78 312c 7832 7d2c 7b79 oup=>{{x1,x2},{y │ │ │ │ -0001c1c0: 7d7d 2c20 2020 2020 2020 2020 2020 7c0a }}, |. │ │ │ │ -0001c1d0: 7c20 2020 2020 2020 2020 2020 2020 4e61 | Na │ │ │ │ -0001c1e0: 6d65 506f 6c79 6e6f 6d69 616c 733d 3e7b mePolynomials=>{ │ │ │ │ -0001c1f0: 6631 2c66 322c 6633 7d2c 2020 2020 2020 f1,f2,f3}, │ │ │ │ -0001c200: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001c210: 2020 2020 2042 2746 756e 6374 696f 6e73 B'Functions │ │ │ │ -0001c220: 3d3e 7b20 2020 2020 2020 2020 2020 2020 =>{ │ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001c240: 2020 2020 2020 2020 2020 207b 582c 7831 {X,x1 │ │ │ │ -0001c250: 2b78 322b 317d 2c20 2020 2020 2020 2020 +x2+1}, │ │ │ │ +0001c160: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c170: 2020 2042 6572 7469 6e69 496e 7075 7443 BertiniInputC │ │ │ │ +0001c180: 6f6e 6669 6775 7261 7469 6f6e 3d3e 7b4d onfiguration=>{M │ │ │ │ +0001c190: 5054 7970 653d 3e32 7d2c 7c0a 7c20 2020 PType=>2},|.| │ │ │ │ +0001c1a0: 2020 2020 2020 4166 6656 6172 6961 626c AffVariabl │ │ │ │ +0001c1b0: 6547 726f 7570 3d3e 7b7b 7831 2c78 327d eGroup=>{{x1,x2} │ │ │ │ +0001c1c0: 2c7b 797d 7d2c 2020 2020 2020 2020 2020 ,{y}}, │ │ │ │ +0001c1d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c1e0: 204e 616d 6550 6f6c 796e 6f6d 6961 6c73 NamePolynomials │ │ │ │ +0001c1f0: 3d3e 7b66 312c 6632 2c66 337d 2c20 2020 =>{f1,f2,f3}, │ │ │ │ +0001c200: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c210: 2020 2020 2020 2020 4227 4675 6e63 7469 B'Functi │ │ │ │ +0001c220: 6f6e 733d 3e7b 2020 2020 2020 2020 2020 ons=>{ │ │ │ │ +0001c230: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c240: 0a7c 2020 2020 2020 2020 2020 2020 7b58 .| {X │ │ │ │ +0001c250: 2c78 312b 7832 2b31 7d2c 2020 2020 2020 ,x1+x2+1}, │ │ │ │ 0001c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c270: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c280: 2020 7b66 312c 792a 585e 322b 317d 2c20 {f1,y*X^2+1}, │ │ │ │ -0001c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001c2b0: 2020 2020 2020 2020 207b 6632 2c78 312d {f2,x1- │ │ │ │ -0001c2c0: 7832 2b31 7d2c 2020 2020 2020 2020 2020 x2+1}, │ │ │ │ +0001c270: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c280: 2020 2020 207b 6631 2c79 2a58 5e32 2b31 {f1,y*X^2+1 │ │ │ │ +0001c290: 7d2c 2020 2020 2020 2020 2020 2020 2020 }, │ │ │ │ +0001c2a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c2b0: 2020 2020 2020 2020 2020 2020 7b66 322c {f2, │ │ │ │ +0001c2c0: 7831 2d78 322b 317d 2c20 2020 2020 2020 x1-x2+1}, │ │ │ │ 0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c2f0: 7b66 332c 792d 327d 7d29 3b20 2020 2020 {f3,y-2}}); │ │ │ │ +0001c2e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001c2f0: 2020 207b 6633 2c79 2d32 7d7d 293b 2020 {f3,y-2}}); │ │ │ │ 0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c310: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001c310: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0001c320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001c350: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c350: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ 0001c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c380: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 523d ------+.|i7 : R= │ │ │ │ -0001c390: 5151 5b78 312c 7832 2c79 2c58 5d20 2020 QQ[x1,x2,y,X] │ │ │ │ +0001c380: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ +0001c390: 2052 3d51 515b 7831 2c78 322c 792c 585d R=QQ[x1,x2,y,X] │ │ │ │ 0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c3c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3f0: 2020 2020 7c0a 7c6f 3720 3d20 5220 2020 |.|o7 = R │ │ │ │ +0001c3f0: 2020 2020 2020 207c 0a7c 6f37 203d 2052 |.|o7 = R │ │ │ │ 0001c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c420: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001c430: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c460: 2020 7c0a 7c6f 3720 3a20 506f 6c79 6e6f |.|o7 : Polyno │ │ │ │ -0001c470: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0001c460: 2020 2020 207c 0a7c 6f37 203a 2050 6f6c |.|o7 : Pol │ │ │ │ +0001c470: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ 0001c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c490: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001c490: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0001c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c4d0: 2b0a 7c69 3820 3a20 6d61 6b65 4227 496e +.|i8 : makeB'In │ │ │ │ -0001c4e0: 7075 7446 696c 6528 7468 6544 6972 2c20 putFile(theDir, │ │ │ │ -0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c500: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0001c510: 2020 2020 2020 2020 4265 7274 696e 6949 BertiniI │ │ │ │ -0001c520: 6e70 7574 436f 6e66 6967 7572 6174 696f nputConfiguratio │ │ │ │ -0001c530: 6e3d 3e7b 4d50 7970 653d 3e32 7d2c 7c0a n=>{MPype=>2},|. │ │ │ │ -0001c540: 7c20 2020 2020 2020 2020 2041 6666 5661 | AffVa │ │ │ │ -0001c550: 7269 6162 6c65 4772 6f75 703d 3e7b 7b78 riableGroup=>{{x │ │ │ │ -0001c560: 312c 7832 7d2c 7b79 7d7d 2c20 2020 2020 1,x2},{y}}, │ │ │ │ -0001c570: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001c580: 2020 2020 2020 2042 2750 6f6c 796e 6f6d B'Polynom │ │ │ │ -0001c590: 6961 6c73 3d3e 7b79 2a58 5e32 2b31 2c78 ials=>{y*X^2+1,x │ │ │ │ -0001c5a0: 312d 7832 2b31 2c79 2d32 7d2c 7c0a 7c20 1-x2+1,y-2},|.| │ │ │ │ -0001c5b0: 2020 2020 2020 2020 2020 2020 2020 2042 B │ │ │ │ -0001c5c0: 2746 756e 6374 696f 6e73 3d3e 7b20 2020 'Functions=>{ │ │ │ │ +0001c4d0: 2d2d 2d2b 0a7c 6938 203a 206d 616b 6542 ---+.|i8 : makeB │ │ │ │ +0001c4e0: 2749 6e70 7574 4669 6c65 2874 6865 4469 'InputFile(theDi │ │ │ │ +0001c4f0: 722c 2020 2020 2020 2020 2020 2020 2020 r, │ │ │ │ +0001c500: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001c510: 2020 2020 2020 2020 2020 2042 6572 7469 Berti │ │ │ │ +0001c520: 6e69 496e 7075 7443 6f6e 6669 6775 7261 niInputConfigura │ │ │ │ +0001c530: 7469 6f6e 3d3e 7b4d 5079 7065 3d3e 327d tion=>{MPype=>2} │ │ │ │ +0001c540: 2c7c 0a7c 2020 2020 2020 2020 2020 4166 ,|.| Af │ │ │ │ +0001c550: 6656 6172 6961 626c 6547 726f 7570 3d3e fVariableGroup=> │ │ │ │ +0001c560: 7b7b 7831 2c78 327d 2c7b 797d 7d2c 2020 {{x1,x2},{y}}, │ │ │ │ +0001c570: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c580: 2020 2020 2020 2020 2020 4227 506f 6c79 B'Poly │ │ │ │ +0001c590: 6e6f 6d69 616c 733d 3e7b 792a 585e 322b nomials=>{y*X^2+ │ │ │ │ +0001c5a0: 312c 7831 2d78 322b 312c 792d 327d 2c7c 1,x1-x2+1,y-2},| │ │ │ │ +0001c5b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001c5c0: 2020 4227 4675 6e63 7469 6f6e 733d 3e7b B'Functions=>{ │ │ │ │ 0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c5f0: 2020 2020 2020 2020 207b 582c 7831 2b78 {X,x1+x │ │ │ │ -0001c600: 322b 317d 7d29 3b20 2020 2020 2020 2020 2+1}}); │ │ │ │ -0001c610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001c5e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 7b58 2c78 {X,x │ │ │ │ +0001c600: 312b 7832 2b31 7d7d 293b 2020 2020 2020 1+x2+1}}); │ │ │ │ +0001c610: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0001c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c650: 2d2b 0a0a 5661 7269 6162 6c65 7320 6d75 -+..Variables mu │ │ │ │ -0001c660: 7374 2062 6567 696e 2077 6974 6820 6120 st begin with a │ │ │ │ -0001c670: 6c65 7474 6572 2028 6c6f 7765 7263 6173 letter (lowercas │ │ │ │ -0001c680: 6520 6f72 2063 6170 6974 616c 2920 616e e or capital) an │ │ │ │ -0001c690: 6420 6361 6e20 6f6e 6c79 2063 6f6e 7461 d can only conta │ │ │ │ -0001c6a0: 696e 0a6c 6574 7465 7273 2c20 6e75 6d62 in.letters, numb │ │ │ │ -0001c6b0: 6572 732c 2075 6e64 6572 7363 6f72 6573 ers, underscores │ │ │ │ -0001c6c0: 2c20 616e 6420 7371 7561 7265 2062 7261 , and square bra │ │ │ │ -0001c6d0: 636b 6574 732e 2022 6a61 6465 2220 7368 ckets. "jade" sh │ │ │ │ -0001c6e0: 6f75 6c64 206e 6f74 2062 6520 7573 6564 ould not be used │ │ │ │ -0001c6f0: 0a69 6e20 616e 7920 6578 7072 6573 7369 .in any expressi │ │ │ │ -0001c700: 6f6e 2e20 2249 2220 6361 6e20 6f6e 6c79 on. "I" can only │ │ │ │ -0001c710: 2062 6520 7573 6564 2074 6f20 7265 7072 be used to repr │ │ │ │ -0001c720: 6573 656e 7420 7468 6520 636f 6d70 6c65 esent the comple │ │ │ │ -0001c730: 7820 6e75 6d62 6572 0a73 7172 7428 2d31 x number.sqrt(-1 │ │ │ │ -0001c740: 292e 0a0a 5761 7973 2074 6f20 7573 6520 )...Ways to use │ │ │ │ -0001c750: 6d61 6b65 4227 496e 7075 7446 696c 653a makeB'InputFile: │ │ │ │ -0001c760: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0001c770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0001c780: 202a 2022 6d61 6b65 4227 496e 7075 7446 * "makeB'InputF │ │ │ │ -0001c790: 696c 6528 5374 7269 6e67 2922 0a0a 466f ile(String)"..Fo │ │ │ │ -0001c7a0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0001c7b0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0001c7c0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0001c7d0: 2a6e 6f74 6520 6d61 6b65 4227 496e 7075 *note makeB'Inpu │ │ │ │ -0001c7e0: 7446 696c 653a 206d 616b 6542 2749 6e70 tFile: makeB'Inp │ │ │ │ -0001c7f0: 7574 4669 6c65 2c20 6973 2061 202a 6e6f utFile, is a *no │ │ │ │ -0001c800: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0001c810: 6f6e 0a77 6974 6820 6f70 7469 6f6e 733a on.with options: │ │ │ │ -0001c820: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0001c830: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ -0001c840: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +0001c650: 2d2d 2d2d 2b0a 0a56 6172 6961 626c 6573 ----+..Variables │ │ │ │ +0001c660: 206d 7573 7420 6265 6769 6e20 7769 7468 must begin with │ │ │ │ +0001c670: 2061 206c 6574 7465 7220 286c 6f77 6572 a letter (lower │ │ │ │ +0001c680: 6361 7365 206f 7220 6361 7069 7461 6c29 case or capital) │ │ │ │ +0001c690: 2061 6e64 2063 616e 206f 6e6c 7920 636f and can only co │ │ │ │ +0001c6a0: 6e74 6169 6e0a 6c65 7474 6572 732c 206e ntain.letters, n │ │ │ │ +0001c6b0: 756d 6265 7273 2c20 756e 6465 7273 636f umbers, undersco │ │ │ │ +0001c6c0: 7265 732c 2061 6e64 2073 7175 6172 6520 res, and square │ │ │ │ +0001c6d0: 6272 6163 6b65 7473 2e20 226a 6164 6522 brackets. "jade" │ │ │ │ +0001c6e0: 2073 686f 756c 6420 6e6f 7420 6265 2075 should not be u │ │ │ │ +0001c6f0: 7365 640a 696e 2061 6e79 2065 7870 7265 sed.in any expre │ │ │ │ +0001c700: 7373 696f 6e2e 2022 4922 2063 616e 206f ssion. "I" can o │ │ │ │ +0001c710: 6e6c 7920 6265 2075 7365 6420 746f 2072 nly be used to r │ │ │ │ +0001c720: 6570 7265 7365 6e74 2074 6865 2063 6f6d epresent the com │ │ │ │ +0001c730: 706c 6578 206e 756d 6265 720a 7371 7274 plex number.sqrt │ │ │ │ +0001c740: 282d 3129 2e0a 0a57 6179 7320 746f 2075 (-1)...Ways to u │ │ │ │ +0001c750: 7365 206d 616b 6542 2749 6e70 7574 4669 se makeB'InputFi │ │ │ │ +0001c760: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ +0001c770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001c780: 0a0a 2020 2a20 226d 616b 6542 2749 6e70 .. * "makeB'Inp │ │ │ │ +0001c790: 7574 4669 6c65 2853 7472 696e 6729 220a utFile(String)". │ │ │ │ +0001c7a0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0001c7b0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0001c7c0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0001c7d0: 6374 202a 6e6f 7465 206d 616b 6542 2749 ct *note makeB'I │ │ │ │ +0001c7e0: 6e70 7574 4669 6c65 3a20 6d61 6b65 4227 nputFile: makeB' │ │ │ │ +0001c7f0: 496e 7075 7446 696c 652c 2069 7320 6120 InputFile, is a │ │ │ │ +0001c800: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ +0001c810: 6374 696f 6e0a 7769 7468 206f 7074 696f ction.with optio │ │ │ │ +0001c820: 6e73 3a20 284d 6163 6175 6c61 7932 446f ns: (Macaulay2Do │ │ │ │ +0001c830: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0001c840: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ 0001c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0001c8a0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0001c8b0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0001c8c0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0001c8d0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0001c8e0: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -0001c8f0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0001c900: 6573 2f42 6572 7469 6e69 2e6d 323a 0a33 es/Bertini.m2:.3 │ │ │ │ -0001c910: 3038 303a 302e 0a1f 0a46 696c 653a 2042 080:0....File: B │ │ │ │ -0001c920: 6572 7469 6e69 2e69 6e66 6f2c 204e 6f64 ertini.info, Nod │ │ │ │ -0001c930: 653a 206d 616b 6542 2753 6563 7469 6f6e e: makeB'Section │ │ │ │ -0001c940: 2c20 4e65 7874 3a20 6d61 6b65 4227 536c , Next: makeB'Sl │ │ │ │ -0001c950: 6963 652c 2050 7265 763a 206d 616b 6542 ice, Prev: makeB │ │ │ │ -0001c960: 2749 6e70 7574 4669 6c65 2c20 5570 3a20 'InputFile, Up: │ │ │ │ -0001c970: 546f 700a 0a6d 616b 6542 2753 6563 7469 Top..makeB'Secti │ │ │ │ -0001c980: 6f6e 202d 2d20 6d61 6b65 4227 5365 6374 on -- makeB'Sect │ │ │ │ -0001c990: 696f 6e20 6372 6561 7465 7320 6120 6861 ion creates a ha │ │ │ │ -0001c9a0: 7368 2074 6162 6c65 2074 6861 7420 7265 sh table that re │ │ │ │ -0001c9b0: 7072 6573 656e 7473 2061 2068 7970 6572 presents a hyper │ │ │ │ -0001c9c0: 706c 616e 652e 0a2a 2a2a 2a2a 2a2a 2a2a plane..********* │ │ │ │ +0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0001c8a0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0001c8b0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0001c8c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0001c8d0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0001c8e0: 6c61 7932 2d31 2e32 352e 3036 2b64 732f lay2-1.25.06+ds/ │ │ │ │ +0001c8f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0001c900: 6b61 6765 732f 4265 7274 696e 692e 6d32 kages/Bertini.m2 │ │ │ │ +0001c910: 3a0a 3330 3830 3a30 2e0a 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**************** │ │ │ │ 0001c9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001ca00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001ca10: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -0001ca20: 6167 653a 200a 2020 2020 2020 2020 6d61 age: . ma │ │ │ │ -0001ca30: 6b65 4227 5365 6374 696f 6e28 6c29 0a20 keB'Section(l). │ │ │ │ -0001ca40: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -0001ca50: 202a 206c 2c20 6120 2a6e 6f74 6520 6c69 * l, a *note li │ │ │ │ -0001ca60: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -0001ca70: 6329 4c69 7374 2c2c 2041 206c 6973 7420 c)List,, A list │ │ │ │ -0001ca80: 6f66 2076 6172 6961 626c 6573 2e0a 2020 of variables.. │ │ │ │ -0001ca90: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c * *note Optional │ │ │ │ -0001caa0: 2069 6e70 7574 733a 2028 4d61 6361 756c inputs: (Macaul │ │ │ │ -0001cab0: 6179 3244 6f63 2975 7369 6e67 2066 756e ay2Doc)using fun │ │ │ │ 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7661 6c75 6520 , default value │ │ │ │ -0001cba0: 6e75 6c6c 0a20 2020 2020 202a 2052 616e null. * Ran │ │ │ │ -0001cbb0: 646f 6d43 6f65 6666 6963 6965 6e74 4765 domCoefficientGe │ │ │ │ -0001cbc0: 6e65 7261 746f 7220 3d3e 202e 2e2e 2c20 nerator => ..., │ │ │ │ -0001cbd0: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ -0001cbe0: 2020 2020 2020 4675 6e63 7469 6f6e 436c FunctionCl │ │ │ │ -0001cbf0: 6f73 7572 655b 2e2e 2f42 6572 7469 6e69 osure[../Bertini │ │ │ │ -0001cc00: 2e6d 323a 3233 3134 3a33 372d 3233 3134 .m2:2314:37-2314 │ │ │ │ -0001cc10: 3a36 365d 0a0a 4465 7363 7269 7074 696f :66]..Descriptio │ │ │ │ -0001cc20: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a6d n.===========..m │ │ │ │ -0001cc30: 616b 6542 2753 6563 7469 6f6e 2061 6c6c akeB'Section all │ │ │ │ -0001cc40: 6f77 7320 666f 7220 6561 7379 2063 7265 ows for easy cre │ │ │ │ -0001cc50: 6174 696f 6e20 6f66 2065 7175 6174 696f ation of equatio │ │ │ │ -0001cc60: 6e73 2074 6861 7420 6465 6669 6e65 2068 ns that define h │ │ │ │ -0001cc70: 7970 6572 706c 616e 6573 2e0a 5468 6520 yperplanes..The │ │ │ │ -0001cc80: 6465 6661 756c 7420 6372 6561 7465 7320 default creates │ │ │ │ -0001cc90: 6120 6861 7368 2074 6162 6c65 2077 6974 a hash table wit │ │ │ │ -0001cca0: 6820 7477 6f20 6b65 7973 3a20 4227 4e75 h two keys: B'Nu │ │ │ │ -0001ccb0: 6d62 6572 436f 6566 6669 6369 656e 7473 mberCoefficients │ │ │ │ -0001ccc0: 2061 6e64 0a42 2753 6563 7469 6f6e 5374 and.B'SectionSt │ │ │ │ -0001ccd0: 7269 6e67 2e20 5468 6520 6669 7273 7420 ring. 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The opt │ │ │ │ -0001cd60: 696f 6e20 5261 6e64 6f6d 436f 6566 6669 ion RandomCoeffi │ │ │ │ -0001cd70: 6369 656e 7447 656e 6572 6174 6f72 2063 cientGenerator c │ │ │ │ -0001cd80: 616e 2062 6520 7365 7420 746f 2061 2066 an be set to a f │ │ │ │ -0001cd90: 756e 6374 696f 6e20 746f 0a67 656e 6572 unction to.gener │ │ │ │ -0001cda0: 6174 6520 7261 6e64 6f6d 206e 756d 6265 ate random numbe │ │ │ │ -0001cdb0: 7273 2066 6f72 2074 6865 2063 6f65 6666 rs for the coeff │ │ │ │ -0001cdc0: 6963 6965 6e74 732e 0a0a 546f 2067 6574 icients...To get │ │ │ │ -0001cdd0: 2061 6666 696e 6520 6c69 6e65 6172 2065 affine linear e │ │ │ │ -0001cde0: 7175 6174 696f 6e73 2069 6e63 6c75 6465 quations include │ │ │ │ -0001cdf0: 2031 2069 6e20 7468 6520 696e 7075 7420 1 in the input │ │ │ │ -0001ce00: 6c69 7374 2e0a 0a54 6f20 6861 7665 2061 list...To have a │ │ │ │ -0001ce10: 6e20 6166 6669 6e65 206c 696e 6561 7220 n affine linear │ │ │ │ -0001ce20: 6571 7561 7469 6f6e 2074 6861 7420 636f equation that co │ │ │ │ -0001ce30: 6e74 6169 6e73 2061 2070 6172 7469 6375 ntains a particu │ │ │ │ -0001ce40: 6c61 7220 706f 696e 7420 7765 2073 6574 lar point we set │ │ │ │ -0001ce50: 2074 6865 0a43 6f6e 7461 696e 7350 6f69 the.ContainsPoi │ │ │ │ -0001ce60: 6e74 206f 7074 696f 6e20 746f 2061 206c nt option to a l │ │ │ │ -0001ce70: 6973 7420 6f66 2063 6f6f 7264 696e 6174 ist of coordinat │ │ │ │ -0001ce80: 6573 206f 7220 6120 706f 696e 742e 2054 es or a point. T │ │ │ │ -0001ce90: 6f20 6765 7420 616e 2068 6f6d 6f67 656e o get an homogen │ │ │ │ -0001cea0: 656f 7573 0a65 7175 6174 696f 6e20 7468 eous.equation th │ │ │ │ -0001ceb0: 6174 2063 6f6e 7461 696e 7320 6120 7072 at contains a pr │ │ │ │ -0001cec0: 6f6a 6563 7469 7665 2070 6f69 6e74 2077 ojective point w │ │ │ │ -0001ced0: 6520 6861 7665 2074 6f20 7365 7420 7468 e have to set th │ │ │ │ -0001cee0: 6520 436f 6e74 6169 6e73 506f 696e 740a e ContainsPoint. │ │ │ │ -0001cef0: 6f70 7469 6f6e 2061 7320 7765 6c6c 2061 option as well a │ │ │ │ -0001cf00: 7320 7468 6520 4227 486f 6d6f 6765 6e69 s the B'Homogeni │ │ │ │ -0001cf10: 7a61 7469 6f6e 206f 7074 696f 6e2e 0a0a zation option... │ │ │ │ -0001cf20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001ca10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0001ca20: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0001ca30: 206d 616b 6542 2753 6563 7469 6f6e 286c makeB'Section(l │ │ │ │ +0001ca40: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0001ca50: 2020 2020 2a20 6c2c 2061 202a 6e6f 7465 * l, a *note │ │ │ │ +0001ca60: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ +0001ca70: 3244 6f63 294c 6973 742c 2c20 4120 6c69 2Doc)List,, A li │ │ │ │ +0001ca80: 7374 206f 6620 7661 7269 6162 6c65 732e st of variables. │ │ │ │ +0001ca90: 0a20 202a 202a 6e6f 7465 204f 7074 696f . * *note Optio │ │ │ │ +0001caa0: 6e61 6c20 696e 7075 7473 3a20 284d 6163 nal inputs: (Mac │ │ │ │ +0001cab0: 6175 6c61 7932 446f 6329 7573 696e 6720 aulay2Doc)using │ │ │ │ +0001cac0: 6675 6e63 7469 6f6e 7320 7769 7468 206f functions with o │ │ │ │ +0001cad0: 7074 696f 6e61 6c20 696e 7075 7473 2c3a ptional inputs,: │ │ │ │ +0001cae0: 0a20 2020 2020 202a 2042 2748 6f6d 6f67 . * B'Homog │ │ │ │ +0001caf0: 656e 697a 6174 696f 6e20 3d3e 202e 2e2e enization => ... │ │ │ │ +0001cb00: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +0001cb10: 310a 2020 2020 2020 2a20 4227 4e75 6d62 1. * B'Numb │ │ │ │ +0001cb20: 6572 436f 6566 6669 6369 656e 7473 203d erCoefficients = │ │ │ │ +0001cb30: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +0001cb40: 616c 7565 207b 7d0a 2020 2020 2020 2a20 alue {}. * │ │ │ │ +0001cb50: 436f 6e74 6169 6e73 506f 696e 7420 3d3e ContainsPoint => │ │ │ │ +0001cb60: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +0001cb70: 6c75 6520 7b7d 0a20 2020 2020 202a 204e lue {}. * N │ │ │ │ +0001cb80: 616d 6542 2753 6563 7469 6f6e 203d 3e20 ameB'Section => │ │ │ │ +0001cb90: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +0001cba0: 7565 206e 756c 6c0a 2020 2020 2020 2a20 ue null. * │ │ │ │ +0001cbb0: 5261 6e64 6f6d 436f 6566 6669 6369 656e RandomCoefficien │ │ │ │ +0001cbc0: 7447 656e 6572 6174 6f72 203d 3e20 2e2e tGenerator => .. │ │ │ │ +0001cbd0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +0001cbe0: 0a20 2020 2020 2020 2046 756e 6374 696f . Functio │ │ │ │ +0001cbf0: 6e43 6c6f 7375 7265 5b2e 2e2f 4265 7274 nClosure[../Bert │ │ │ │ +0001cc00: 696e 692e 6d32 3a32 3331 343a 3337 2d32 ini.m2:2314:37-2 │ │ │ │ +0001cc10: 3331 343a 3636 5d0a 0a44 6573 6372 6970 314:66]..Descrip │ │ │ │ +0001cc20: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0001cc30: 0a0a 6d61 6b65 4227 5365 6374 696f 6e20 ..makeB'Section │ │ │ │ +0001cc40: 616c 6c6f 7773 2066 6f72 2065 6173 7920 allows for easy │ │ │ │ +0001cc50: 6372 6561 7469 6f6e 206f 6620 6571 7561 creation of equa │ │ │ │ +0001cc60: 7469 6f6e 7320 7468 6174 2064 6566 696e tions that defin │ │ │ │ +0001cc70: 6520 6879 7065 7270 6c61 6e65 732e 0a54 e hyperplanes..T │ │ │ │ +0001cc80: 6865 2064 6566 6175 6c74 2063 7265 6174 he default creat │ │ │ │ +0001cc90: 6573 2061 2068 6173 6820 7461 626c 6520 es a hash table │ │ │ │ +0001cca0: 7769 7468 2074 776f 206b 6579 733a 2042 with two keys: B │ │ │ │ +0001ccb0: 274e 756d 6265 7243 6f65 6666 6963 6965 'NumberCoefficie │ │ │ │ +0001ccc0: 6e74 7320 616e 640a 4227 5365 6374 696f nts and.B'Sectio │ │ │ │ +0001ccd0: 6e53 7472 696e 672e 2054 6865 2066 6972 nString. 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│ │ -0001cf70: 7c69 3120 3a20 733d 6d61 6b65 4227 5365 |i1 : s=makeB'Se │ │ │ │ -0001cf80: 6374 696f 6e28 7b78 2c79 2c7a 7d29 2020 ction({x,y,z}) │ │ │ │ -0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf70: 2d2b 0a7c 6931 203a 2073 3d6d 616b 6542 -+.|i1 : s=makeB │ │ │ │ +0001cf80: 2753 6563 7469 6f6e 287b 782c 792c 7a7d 'Section({x,y,z} │ │ │ │ +0001cf90: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0001cfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cfb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001cfc0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cfc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d010: 7c6f 3120 3d20 4227 5365 6374 696f 6e7b |o1 = B'Section{ │ │ │ │ -0001d020: 2e2e 2e32 2e2e 2e7d 2020 2020 2020 2020 ...2...} │ │ │ │ +0001d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d010: 207c 0a7c 6f31 203d 2042 2753 6563 7469 |.|o1 = B'Secti │ │ │ │ +0001d020: 6f6e 7b2e 2e2e 322e 2e2e 7d20 2020 2020 on{...2...} │ │ │ │ 0001d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d060: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d060: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d0b0: 7c6f 3120 3a20 4227 5365 6374 696f 6e20 |o1 : B'Section │ │ │ │ -0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d0b0: 207c 0a7c 6f31 203a 2042 2753 6563 7469 |.|o1 : B'Secti │ │ │ │ +0001d0c0: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ 0001d0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d100: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d100: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001d110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001d150: 7c69 3220 3a20 636c 6173 7320 7320 2020 |i2 : class s │ │ │ │ +0001d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d150: 2d2b 0a7c 6932 203a 2063 6c61 7373 2073 -+.|i2 : class s │ │ │ │ 0001d160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d190: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d1a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d1a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d1f0: 7c6f 3220 3d20 4227 5365 6374 696f 6e20 |o2 = B'Section │ │ │ │ -0001d200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d1f0: 207c 0a7c 6f32 203d 2042 2753 6563 7469 |.|o2 = B'Secti │ │ │ │ +0001d200: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ 0001d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d240: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d240: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d290: 7c6f 3220 3a20 5479 7065 2020 2020 2020 |o2 : Type │ │ │ │ +0001d280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d290: 207c 0a7c 6f32 203a 2054 7970 6520 2020 |.|o2 : Type │ │ │ │ 0001d2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d2e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d2e0: 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001d330: 7c69 3320 3a20 7261 6e64 6f6d 5265 616c |i3 : randomReal │ │ │ │ -0001d340: 436f 6566 6669 6369 656e 7447 656e 6572 CoefficientGener │ │ │ │ -0001d350: 6174 6f72 3d28 292d 3e72 616e 646f 6d28 ator=()->random( │ │ │ │ -0001d360: 5252 2920 2020 2020 2020 2020 2020 2020 RR) │ │ │ │ -0001d370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d380: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d330: 2d2b 0a7c 6933 203a 2072 616e 646f 6d52 -+.|i3 : randomR │ │ │ │ +0001d340: 6561 6c43 6f65 6666 6963 6965 6e74 4765 ealCoefficientGe │ │ │ │ +0001d350: 6e65 7261 746f 723d 2829 2d3e 7261 6e64 nerator=()->rand │ │ │ │ +0001d360: 6f6d 2852 5229 2020 2020 2020 2020 2020 om(RR) │ │ │ │ +0001d370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d380: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d3d0: 7c6f 3320 3d20 7261 6e64 6f6d 5265 616c |o3 = randomReal │ │ │ │ -0001d3e0: 436f 6566 6669 6369 656e 7447 656e 6572 CoefficientGener │ │ │ │ -0001d3f0: 6174 6f72 2020 2020 2020 2020 2020 2020 ator │ │ │ │ +0001d3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d3d0: 207c 0a7c 6f33 203d 2072 616e 646f 6d52 |.|o3 = randomR │ │ │ │ +0001d3e0: 6561 6c43 6f65 6666 6963 6965 6e74 4765 ealCoefficientGe │ │ │ │ +0001d3f0: 6e65 7261 746f 7220 2020 2020 2020 2020 nerator │ │ │ │ 0001d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d410: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d420: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d420: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d460: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d470: 7c6f 3320 3a20 4675 6e63 7469 6f6e 436c |o3 : FunctionCl │ │ │ │ -0001d480: 6f73 7572 6520 2020 2020 2020 2020 2020 osure │ │ │ │ +0001d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d470: 207c 0a7c 6f33 203a 2046 756e 6374 696f |.|o3 : Functio │ │ │ │ +0001d480: 6e43 6c6f 7375 7265 2020 2020 2020 2020 nClosure │ │ │ │ 0001d490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d4c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001d4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d4c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001d4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001d510: 7c69 3420 3a20 7352 6561 6c3d 6d61 6b65 |i4 : sReal=make │ │ │ │ -0001d520: 4227 5365 6374 696f 6e28 7b78 2c79 2c7a B'Section({x,y,z │ │ │ │ -0001d530: 7d2c 5261 6e64 6f6d 436f 6566 6669 6369 },RandomCoeffici │ │ │ │ -0001d540: 656e 7447 656e 6572 6174 6f72 3d3e 2020 entGenerator=> │ │ │ │ -0001d550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d560: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d510: 2d2b 0a7c 6934 203a 2073 5265 616c 3d6d -+.|i4 : sReal=m │ │ │ │ +0001d520: 616b 6542 2753 6563 7469 6f6e 287b 782c akeB'Section({x, │ │ │ │ +0001d530: 792c 7a7d 2c52 616e 646f 6d43 6f65 6666 y,z},RandomCoeff │ │ │ │ +0001d540: 6963 6965 6e74 4765 6e65 7261 746f 723d icientGenerator= │ │ │ │ +0001d550: 3e20 2020 2020 2020 2020 2020 2020 2020 > │ │ │ │ +0001d560: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d5a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d5b0: 7c6f 3420 3d20 4227 5365 6374 696f 6e7b |o4 = B'Section{ │ │ │ │ -0001d5c0: 2e2e 2e32 2e2e 2e7d 2020 2020 2020 2020 ...2...} │ │ │ │ +0001d5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d5b0: 207c 0a7c 6f34 203d 2042 2753 6563 7469 |.|o4 = B'Secti │ │ │ │ +0001d5c0: 6f6e 7b2e 2e2e 322e 2e2e 7d20 2020 2020 on{...2...} │ │ │ │ 0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d5f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d600: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d600: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d640: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d650: 7c6f 3420 3a20 4227 5365 6374 696f 6e20 |o4 : B'Section │ │ │ │ -0001d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d650: 207c 0a7c 6f34 203a 2042 2753 6563 7469 |.|o4 : B'Secti │ │ │ │ +0001d660: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ 0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d6a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +0001d690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 0001d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0001d6f0: 7c72 616e 646f 6d52 6561 6c43 6f65 6666 |randomRealCoeff │ │ │ │ -0001d700: 6963 6965 6e74 4765 6e65 7261 746f 7229 icientGenerator) │ │ │ │ -0001d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d6f0: 2d7c 0a7c 7261 6e64 6f6d 5265 616c 436f -|.|randomRealCo │ │ │ │ +0001d700: 6566 6669 6369 656e 7447 656e 6572 6174 efficientGenerat │ │ │ │ +0001d710: 6f72 2920 2020 2020 2020 2020 2020 2020 or) │ │ │ │ 0001d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d740: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001d730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d740: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001d790: 7c69 3520 3a20 7352 6561 6c23 4227 4e75 |i5 : sReal#B'Nu │ │ │ │ -0001d7a0: 6d62 6572 436f 6566 6669 6369 656e 7473 mberCoefficients │ │ │ │ -0001d7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d790: 2d2b 0a7c 6935 203a 2073 5265 616c 2342 -+.|i5 : sReal#B │ │ │ │ +0001d7a0: 274e 756d 6265 7243 6f65 6666 6963 6965 'NumberCoefficie │ │ │ │ +0001d7b0: 6e74 7320 2020 2020 2020 2020 2020 2020 nts │ │ │ │ 0001d7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d7d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d7e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d820: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d830: 7c6f 3520 3d20 7b2e 3037 3431 3833 352c |o5 = {.0741835, │ │ │ │ -0001d840: 202e 3830 3836 3934 2c20 2e33 3632 3833 .808694, .36283 │ │ │ │ -0001d850: 357d 2020 2020 2020 2020 2020 2020 2020 5} │ │ │ │ +0001d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d830: 207c 0a7c 6f35 203d 207b 2e30 3734 3138 |.|o5 = {.07418 │ │ │ │ +0001d840: 3335 2c20 2e38 3038 3639 342c 202e 3336 35, .808694, .36 │ │ │ │ +0001d850: 3238 3335 7d20 2020 2020 2020 2020 2020 2835} │ │ │ │ 0001d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d880: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d880: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d8d0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d8d0: 207c 0a7c 6f35 203a 204c 6973 7420 2020 |.|o5 : List │ │ │ │ 0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d920: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001d970: 7c69 3620 3a20 7261 6e64 6f6d 5261 7469 |i6 : randomRati │ │ │ │ -0001d980: 6f6e 616c 436f 6566 6669 6369 656e 7447 onalCoefficientG │ │ │ │ -0001d990: 656e 6572 6174 6f72 3d28 292d 3e72 616e enerator=()->ran │ │ │ │ -0001d9a0: 646f 6d28 5151 2920 2020 2020 2020 2020 dom(QQ) │ │ │ │ -0001d9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d9c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d970: 2d2b 0a7c 6936 203a 2072 616e 646f 6d52 -+.|i6 : randomR │ │ │ │ +0001d980: 6174 696f 6e61 6c43 6f65 6666 6963 6965 ationalCoefficie │ │ │ │ +0001d990: 6e74 4765 6e65 7261 746f 723d 2829 2d3e ntGenerator=()-> │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001da60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001daa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001dab0: 7c6f 3620 3a20 4675 6e63 7469 6f6e 436c |o6 : FunctionCl │ │ │ │ -0001dac0: 6f73 7572 6520 2020 2020 2020 2020 2020 osure │ │ │ │ +0001daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dab0: 207c 0a7c 6f36 203a 2046 756e 6374 696f |.|o6 : Functio │ │ │ │ +0001dac0: 6e43 6c6f 7375 7265 2020 2020 2020 2020 nClosure │ │ │ │ 0001dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001daf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 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---------------- │ │ │ │ +0001db50: 2d2b 0a7c 6937 203a 2073 5261 7469 6f6e -+.|i7 : sRation │ │ │ │ +0001db60: 616c 3d6d 616b 6542 2753 6563 7469 6f6e al=makeB'Section │ │ │ │ +0001db70: 287b 782c 792c 7a7d 2c52 616e 646f 6d43 ({x,y,z},RandomC │ │ │ │ +0001db80: 6f65 6666 6963 6965 6e74 4765 6e65 7261 oefficientGenera │ │ │ │ +0001db90: 746f 723d 3e20 2020 2020 2020 2020 2020 tor=> │ │ │ │ +0001dba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001dbe0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001dbf0: 7c6f 3720 3d20 4227 5365 6374 696f 6e7b |o7 = B'Section{ │ │ │ │ -0001dc00: 2e2e 2e32 2e2e 2e7d 2020 2020 2020 2020 ...2...} │ │ │ │ +0001dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dbf0: 207c 0a7c 6f37 203d 2042 2753 6563 7469 |.|o7 = B'Secti │ │ │ │ +0001dc00: 6f6e 7b2e 2e2e 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2020 2020 2020 2020 2020 │ │ │ │ 0001dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001dcd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001dce0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +0001dcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dce0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 0001dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0001dd30: 7c72 616e 646f 6d52 6174 696f 6e61 6c43 |randomRationalC │ │ │ │ -0001dd40: 6f65 6666 6963 6965 6e74 4765 6e65 7261 oefficientGenera │ │ │ │ -0001dd50: 746f 7229 2020 2020 2020 2020 2020 2020 tor) │ │ │ │ +0001dd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dd30: 2d7c 0a7c 7261 6e64 6f6d 5261 7469 6f6e -|.|randomRation │ │ │ │ +0001dd40: 616c 436f 6566 6669 6369 656e 7447 656e alCoefficientGen │ │ │ │ +0001dd50: 6572 6174 6f72 2920 2020 2020 2020 2020 erator) │ │ │ │ 0001dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001dd70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001dd80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dd80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ddd0: 7c69 3820 3a20 7352 6174 696f 6e61 6c23 |i8 : sRational# │ │ │ │ -0001dde0: 4227 4e75 6d62 6572 436f 6566 6669 6369 B'NumberCoeffici │ │ │ │ -0001ddf0: 656e 7473 2020 2020 2020 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|.| │ │ │ │ +0001e230: 2020 207c 0a7c 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 2020 │ │ │ │ 0001e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e280: 7c0a 7c6f 3130 203d 207b 782c 2079 2c20 |.|o10 = {x, y, │ │ │ │ -0001e290: 7a7d 2020 2020 2020 2020 2020 2020 2020 z} │ │ │ │ +0001e280: 2020 207c 0a7c 6f31 3020 3d20 7b78 2c20 |.|o10 = {x, │ │ │ │ +0001e290: 792c 207a 7d20 2020 2020 2020 2020 2020 y, z} │ │ │ │ 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: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e2d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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|.+------------- │ │ │ │ +0001e550: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0001e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e5a0: 2b0a 7c69 3132 203a 2061 6666 696e 6543 +.|i12 : affineC │ │ │ │ -0001e5b0: 6f6e 7461 696e 696e 6750 6f69 6e74 3d6d ontainingPoint=m │ │ │ │ -0001e5c0: 616b 6542 2753 6563 7469 6f6e 287b 782c akeB'Section({x, │ │ │ │ -0001e5d0: 792c 7a7d 2c43 6f6e 7461 696e 7350 6f69 y,z},ContainsPoi │ │ │ │ -0001e5e0: 6e74 3d3e 5029 2020 2020 2020 2020 2020 nt=>P) │ │ │ │ -0001e5f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e5a0: 2d2d 2d2b 0a7c 6931 3220 3a20 6166 6669 ---+.|i12 : affi │ │ │ │ +0001e5b0: 6e65 436f 6e74 6169 6e69 6e67 506f 696e neContainingPoin │ │ │ │ +0001e5c0: 743d 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2020 2020 2020 2020 |.| │ │ │ │ +0001e690: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e6e0: 7c0a 7c6f 3132 203a 2042 2753 6563 7469 |.|o12 : B'Secti │ │ │ │ -0001e6f0: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ +0001e6e0: 2020 207c 0a7c 6f31 3220 3a20 4227 5365 |.|o12 : B'Se │ │ │ │ +0001e6f0: 6374 696f 6e20 2020 2020 2020 2020 2020 ction │ │ │ │ 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: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001e730: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0001e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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849539*ii)*(x-(1 │ │ │ │ +0001e980: 292a 2831 2929 2b28 2d2e 3534 3233 3731 )*(1))+(-.542371 │ │ │ │ +0001e990: 2b2e 3330 3731 3337 2a69 6929 2a28 792d +.307137*ii)*(y- │ │ │ │ +0001e9a0: 2831 292a 2832 2929 2b28 312e 3336 3934 (1)*(2))+(1.3694 │ │ │ │ +0001e9b0: 3520 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 5 |.|---------- │ │ │ │ 0001e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea00: 7c0a 7c2b 2e30 3135 3633 332a 6969 292a |.|+.015633*ii)* │ │ │ │ -0001ea10: 287a 2d28 3129 2a28 3329 2920 2020 2020 (z-(1)*(3)) │ │ │ │ +0001ea00: 2d2d 2d7c 0a7c 2b2e 3031 3536 3333 2a69 ---|.|+.015633*i │ │ │ │ +0001ea10: 6929 2a28 7a2d 2831 292a 2833 2929 2020 i)*(z-(1)*(3)) │ │ │ │ 0001ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea30: 2020 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│ │ │ 0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb90: 7c0a 7c6f 3135 203d 2042 2753 6563 7469 |.|o15 = B'Secti │ │ │ │ -0001eba0: 6f6e 7b2e 2e2e 332e 2e2e 7d20 2020 2020 on{...3...} │ │ │ │ +0001eb90: 2020 207c 0a7c 6f31 3520 3d20 4227 5365 |.|o15 = B'Se │ │ │ │ +0001eba0: 6374 696f 6e7b 2e2e 2e33 2e2e 2e7d 2020 ction{...3...} │ │ │ │ 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 2020 │ │ │ │ -0001ebe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ebe0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec30: 7c0a 7c6f 3135 203a 2042 2753 6563 7469 |.|o15 : B'Secti │ │ │ │ -0001ec40: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ +0001ec30: 2020 207c 0a7c 6f31 3520 3a20 4227 5365 |.|o15 : B'Se │ │ │ │ +0001ec40: 6374 696f 6e20 2020 2020 2020 2020 2020 ction │ │ │ │ 0001ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec80: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ +0001ec80: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ecd0: 7c0a 7c3d 3e22 782b 792b 7a22 2920 2020 |.|=>"x+y+z") │ │ │ │ +0001ecd0: 2d2d 2d7c 0a7c 3d3e 2278 2b79 2b7a 2229 ---|.|=>"x+y+z") │ │ │ │ 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2020 2020 │ │ │ │ 0001ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001ee70: 2020 2042 274e 756d 6265 7243 6f65 6666 B'NumberCoeff │ │ │ │ -0001ee80: 6963 6965 6e74 7320 3d3e 207b 2e35 3334 icients => {.534 │ │ │ │ -0001ee90: 3631 342d 2e31 3735 3934 352a 6969 2c20 614-.175945*ii, │ │ │ │ -0001eea0: 2e34 3236 3730 3420 2020 2020 2020 2020 .426704 │ │ │ │ -0001eeb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001eec0: 2020 2042 2753 6563 7469 6f6e 5374 7269 B'SectionStri │ │ │ │ -0001eed0: 6e67 203d 3e20 282e 3533 3436 3134 2d2e ng => (.534614-. │ │ │ │ -0001eee0: 3137 3539 3435 2a69 6929 2a28 782d 2878 175945*ii)*(x-(x │ │ │ │ -0001eef0: 2b79 2b7a 292a 2820 2020 2020 2020 2020 +y+z)*( │ │ │ │ -0001ef00: 7c0a 7c20 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +0001ee60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ee70: 2020 2020 2020 4227 4e75 6d62 6572 436f B'NumberCo │ │ │ │ +0001ee80: 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4227 ys to use makeB' │ │ │ │ -0001f750: 5365 6374 696f 6e3a 0a3d 3d3d 3d3d 3d3d Section:.======= │ │ │ │ +0001f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001f740: 0a57 6179 7320 746f 2075 7365 206d 616b .Ways to use mak │ │ │ │ +0001f750: 6542 2753 6563 7469 6f6e 3a0a 3d3d 3d3d eB'Section:.==== │ │ │ │ 0001f760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001f770: 3d3d 3d0a 0a20 202a 2022 6d61 6b65 4227 ===.. * "makeB' │ │ │ │ -0001f780: 5365 6374 696f 6e28 4c69 7374 2922 0a0a Section(List)".. │ │ │ │ -0001f790: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0001f7a0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0001f7b0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0001f7c0: 7420 2a6e 6f74 6520 6d61 6b65 4227 5365 t *note makeB'Se │ │ │ │ -0001f7d0: 6374 696f 6e3a 206d 616b 6542 2753 6563 ction: makeB'Sec │ │ │ │ -0001f7e0: 7469 6f6e 2c20 6973 2061 202a 6e6f 7465 tion, is a *note │ │ │ │ -0001f7f0: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -0001f800: 2077 6974 680a 6f70 7469 6f6e 733a 2028 with.options: ( │ │ │ │ -0001f810: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0001f820: 686f 6446 756e 6374 696f 6e57 6974 684f hodFunctionWithO │ │ │ │ -0001f830: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d ptions,...------ │ │ │ │ +0001f770: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ +0001f780: 6542 2753 6563 7469 6f6e 284c 6973 7429 eB'Section(List) │ │ │ │ +0001f790: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0001f7a0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0001f7b0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0001f7c0: 6a65 6374 202a 6e6f 7465 206d 616b 6542 ject *note makeB │ │ │ │ +0001f7d0: 2753 6563 7469 6f6e 3a20 6d61 6b65 4227 'Section: makeB' │ │ │ │ +0001f7e0: 5365 6374 696f 6e2c 2069 7320 6120 2a6e Section, is a *n │ │ │ │ +0001f7f0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +0001f800: 696f 6e20 7769 7468 0a6f 7074 696f 6e73 ion with.options │ │ │ │ +0001f810: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0001f820: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +0001f830: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ 0001f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f880: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0001f890: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0001f8a0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0001f8b0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0001f8c0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0001f8d0: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ -0001f8e0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0001f8f0: 2f42 6572 7469 6e69 2e6d 323a 0a33 3631 /Bertini.m2:.361 │ │ │ │ -0001f900: 313a 302e 0a1f 0a46 696c 653a 2042 6572 1:0....File: Ber │ │ │ │ -0001f910: 7469 6e69 2e69 6e66 6f2c 204e 6f64 653a tini.info, Node: │ │ │ │ -0001f920: 206d 616b 6542 2753 6c69 6365 2c20 4e65 makeB'Slice, Ne │ │ │ │ -0001f930: 7874 3a20 6d6f 7665 4227 4669 6c65 2c20 xt: moveB'File, │ │ │ │ -0001f940: 5072 6576 3a20 6d61 6b65 4227 5365 6374 Prev: makeB'Sect │ │ │ │ -0001f950: 696f 6e2c 2055 703a 2054 6f70 0a0a 6d61 ion, Up: Top..ma │ │ │ │ -0001f960: 6b65 4227 536c 6963 6520 2d2d 206d 616b keB'Slice -- mak │ │ │ │ -0001f970: 6542 2753 6c69 6365 2063 7265 6174 6573 eB'Slice creates │ │ │ │ -0001f980: 2061 2068 6173 6820 7461 626c 6520 7468 a hash table th │ │ │ │ -0001f990: 6174 2072 6570 7265 7365 6e74 7320 6120 at represents a │ │ │ │ -0001f9a0: 6c69 6e65 6172 2073 6c69 6365 2e0a 2a2a linear slice..** │ │ │ │ -0001f9b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +0001f890: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +0001f8a0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +0001f8b0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +0001f8c0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +0001f8d0: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ +0001f8e0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +0001f8f0: 6765 732f 4265 7274 696e 692e 6d32 3a0a ges/Bertini.m2:. │ │ │ │ +0001f900: 3336 3131 3a30 2e0a 1f0a 4669 6c65 3a20 3611:0....File: │ │ │ │ +0001f910: 4265 7274 696e 692e 696e 666f 2c20 4e6f Bertini.info, No │ │ │ │ +0001f920: 6465 3a20 6d61 6b65 4227 536c 6963 652c de: makeB'Slice, │ │ │ │ +0001f930: 204e 6578 743a 206d 6f76 6542 2746 696c Next: moveB'Fil │ │ │ │ +0001f940: 652c 2050 7265 763a 206d 616b 6542 2753 e, Prev: makeB'S │ │ │ │ +0001f950: 6563 7469 6f6e 2c20 5570 3a20 546f 700a ection, Up: Top. │ │ │ │ +0001f960: 0a6d 616b 6542 2753 6c69 6365 202d 2d20 .makeB'Slice -- │ │ │ │ +0001f970: 6d61 6b65 4227 536c 6963 6520 6372 6561 makeB'Slice crea │ │ │ │ +0001f980: 7465 7320 6120 6861 7368 2074 6162 6c65 tes a hash table │ │ │ │ +0001f990: 2074 6861 7420 7265 7072 6573 656e 7473 that represents │ │ │ │ +0001f9a0: 2061 206c 696e 6561 7220 736c 6963 652e a linear slice. │ │ │ │ +0001f9b0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0001f9c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001f9d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001f9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001f9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -0001fa00: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -0001fa10: 2020 206d 616b 6542 2753 6c69 6365 2873 makeB'Slice(s │ │ │ │ -0001fa20: 6c69 6365 5479 7065 2c76 6172 6961 626c liceType,variabl │ │ │ │ -0001fa30: 6547 726f 7570 7329 0a20 202a 2049 6e70 eGroups). * Inp │ │ │ │ -0001fa40: 7574 733a 0a20 2020 2020 202a 2073 6c69 uts:. * sli │ │ │ │ -0001fa50: 6365 5479 7065 2c20 6120 2a6e 6f74 6520 ceType, a *note │ │ │ │ -0001fa60: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ -0001fa70: 446f 6329 4c69 7374 2c2c 2041 206c 6973 Doc)List,, A lis │ │ │ │ -0001fa80: 7420 6f66 2069 6e74 6567 6572 7320 6f72 t of integers or │ │ │ │ -0001fa90: 0a20 2020 2020 2020 2069 6e74 6567 6572 . integer │ │ │ │ -0001faa0: 2e0a 2020 2020 2020 2a20 7661 7269 6162 .. * variab │ │ │ │ -0001fab0: 6c65 4772 6f75 7073 2c20 6120 2a6e 6f74 leGroups, a *not │ │ │ │ -0001fac0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -0001fad0: 7932 446f 6329 4c69 7374 2c2c 2041 206c y2Doc)List,, A l │ │ │ │ -0001fae0: 6973 7420 6f66 206c 6973 7420 6f66 0a20 ist of list of. │ │ │ │ -0001faf0: 2020 2020 2020 2076 6172 6961 626c 6573 variables │ │ │ │ -0001fb00: 206f 7220 6c69 7374 206f 6620 7661 7269 or list of vari │ │ │ │ -0001fb10: 6162 6c65 732e 0a20 202a 202a 6e6f 7465 ables.. * *note │ │ │ │ -0001fb20: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -0001fb30: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001fb40: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ -0001fb50: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ -0001fb60: 7075 7473 2c3a 0a20 2020 2020 202a 202a puts,:. * * │ │ │ │ -0001fb70: 6e6f 7465 2042 2748 6f6d 6f67 656e 697a note B'Homogeniz │ │ │ │ -0001fb80: 6174 696f 6e3a 206d 616b 6542 2753 6563 ation: makeB'Sec │ │ │ │ -0001fb90: 7469 6f6e 2c20 3d3e 202e 2e2e 2c20 6465 tion, => ..., de │ │ │ │ -0001fba0: 6661 756c 7420 7661 6c75 6520 7b7d 2c0a fault value {},. │ │ │ │ -0001fbb0: 2020 2020 2020 2020 6d61 6b65 4227 5365 makeB'Se │ │ │ │ -0001fbc0: 6374 696f 6e20 6372 6561 7465 7320 6120 ction creates a │ │ │ │ -0001fbd0: 6861 7368 2074 6162 6c65 2074 6861 7420 hash table that │ │ │ │ -0001fbe0: 7265 7072 6573 656e 7473 2061 2068 7970 represents a hyp │ │ │ │ -0001fbf0: 6572 706c 616e 652e 0a20 2020 2020 202a erplane.. * │ │ │ │ -0001fc00: 2042 274e 756d 6265 7243 6f65 6666 6963 B'NumberCoeffic │ │ │ │ -0001fc10: 6965 6e74 7320 3d3e 202e 2e2e 2c20 6465 ients => ..., de │ │ │ │ -0001fc20: 6661 756c 7420 7661 6c75 6520 7b7d 0a20 fault value {}. │ │ │ │ -0001fc30: 2020 2020 202a 2043 6f6e 7461 696e 734d * ContainsM │ │ │ │ -0001fc40: 756c 7469 5072 6f6a 6563 7469 7665 506f ultiProjectivePo │ │ │ │ -0001fc50: 696e 7420 3d3e 202e 2e2e 2c20 6465 6661 int => ..., defa │ │ │ │ -0001fc60: 756c 7420 7661 6c75 6520 7b7d 0a20 2020 ult value {}. │ │ │ │ -0001fc70: 2020 202a 2043 6f6e 7461 696e 7350 6f69 * ContainsPoi │ │ │ │ -0001fc80: 6e74 203d 3e20 2e2e 2e2c 2064 6566 6175 nt => ..., defau │ │ │ │ -0001fc90: 6c74 2076 616c 7565 207b 7d0a 2020 2020 lt value {}. │ │ │ │ -0001fca0: 2020 2a20 4e61 6d65 4227 536c 6963 6520 * NameB'Slice │ │ │ │ -0001fcb0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -0001fcc0: 7661 6c75 6520 6e75 6c6c 0a20 2020 2020 value null. │ │ │ │ -0001fcd0: 202a 2052 616e 646f 6d43 6f65 6666 6963 * RandomCoeffic │ │ │ │ -0001fce0: 6965 6e74 4765 6e65 7261 746f 7220 3d3e ientGenerator => │ │ │ │ -0001fcf0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0001fd00: 6c75 650a 2020 2020 2020 2020 4675 6e63 lue. Func │ │ │ │ -0001fd10: 7469 6f6e 436c 6f73 7572 655b 2e2e 2f42 tionClosure[../B │ │ │ │ -0001fd20: 6572 7469 6e69 2e6d 323a 3233 3536 3a33 ertini.m2:2356:3 │ │ │ │ -0001fd30: 372d 3233 3536 3a36 365d 0a0a 4465 7363 7-2356:66]..Desc │ │ │ │ -0001fd40: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0001fd50: 3d3d 3d0a 0a6d 616b 6542 2753 6c69 6365 ===..makeB'Slice │ │ │ │ -0001fd60: 2061 6c6c 6f77 7320 666f 7220 6561 7379 allows for easy │ │ │ │ -0001fd70: 2063 7265 6174 696f 6e20 6f66 2065 7175 creation of equ │ │ │ │ -0001fd80: 6174 696f 6e73 2074 6861 7420 6465 6669 ations that defi │ │ │ │ -0001fd90: 6e65 206c 696e 6561 7220 7370 6163 6573 ne linear spaces │ │ │ │ -0001fda0: 2c0a 692e 652e 2073 6c69 6365 732e 2054 ,.i.e. slices. T │ │ │ │ -0001fdb0: 6865 2064 6566 6175 6c74 2063 7265 6174 he default creat │ │ │ │ -0001fdc0: 6573 2061 2068 6173 6820 7461 626c 6520 es a hash table │ │ │ │ -0001fdd0: 7769 7468 2074 776f 206b 6579 733a 0a42 with two keys:.B │ │ │ │ -0001fde0: 274e 756d 6265 7243 6f65 6666 6963 6965 'NumberCoefficie │ │ │ │ -0001fdf0: 6e74 7320 616e 6420 4227 5365 6374 696f nts and B'Sectio │ │ │ │ -0001fe00: 6e53 7472 696e 672e 2057 6865 6e20 7765 nString. When we │ │ │ │ -0001fe10: 2068 6176 6520 6120 6d75 6c74 6970 726f have a multipro │ │ │ │ -0001fe20: 6a65 6374 6976 650a 7661 7269 6574 7920 jective.variety │ │ │ │ -0001fe30: 7765 2063 616e 2064 6966 6665 7265 6e74 we can different │ │ │ │ -0001fe40: 2074 7970 6573 206f 6620 736c 6963 6573 types of slices │ │ │ │ -0001fe50: 2e20 546f 206d 616b 6520 6120 736c 6963 . To make a slic │ │ │ │ -0001fe60: 6520 7765 206e 6565 6420 746f 2073 7065 e we need to spe │ │ │ │ -0001fe70: 6369 6679 0a74 6865 2074 7970 6520 6f66 cify.the type of │ │ │ │ -0001fe80: 2073 6c69 6365 2077 6520 7761 6e74 2066 slice we want f │ │ │ │ -0001fe90: 6f6c 6c6f 7765 6420 6279 2076 6172 6961 ollowed by varia │ │ │ │ -0001fea0: 626c 6520 6772 6f75 7073 2e0a 0a2b 2d2d ble groups...+-- │ │ │ │ -0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fa00: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0001fa10: 2020 2020 2020 6d61 6b65 4227 536c 6963 makeB'Slic │ │ │ │ +0001fa20: 6528 736c 6963 6554 7970 652c 7661 7269 e(sliceType,vari │ │ │ │ +0001fa30: 6162 6c65 4772 6f75 7073 290a 2020 2a20 ableGroups). * │ │ │ │ +0001fa40: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +0001fa50: 736c 6963 6554 7970 652c 2061 202a 6e6f sliceType, a *no │ │ │ │ +0001fa60: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ +0001fa70: 6179 3244 6f63 294c 6973 742c 2c20 4120 ay2Doc)List,, A │ │ │ │ +0001fa80: 6c69 7374 206f 6620 696e 7465 6765 7273 list of integers │ │ │ │ +0001fa90: 206f 720a 2020 2020 2020 2020 696e 7465 or. inte │ │ │ │ +0001faa0: 6765 722e 0a20 2020 2020 202a 2076 6172 ger.. * var │ │ │ │ +0001fab0: 6961 626c 6547 726f 7570 732c 2061 202a iableGroups, a * │ │ │ │ +0001fac0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +0001fad0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +0001fae0: 4120 6c69 7374 206f 6620 6c69 7374 206f A list of list o │ │ │ │ +0001faf0: 660a 2020 2020 2020 2020 7661 7269 6162 f. variab │ │ │ │ +0001fb00: 6c65 7320 6f72 206c 6973 7420 6f66 2076 les or list of v │ │ │ │ +0001fb10: 6172 6961 626c 6573 2e0a 2020 2a20 2a6e ariables.. * *n │ │ │ │ +0001fb20: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +0001fb30: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ +0001fb40: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ +0001fb50: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +0001fb60: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ +0001fb70: 2a20 2a6e 6f74 6520 4227 486f 6d6f 6765 * *note B'Homoge │ │ │ │ +0001fb80: 6e69 7a61 7469 6f6e 3a20 6d61 6b65 4227 nization: makeB' │ │ │ │ +0001fb90: 5365 6374 696f 6e2c 203d 3e20 2e2e 2e2c Section, => ..., │ │ │ │ +0001fba0: 2064 6566 6175 6c74 2076 616c 7565 207b default value { │ │ │ │ +0001fbb0: 7d2c 0a20 2020 2020 2020 206d 616b 6542 },. makeB │ │ │ │ +0001fbc0: 2753 6563 7469 6f6e 2063 7265 6174 6573 'Section creates │ │ │ │ +0001fbd0: 2061 2068 6173 6820 7461 626c 6520 7468 a hash table th │ │ │ │ +0001fbe0: 6174 2072 6570 7265 7365 6e74 7320 6120 at represents a │ │ │ │ +0001fbf0: 6879 7065 7270 6c61 6e65 2e0a 2020 2020 hyperplane.. │ │ │ │ +0001fc00: 2020 2a20 4227 4e75 6d62 6572 436f 6566 * B'NumberCoef │ │ │ │ +0001fc10: 6669 6369 656e 7473 203d 3e20 2e2e 2e2c ficients => ..., │ │ │ │ +0001fc20: 2064 6566 6175 6c74 2076 616c 7565 207b default value { │ │ │ │ +0001fc30: 7d0a 2020 2020 2020 2a20 436f 6e74 6169 }. * Contai │ │ │ │ +0001fc40: 6e73 4d75 6c74 6950 726f 6a65 6374 6976 nsMultiProjectiv │ │ │ │ +0001fc50: 6550 6f69 6e74 203d 3e20 2e2e 2e2c 2064 ePoint => ..., d │ │ │ │ +0001fc60: 6566 6175 6c74 2076 616c 7565 207b 7d0a efault value {}. │ │ │ │ +0001fc70: 2020 2020 2020 2a20 436f 6e74 6169 6e73 * Contains │ │ │ │ +0001fc80: 506f 696e 7420 3d3e 202e 2e2e 2c20 6465 Point => ..., de │ │ │ │ +0001fc90: 6661 756c 7420 7661 6c75 6520 7b7d 0a20 fault value {}. │ │ │ │ +0001fca0: 2020 2020 202a 204e 616d 6542 2753 6c69 * NameB'Sli │ │ │ │ +0001fcb0: 6365 203d 3e20 2e2e 2e2c 2064 6566 6175 ce => ..., defau │ │ │ │ +0001fcc0: 6c74 2076 616c 7565 206e 756c 6c0a 2020 lt value null. │ │ │ │ +0001fcd0: 2020 2020 2a20 5261 6e64 6f6d 436f 6566 * RandomCoef │ │ │ │ +0001fce0: 6669 6369 656e 7447 656e 6572 6174 6f72 ficientGenerator │ │ │ │ +0001fcf0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +0001fd00: 2076 616c 7565 0a20 2020 2020 2020 2046 value. F │ │ │ │ +0001fd10: 756e 6374 696f 6e43 6c6f 7375 7265 5b2e unctionClosure[. │ │ │ │ +0001fd20: 2e2f 4265 7274 696e 692e 6d32 3a32 3335 ./Bertini.m2:235 │ │ │ │ +0001fd30: 363a 3337 2d32 3335 363a 3636 5d0a 0a44 6:37-2356:66]..D │ │ │ │ +0001fd40: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0001fd50: 3d3d 3d3d 3d3d 0a0a 6d61 6b65 4227 536c ======..makeB'Sl │ │ │ │ +0001fd60: 6963 6520 616c 6c6f 7773 2066 6f72 2065 ice allows for e │ │ │ │ +0001fd70: 6173 7920 6372 6561 7469 6f6e 206f 6620 asy creation of │ │ │ │ +0001fd80: 6571 7561 7469 6f6e 7320 7468 6174 2064 equations that d │ │ │ │ +0001fd90: 6566 696e 6520 6c69 6e65 6172 2073 7061 efine linear spa │ │ │ │ +0001fda0: 6365 732c 0a69 2e65 2e20 736c 6963 6573 ces,.i.e. slices │ │ │ │ +0001fdb0: 2e20 5468 6520 6465 6661 756c 7420 6372 . The default cr │ │ │ │ +0001fdc0: 6561 7465 7320 6120 6861 7368 2074 6162 eates a hash tab │ │ │ │ +0001fdd0: 6c65 2077 6974 6820 7477 6f20 6b65 7973 le with two keys │ │ │ │ +0001fde0: 3a0a 4227 4e75 6d62 6572 436f 6566 6669 :.B'NumberCoeffi │ │ │ │ +0001fdf0: 6369 656e 7473 2061 6e64 2042 2753 6563 cients and B'Sec │ │ │ │ +0001fe00: 7469 6f6e 5374 7269 6e67 2e20 5768 656e tionString. When │ │ │ │ +0001fe10: 2077 6520 6861 7665 2061 206d 756c 7469 we have a multi │ │ │ │ +0001fe20: 7072 6f6a 6563 7469 7665 0a76 6172 6965 projective.varie │ │ │ │ +0001fe30: 7479 2077 6520 6361 6e20 6469 6666 6572 ty we can differ │ │ │ │ +0001fe40: 656e 7420 7479 7065 7320 6f66 2073 6c69 ent types of sli │ │ │ │ +0001fe50: 6365 732e 2054 6f20 6d61 6b65 2061 2073 ces. 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2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00020450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00020460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -000204a0: 203a 2070 6565 6b20 7879 536c 6963 6520 : peek xySlice │ │ │ │ -000204b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000204a0: 7c69 3420 3a20 7065 656b 2078 7953 6c69 |i4 : peek xySli │ │ │ │ +000204b0: 6365 2020 2020 2020 2020 2020 2020 2020 ce │ │ │ │ 000204c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000204d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000204e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000204f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000204f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00020500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020530: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -00020540: 203d 2042 2753 6c69 6365 7b42 274e 756d = B'Slice{B'Num │ │ │ │ -00020550: 6265 7243 6f65 6666 6963 6965 6e74 7320 berCoefficients │ │ │ │ -00020560: 3d3e 207b 7b31 2e34 3931 3434 2b2e 3731 => {{1.49144+.71 │ │ │ │ -00020570: 3338 3436 2a69 692c 202d 2e38 3430 3131 3846*ii, -.84011 │ │ │ │ -00020580: 332b 312e 3139 3836 2a20 207c 0a7c 2020 3+1.1986* |.| │ │ │ │ -00020590: 2020 2020 2020 2020 2020 2042 2753 6563 B'Sec │ │ │ │ -000205a0: 7469 6f6e 5374 7269 6e67 203d 3e20 7b28 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2020 ) │ │ │ │ -00020940: 2020 2020 2020 2020 2020 207c 0a7c 282e |.|(. │ │ │ │ -00020950: 3031 3438 3432 2b31 2e32 3335 3438 2a69 014842+1.23548*i │ │ │ │ -00020960: 6929 2a28 7930 292b 282d 2e32 3134 3436 i)*(y0)+(-.21446 │ │ │ │ -00020970: 382b 2e39 3131 3239 332a 6969 292a 2879 8+.911293*ii)*(y │ │ │ │ -00020980: 3129 2b28 2d2e 3438 3631 3736 2b2e 3430 1)+(-.486176+.40 │ │ │ │ -00020990: 3035 3737 2a69 6929 2a28 207c 0a7c 2d2d 0577*ii)*( |.|-- │ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000208a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000208b0: 7c69 3520 3a20 666f 7220 6920 696e 2020 |i5 : for i in │ │ │ │ +000208c0: 7879 536c 6963 6523 4227 5365 6374 696f xySlice#B'Sectio │ │ │ │ +000208d0: 6e53 7472 696e 6720 646f 2070 7269 6e74 nString do print │ │ │ │ +000208e0: 2069 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ +000208f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020900: 7c28 312e 3439 3134 342b 2e37 3133 3834 |(1.49144+.71384 │ │ │ │ +00020910: 362a 6969 292a 2878 3029 2b28 2d2e 3834 6*ii)*(x0)+(-.84 │ │ │ │ +00020920: 3031 3133 2b31 2e31 3938 362a 6969 292a 0113+1.1986*ii)* │ │ │ │ +00020930: 2878 3129 2020 2020 2020 2020 2020 2020 (x1) │ │ │ │ +00020940: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020950: 7c28 2e30 3134 3834 322b 312e 3233 3534 |(.014842+1.2354 │ │ │ │ +00020960: 382a 6969 292a 2879 3029 2b28 2d2e 3231 8*ii)*(y0)+(-.21 │ │ │ │ +00020970: 3434 3638 2b2e 3931 3132 3933 2a69 6929 4468+.911293*ii) │ │ │ │ +00020980: 2a28 7931 292b 282d 2e34 3836 3137 362b *(y1)+(-.486176+ │ │ │ │ +00020990: 2e34 3030 3537 372a 6969 292a 2820 7c0a .400577*ii)*( |. │ │ │ │ +000209a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ 000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000209c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2d2d -----------+.+-- │ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -00020ae0: 203a 2061 536c 6963 653d 6d61 6b65 4227 : aSlice=makeB' │ │ │ │ -00020af0: 536c 6963 6528 332c 7b78 2c79 2c7a 2c31 Slice(3,{x,y,z,1 │ │ │ │ -00020b00: 7d2c 4e61 6d65 4227 536c 6963 653d 3e22 },NameB'Slice=>" │ │ │ │ -00020b10: 6622 293b 2020 2020 2020 2020 2020 2020 f"); │ │ │ │ -00020b20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00020b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020ae0: 7c69 3620 3a20 6153 6c69 6365 3d6d 616b |i6 : aSlice=mak │ │ │ │ +00020af0: 6542 2753 6c69 6365 2833 2c7b 782c 792c eB'Slice(3,{x,y, │ │ │ │ +00020b00: 7a2c 317d 2c4e 616d 6542 2753 6c69 6365 z,1},NameB'Slice │ │ │ │ +00020b10: 3d3e 2266 2229 3b20 2020 2020 2020 2020 =>"f"); │ │ │ │ +00020b20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020b30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00020b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -00020b80: 203a 2061 536c 6963 6523 4e61 6d65 4227 : aSlice#NameB' │ │ │ │ -00020b90: 536c 6963 6520 2020 2020 2020 2020 2020 Slice │ │ │ │ +00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020b80: 7c69 3720 3a20 6153 6c69 6365 234e 616d |i7 : aSlice#Nam │ │ │ │ +00020b90: 6542 2753 6c69 6365 2020 2020 2020 2020 eB'Slice │ │ │ │ 00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020bc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020bd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00020be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c10: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00020c20: 203d 207b 6630 2c20 6631 2c20 6632 7d20 = {f0, f1, f2} │ │ │ │ -00020c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020c10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020c20: 7c6f 3720 3d20 7b66 302c 2066 312c 2066 |o7 = {f0, f1, f │ │ │ │ +00020c30: 327d 2020 2020 2020 2020 2020 2020 2020 2} │ │ │ │ 00020c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020c60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020c70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00020c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020cb0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00020cc0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +00020cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020cc0: 7c6f 3720 3a20 4c69 7374 2020 2020 2020 |o7 : List │ │ │ │ 00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00020d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020d00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020d10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00020d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -00020d60: 203a 206d 616b 6542 2749 6e70 7574 4669 : makeB'InputFi │ │ │ │ -00020d70: 6c65 2873 746f 7265 424d 3246 696c 6573 le(storeBM2Files │ │ │ │ -00020d80: 2c41 6666 5661 7269 6162 6c65 4772 6f75 ,AffVariableGrou │ │ │ │ -00020d90: 703d 3e7b 782c 792c 7a7d 2c42 2746 756e p=>{x,y,z},B'Fun │ │ │ │ -00020da0: 6374 696f 6e73 3d3e 7b20 207c 0a7c 2d2d ctions=>{ |.|-- │ │ │ │ -00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020d60: 7c69 3820 3a20 6d61 6b65 4227 496e 7075 |i8 : makeB'Inpu │ │ │ │ +00020d70: 7446 696c 6528 7374 6f72 6542 4d32 4669 tFile(storeBM2Fi │ │ │ │ +00020d80: 6c65 732c 4166 6656 6172 6961 626c 6547 les,AffVariableG │ │ │ │ +00020d90: 726f 7570 3d3e 7b78 2c79 2c7a 7d2c 4227 roup=>{x,y,z},B' │ │ │ │ +00020da0: 4675 6e63 7469 6f6e 733d 3e7b 2020 7c0a Functions=>{ |. │ │ │ │ +00020db0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ 00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6153 -----------|.|aS │ │ │ │ -00020e00: 6c69 6365 7d2c 4e61 6d65 506f 6c79 6e6f lice},NamePolyno │ │ │ │ -00020e10: 6d69 616c 733d 3e7b 2266 3022 2c22 6631 mials=>{"f0","f1 │ │ │ │ -00020e20: 222c 2266 3222 7d29 3b20 2020 2020 2020 ","f2"}); │ │ │ │ +00020df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00020e00: 7c61 536c 6963 657d 2c4e 616d 6550 6f6c |aSlice},NamePol │ │ │ │ +00020e10: 796e 6f6d 6961 6c73 3d3e 7b22 6630 222c ynomials=>{"f0", │ │ │ │ +00020e20: 2266 3122 2c22 6632 227d 293b 2020 2020 "f1","f2"}); │ │ │ │ 00020e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00020e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020e40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020e50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a2b 2d2d -----------+.+-- │ │ │ │ -00020ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020ea0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00020eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020ed0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2066 -------+.|i9 : f │ │ │ │ -00020ee0: 313d 2278 302a 7930 2b78 312a 7930 2b78 1="x0*y0+x1*y0+x │ │ │ │ -00020ef0: 322a 7932 2220 2020 2020 2020 2020 2020 2*y2" │ │ │ │ +00020ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +00020ee0: 3a20 6631 3d22 7830 2a79 302b 7831 2a79 : f1="x0*y0+x1*y │ │ │ │ +00020ef0: 302b 7832 2a79 3222 2020 2020 2020 2020 0+x2*y2" │ │ │ │ 00020f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020f10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020f10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00020f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020f40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020f50: 0a7c 6f39 203d 2078 302a 7930 2b78 312a .|o9 = x0*y0+x1* │ │ │ │ -00020f60: 7930 2b78 322a 7932 2020 2020 2020 2020 y0+x2*y2 │ │ │ │ +00020f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020f50: 2020 7c0a 7c6f 3920 3d20 7830 2a79 302b |.|o9 = x0*y0+ │ │ │ │ +00020f60: 7831 2a79 302b 7832 2a79 3220 2020 2020 x1*y0+x2*y2 │ │ │ │ 00020f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020f80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00020f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020f80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020f90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00020fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020fc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ -00020fd0: 6632 3d22 7830 2a79 305e 322b 7831 2a79 f2="x0*y0^2+x1*y │ │ │ │ -00020fe0: 312a 7932 2b78 322a 7930 2a79 3222 2020 1*y2+x2*y0*y2" │ │ │ │ -00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021000: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ +00020fd0: 203a 2066 323d 2278 302a 7930 5e32 2b78 : f2="x0*y0^2+x │ │ │ │ +00020fe0: 312a 7931 2a79 322b 7832 2a79 302a 7932 1*y1*y2+x2*y0*y2 │ │ │ │ +00020ff0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +00021000: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021040: 0a7c 6f31 3020 3d20 7830 2a79 305e 322b .|o10 = x0*y0^2+ │ │ │ │ -00021050: 7831 2a79 312a 7932 2b78 322a 7930 2a79 x1*y1*y2+x2*y0*y │ │ │ │ -00021060: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00021070: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00021080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021040: 2020 7c0a 7c6f 3130 203d 2078 302a 7930 |.|o10 = x0*y0 │ │ │ │ +00021050: 5e32 2b78 312a 7931 2a79 322b 7832 2a79 ^2+x1*y1*y2+x2*y │ │ │ │ +00021060: 302a 7932 2020 2020 2020 2020 2020 2020 0*y2 │ │ │ │ +00021070: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021080: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000210a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -000210c0: 7661 7269 6162 6c65 4772 6f75 7073 3d7b variableGroups={ │ │ │ │ -000210d0: 7b78 302c 7831 2c78 327d 2c7b 7930 2c79 {x0,x1,x2},{y0,y │ │ │ │ -000210e0: 312c 7932 7d7d 2020 2020 2020 2020 2020 1,y2}} │ │ │ │ -000210f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000210b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ +000210c0: 203a 2076 6172 6961 626c 6547 726f 7570 : variableGroup │ │ │ │ +000210d0: 733d 7b7b 7830 2c78 312c 7832 7d2c 7b79 s={{x0,x1,x2},{y │ │ │ │ +000210e0: 302c 7931 2c79 327d 7d20 2020 2020 2020 0,y1,y2}} │ │ │ │ +000210f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021130: 0a7c 6f31 3120 3d20 7b7b 7830 2c20 7831 .|o11 = {{x0, x1 │ │ │ │ -00021140: 2c20 7832 7d2c 207b 7930 2c20 7931 2c20 , x2}, {y0, y1, │ │ │ │ -00021150: 7932 7d7d 2020 2020 2020 2020 2020 2020 y2}} │ │ │ │ -00021160: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00021170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021130: 2020 7c0a 7c6f 3131 203d 207b 7b78 302c |.|o11 = {{x0, │ │ │ │ +00021140: 2078 312c 2078 327d 2c20 7b79 302c 2079 x1, x2}, {y0, y │ │ │ │ +00021150: 312c 2079 327d 7d20 2020 2020 2020 2020 1, y2}} │ │ │ │ +00021160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021170: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000211a0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20 |.|o11 : │ │ │ │ -000211b0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +000211a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ +000211b0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 000211c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000211d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000211e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000211e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000211f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00021220: 0a7c 6931 3220 3a20 7878 536c 6963 653d .|i12 : xxSlice= │ │ │ │ -00021230: 6d61 6b65 4227 536c 6963 6528 7b32 2c30 makeB'Slice({2,0 │ │ │ │ -00021240: 7d2c 7661 7269 6162 6c65 4772 6f75 7073 },variableGroups │ │ │ │ -00021250: 2920 2020 2020 2020 2020 207c 0a7c 2020 ) |.| │ │ │ │ -00021260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021220: 2d2d 2b0a 7c69 3132 203a 2078 7853 6c69 --+.|i12 : xxSli │ │ │ │ +00021230: 6365 3d6d 616b 6542 2753 6c69 6365 287b ce=makeB'Slice({ │ │ │ │ +00021240: 322c 307d 2c76 6172 6961 626c 6547 726f 2,0},variableGro │ │ │ │ +00021250: 7570 7329 2020 2020 2020 2020 2020 7c0a ups) |. │ │ │ │ +00021260: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021290: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ -000212a0: 4227 536c 6963 657b 2e2e 2e34 2e2e 2e7d B'Slice{...4...} │ │ │ │ -000212b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021290: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ +000212a0: 203d 2042 2753 6c69 6365 7b2e 2e2e 342e = B'Slice{...4. │ │ │ │ +000212b0: 2e2e 7d20 2020 2020 2020 2020 2020 2020 ..} │ │ │ │ 000212c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000212d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000212e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000212f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021310: 0a7c 6f31 3220 3a20 4227 536c 6963 6520 .|o12 : B'Slice │ │ │ │ -00021320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021310: 2020 7c0a 7c6f 3132 203a 2042 2753 6c69 |.|o12 : B'Sli │ │ │ │ +00021320: 6365 2020 2020 2020 2020 2020 2020 2020 ce │ │ │ │ 00021330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021340: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00021350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021380: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ -00021390: 7879 536c 6963 653d 6d61 6b65 4227 536c xySlice=makeB'Sl │ │ │ │ -000213a0: 6963 6528 7b31 2c31 7d2c 7661 7269 6162 ice({1,1},variab │ │ │ │ -000213b0: 6c65 4772 6f75 7073 2920 2020 2020 2020 leGroups) │ │ │ │ -000213c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021380: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +00021390: 203a 2078 7953 6c69 6365 3d6d 616b 6542 : xySlice=makeB │ │ │ │ +000213a0: 2753 6c69 6365 287b 312c 317d 2c76 6172 'Slice({1,1},var │ │ │ │ +000213b0: 6961 626c 6547 726f 7570 7329 2020 2020 iableGroups) │ │ │ │ +000213c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000213d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000213e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021400: 0a7c 6f31 3320 3d20 4227 536c 6963 657b .|o13 = B'Slice{ │ │ │ │ -00021410: 2e2e 2e34 2e2e 2e7d 2020 2020 2020 2020 ...4...} │ │ │ │ +000213f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021400: 2020 7c0a 7c6f 3133 203d 2042 2753 6c69 |.|o13 = B'Sli │ │ │ │ +00021410: 6365 7b2e 2e2e 342e 2e2e 7d20 2020 2020 ce{...4...} │ │ │ │ 00021420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021430: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00021440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021440: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021470: 2020 2020 2020 207c 0a7c 6f31 3320 3a20 |.|o13 : │ │ │ │ -00021480: 4227 536c 6963 6520 2020 2020 2020 2020 B'Slice │ │ │ │ +00021470: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +00021480: 203a 2042 2753 6c69 6365 2020 2020 2020 : B'Slice │ │ │ │ 00021490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000214b0: 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000214f0: 0a7c 6931 3420 3a20 7979 536c 6963 653d .|i14 : yySlice= │ │ │ │ -00021500: 6d61 6b65 4227 536c 6963 6528 7b30 2c32 makeB'Slice({0,2 │ │ │ │ -00021510: 7d2c 7661 7269 6162 6c65 4772 6f75 7073 },variableGroups │ │ │ │ -00021520: 2920 2020 2020 2020 2020 207c 0a7c 2020 ) |.| │ │ │ │ -00021530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000214f0: 2d2d 2b0a 7c69 3134 203a 2079 7953 6c69 --+.|i14 : yySli │ │ │ │ +00021500: 6365 3d6d 616b 6542 2753 6c69 6365 287b ce=makeB'Slice({ │ │ │ │ +00021510: 302c 327d 2c76 6172 6961 626c 6547 726f 0,2},variableGro │ │ │ │ +00021520: 7570 7329 2020 2020 2020 2020 2020 7c0a ups) |. │ │ │ │ +00021530: 7c20 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 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ -00021570: 4227 536c 6963 657b 2e2e 2e34 2e2e 2e7d B'Slice{...4...} │ │ │ │ -00021580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021560: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +00021570: 203d 2042 2753 6c69 6365 7b2e 2e2e 342e = B'Slice{...4. │ │ │ │ +00021580: 2e2e 7d20 2020 2020 2020 2020 2020 2020 ..} │ │ │ │ 00021590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000215a0: 2020 2020 2020 7c0a 7c20 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 3420 3a20 4227 536c 6963 6520 .|o14 : B'Slice │ │ │ │ -000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000215d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000215e0: 2020 7c0a 7c6f 3134 203a 2042 2753 6c69 |.|o14 : B'Sli │ │ │ │ +000215f0: 6365 2020 2020 2020 2020 2020 2020 2020 ce │ │ │ │ 00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021610: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00021620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021620: 2b2d 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 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ -00021660: 6d61 6b65 4227 496e 7075 7446 696c 6528 makeB'InputFile( │ │ │ │ -00021670: 7374 6f72 6542 4d32 4669 6c65 732c 2020 storeBM2Files, │ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021690: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000216a0: 486f 6d56 6172 6961 626c 6547 726f 7570 HomVariableGroup │ │ │ │ -000216b0: 3d3e 7661 7269 6162 6c65 4772 6f75 7073 =>variableGroups │ │ │ │ -000216c0: 2c20 2020 2020 2020 2020 2020 2020 207c , | │ │ │ │ -000216d0: 0a7c 2020 2020 2020 2020 2020 4227 506f .| B'Po │ │ │ │ -000216e0: 6c79 6e6f 6d69 616c 733d 3e7b 6631 2c66 lynomials=>{f1,f │ │ │ │ -000216f0: 327d 7c78 7853 6c69 6365 234c 6973 7442 2}|xxSlice#ListB │ │ │ │ -00021700: 2753 6563 7469 6f6e 7329 3b7c 0a2b 2d2d 'Sections);|.+-- │ │ │ │ -00021710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021650: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 ----------+.|i15 │ │ │ │ +00021660: 203a 206d 616b 6542 2749 6e70 7574 4669 : makeB'InputFi │ │ │ │ +00021670: 6c65 2873 746f 7265 424d 3246 696c 6573 le(storeBM2Files │ │ │ │ +00021680: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ +00021690: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000216a0: 2020 2048 6f6d 5661 7269 6162 6c65 4772 HomVariableGr │ │ │ │ +000216b0: 6f75 703d 3e76 6172 6961 626c 6547 726f oup=>variableGro │ │ │ │ +000216c0: 7570 732c 2020 2020 2020 2020 2020 2020 ups, │ │ │ │ +000216d0: 2020 7c0a 7c20 2020 2020 2020 2020 2042 |.| B │ │ │ │ +000216e0: 2750 6f6c 796e 6f6d 6961 6c73 3d3e 7b66 'Polynomials=>{f │ │ │ │ +000216f0: 312c 6632 7d7c 7878 536c 6963 6523 4c69 1,f2}|xxSlice#Li │ │ │ │ +00021700: 7374 4227 5365 6374 696f 6e73 293b 7c0a stB'Sections);|. │ │ │ │ +00021710: 2b2d 2d2d 2d2d 2d2d 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 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ -00021750: 7275 6e42 6572 7469 6e69 2873 746f 7265 runBertini(store │ │ │ │ -00021760: 424d 3246 696c 6573 2920 2020 2020 2020 BM2Files) │ │ │ │ +00021740: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ +00021750: 203a 2072 756e 4265 7274 696e 6928 7374 : runBertini(st │ │ │ │ +00021760: 6f72 6542 4d32 4669 6c65 7329 2020 2020 oreBM2Files) │ │ │ │ 00021770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021780: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00021780: 2020 2020 2020 7c0a 2b2d 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 2d2b ---------------+ │ │ │ │ -000217c0: 0a7c 6931 3720 3a20 7878 4465 6772 6565 .|i17 : xxDegree │ │ │ │ -000217d0: 3d23 696d 706f 7274 536f 6c75 7469 6f6e =#importSolution │ │ │ │ -000217e0: 7346 696c 6528 7374 6f72 6542 4d32 4669 sFile(storeBM2Fi │ │ │ │ -000217f0: 6c65 7329 2020 2020 2020 207c 0a7c 2020 les) |.| │ │ │ │ -00021800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000217c0: 2d2d 2b0a 7c69 3137 203a 2078 7844 6567 --+.|i17 : xxDeg │ │ │ │ +000217d0: 7265 653d 2369 6d70 6f72 7453 6f6c 7574 ree=#importSolut │ │ │ │ +000217e0: 696f 6e73 4669 6c65 2873 746f 7265 424d ionsFile(storeBM │ │ │ │ +000217f0: 3246 696c 6573 2920 2020 2020 2020 7c0a 2Files) |. │ │ │ │ +00021800: 7c20 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 207c 0a7c 6f31 3720 3d20 |.|o17 = │ │ │ │ -00021840: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00021830: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ +00021840: 203d 2032 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ 00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021870: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00021870: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000218a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000218b0: 0a7c 6931 3820 3a20 6d61 6b65 4227 496e .|i18 : makeB'In │ │ │ │ -000218c0: 7075 7446 696c 6528 7374 6f72 6542 4d32 putFile(storeBM2 │ │ │ │ -000218d0: 4669 6c65 732c 2020 2020 2020 2020 2020 Files, │ │ │ │ -000218e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000218f0: 2020 2020 2020 2020 486f 6d56 6172 6961 HomVaria │ │ │ │ -00021900: 626c 6547 726f 7570 3d3e 7661 7269 6162 bleGroup=>variab │ │ │ │ -00021910: 6c65 4772 6f75 7073 2c20 2020 2020 2020 leGroups, │ │ │ │ -00021920: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00021930: 2020 2020 4227 506f 6c79 6e6f 6d69 616c B'Polynomial │ │ │ │ -00021940: 733d 3e7b 6631 2c66 327d 7c78 7953 6c69 s=>{f1,f2}|xySli │ │ │ │ -00021950: 6365 234c 6973 7442 2753 6563 7469 6f6e ce#ListB'Section │ │ │ │ -00021960: 7329 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d s);|.+---------- │ │ │ │ +000218a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000218b0: 2d2d 2b0a 7c69 3138 203a 206d 616b 6542 --+.|i18 : makeB │ │ │ │ +000218c0: 2749 6e70 7574 4669 6c65 2873 746f 7265 'InputFile(store │ │ │ │ +000218d0: 424d 3246 696c 6573 2c20 2020 2020 2020 BM2Files, │ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000218f0: 7c20 2020 2020 2020 2020 2048 6f6d 5661 | HomVa │ │ │ │ +00021900: 7269 6162 6c65 4772 6f75 703d 3e76 6172 riableGroup=>var │ │ │ │ +00021910: 6961 626c 6547 726f 7570 732c 2020 2020 iableGroups, │ │ │ │ +00021920: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00021930: 2020 2020 2020 2042 2750 6f6c 796e 6f6d B'Polynom │ │ │ │ +00021940: 6961 6c73 3d3e 7b66 312c 6632 7d7c 7879 ials=>{f1,f2}|xy │ │ │ │ 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7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021a90: 0a7c 6f32 3020 3d20 3320 2020 2020 2020 .|o20 = 3 │ │ │ │ +00021a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021a90: 2020 7c0a 7c6f 3230 203d 2033 2020 2020 |.|o20 = 3 │ │ │ │ 00021aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021ac0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021ad0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021b00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 -------+.|i21 : │ │ │ │ -00021b10: 6d61 6b65 4227 496e 7075 7446 696c 6528 makeB'InputFile( │ │ │ │ -00021b20: 7374 6f72 6542 4d32 4669 6c65 732c 2020 storeBM2Files, │ │ │ │ -00021b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021b50: 486f 6d56 6172 6961 626c 6547 726f 7570 HomVariableGroup │ │ │ │ -00021b60: 3d3e 7661 7269 6162 6c65 4772 6f75 7073 =>variableGroups │ │ │ │ -00021b70: 2c20 2020 2020 2020 2020 2020 2020 207c , | │ │ │ │ -00021b80: 0a7c 2020 2020 2020 2020 2020 4227 506f .| B'Po │ │ │ │ -00021b90: 6c79 6e6f 6d69 616c 733d 3e7b 6631 2c66 lynomials=>{f1,f │ │ │ │ -00021ba0: 327d 7c79 7953 6c69 6365 234c 6973 7442 2}|yySlice#ListB │ │ │ │ -00021bb0: 2753 6563 7469 6f6e 7329 3b7c 0a2b 2d2d 'Sections);|.+-- │ │ │ │ -00021bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 ----------+.|i21 │ │ │ │ +00021b10: 203a 206d 616b 6542 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= 1 │ │ │ │ 00021d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00021d20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00021d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00021d60: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ -00021d70: 6b65 4227 536c 6963 653a 0a3d 3d3d 3d3d keB'Slice:.===== │ │ │ │ +00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d60: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ +00021d70: 206d 616b 6542 2753 6c69 6365 3a0a 3d3d makeB'Slice:.== │ │ │ │ 00021d80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00021d90: 3d3d 3d0a 0a20 202a 2022 6d61 6b65 4227 ===.. * "makeB' │ │ │ │ -00021da0: 536c 6963 6528 5468 696e 672c 4c69 7374 Slice(Thing,List │ │ │ │ -00021db0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -00021dc0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00021dd0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -00021de0: 626a 6563 7420 2a6e 6f74 6520 6d61 6b65 bject *note make │ │ │ │ -00021df0: 4227 536c 6963 653a 206d 616b 6542 2753 B'Slice: makeB'S │ │ │ │ -00021e00: 6c69 6365 2c20 6973 2061 202a 6e6f 7465 lice, is a *note │ │ │ │ -00021e10: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -00021e20: 2077 6974 680a 6f70 7469 6f6e 733a 2028 with.options: ( │ │ │ │ -00021e30: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00021e40: 686f 6446 756e 6374 696f 6e57 6974 684f hodFunctionWithO │ │ │ │ -00021e50: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d ptions,...------ │ │ │ │ +00021d90: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ +00021da0: 6542 2753 6c69 6365 2854 6869 6e67 2c4c eB'Slice(Thing,L │ │ │ │ +00021db0: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +00021dc0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00021dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00021de0: 6520 6f62 6a65 6374 202a 6e6f 7465 206d e object *note m │ │ │ │ +00021df0: 616b 6542 2753 6c69 6365 3a20 6d61 6b65 akeB'Slice: make │ │ │ │ +00021e00: 4227 536c 6963 652c 2069 7320 6120 2a6e B'Slice, is a *n │ │ │ │ +00021e10: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +00021e20: 696f 6e20 7769 7468 0a6f 7074 696f 6e73 ion with.options │ │ │ │ +00021e30: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021e40: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +00021e50: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ 00021e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ea0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00021eb0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00021ec0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00021ed0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00021ee0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00021ef0: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ -00021f00: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00021f10: 2f42 6572 7469 6e69 2e6d 323a 0a33 3637 /Bertini.m2:.367 │ │ │ │ -00021f20: 363a 302e 0a1f 0a46 696c 653a 2042 6572 6:0....File: Ber │ │ │ │ -00021f30: 7469 6e69 2e69 6e66 6f2c 204e 6f64 653a tini.info, Node: │ │ │ │ -00021f40: 206d 6f76 6542 2746 696c 652c 204e 6578 moveB'File, Nex │ │ │ │ -00021f50: 743a 204e 756d 6265 7254 6f42 2753 7472 t: NumberToB'Str │ │ │ │ -00021f60: 696e 672c 2050 7265 763a 206d 616b 6542 ing, Prev: makeB │ │ │ │ -00021f70: 2753 6c69 6365 2c20 5570 3a20 546f 700a 'Slice, Up: Top. │ │ │ │ -00021f80: 0a6d 6f76 6542 2746 696c 6520 2d2d 204d .moveB'File -- M │ │ │ │ -00021f90: 6f76 6520 6f72 2063 6f70 7920 6669 6c65 ove or copy file │ │ │ │ -00021fa0: 732e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a s..************* │ │ │ │ +00021ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00021eb0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00021ec0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00021ed0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00021ee0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00021ef0: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ +00021f00: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00021f10: 6765 732f 4265 7274 696e 692e 6d32 3a0a ges/Bertini.m2:. │ │ │ │ +00021f20: 3336 3736 3a30 2e0a 1f0a 4669 6c65 3a20 3676:0....File: │ │ │ │ +00021f30: 4265 7274 696e 692e 696e 666f 2c20 4e6f Bertini.info, No │ │ │ │ +00021f40: 6465 3a20 6d6f 7665 4227 4669 6c65 2c20 de: moveB'File, │ │ │ │ +00021f50: 4e65 7874 3a20 4e75 6d62 6572 546f 4227 Next: NumberToB' │ │ │ │ +00021f60: 5374 7269 6e67 2c20 5072 6576 3a20 6d61 String, Prev: ma │ │ │ │ +00021f70: 6b65 4227 536c 6963 652c 2055 703a 2054 keB'Slice, Up: T │ │ │ │ +00021f80: 6f70 0a0a 6d6f 7665 4227 4669 6c65 202d op..moveB'File - │ │ │ │ +00021f90: 2d20 4d6f 7665 206f 7220 636f 7079 2066 - Move or copy f │ │ │ │ +00021fa0: 696c 6573 2e0a 2a2a 2a2a 2a2a 2a2a 2a2a iles..********** │ │ │ │ 00021fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021fc0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00021fd0: 200a 2020 2020 2020 2020 6d6f 7665 4227 . moveB' │ │ │ │ -00021fe0: 4669 6c65 2873 2c66 2c6e 290a 2020 2a20 File(s,f,n). * │ │ │ │ -00021ff0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00022000: 732c 2061 202a 6e6f 7465 2073 7472 696e s, a *note strin │ │ │ │ -00022010: 673a 2028 4d61 6361 756c 6179 3244 6f63 g: (Macaulay2Doc │ │ │ │ -00022020: 2953 7472 696e 672c 2c20 4120 7374 7269 )String,, A stri │ │ │ │ -00022030: 6e67 2067 6976 696e 6720 6120 6469 7265 ng giving a dire │ │ │ │ -00022040: 6374 6f72 792e 0a20 2020 2020 202a 2066 ctory.. * f │ │ │ │ -00022050: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67 , a *note string │ │ │ │ -00022060: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00022070: 5374 7269 6e67 2c2c 2041 206e 616d 6520 String,, A name │ │ │ │ -00022080: 6f66 2061 2066 696c 652e 0a20 2020 2020 of a file.. │ │ │ │ -00022090: 202a 2073 2c20 6120 2a6e 6f74 6520 7374 * s, a *note st │ │ │ │ -000220a0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ -000220b0: 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2020 2020 │ │ │ │ -000228d0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20 |.|o7 = │ │ │ │ -000228e0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +000228d0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ +000228e0: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ 000228f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022920: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00022920: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00022930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022970: 2d2d 2d2d 2b0a 7c69 3820 3a20 6669 6c65 ----+.|i8 : file │ │ │ │ -00022980: 4578 6973 7473 2873 746f 7265 424d 3246 Exists(storeBM2F │ │ │ │ -00022990: 696c 6573 7c22 2f62 6163 6b75 7022 2920 iles|"/backup") │ │ │ │ -000229a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022970: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2066 -------+.|i8 : f │ │ │ │ +00022980: 696c 6545 7869 7374 7328 7374 6f72 6542 ileExists(storeB │ │ │ │ +00022990: 4d32 4669 6c65 737c 222f 6261 636b 7570 M2Files|"/backup │ │ │ │ +000229a0: 2229 2020 2020 2020 2020 2020 2020 2020 ") │ │ │ │ 000229b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000229c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a10: 7c0a 7c6f 3820 3d20 7472 7565 2020 2020 |.|o8 = true │ │ │ │ +00022a10: 2020 207c 0a7c 6f38 203d 2074 7275 6520 |.|o8 = true │ │ │ │ 00022a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022a60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022a60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00022a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 ------------+..T │ │ │ │ -00022ab0: 6865 206f 7074 696f 6e73 204d 6f76 6554 he options MoveT │ │ │ │ -00022ac0: 6f44 6972 6563 746f 7279 2061 6e64 2053 oDirectory and S │ │ │ │ -00022ad0: 7562 466f 6c64 6572 2067 6976 6520 6772 ubFolder give gr │ │ │ │ -00022ae0: 6561 7465 7220 636f 6e74 726f 6c20 666f eater control fo │ │ │ │ -00022af0: 7220 7768 6572 6520 746f 0a6d 6f76 6520 r where to.move │ │ │ │ -00022b00: 7468 6520 6669 6c65 2e0a 0a2b 2d2d 2d2d the file...+---- │ │ │ │ +00022aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00022ab0: 0a0a 5468 6520 6f70 7469 6f6e 7320 4d6f ..The options Mo │ │ │ │ +00022ac0: 7665 546f 4469 7265 6374 6f72 7920 616e veToDirectory an │ │ │ │ +00022ad0: 6420 5375 6246 6f6c 6465 7220 6769 7665 d SubFolder give │ │ │ │ +00022ae0: 2067 7265 6174 6572 2063 6f6e 7472 6f6c greater control │ │ │ │ +00022af0: 2066 6f72 2077 6865 7265 2074 6f0a 6d6f for where to.mo │ │ │ │ +00022b00: 7665 2074 6865 2066 696c 652e 0a0a 2b2d ve the file...+- │ │ │ │ 00022b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022b50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ -00022b60: 2044 6972 3120 3d20 7465 6d70 6f72 6172 Dir1 = temporar │ │ │ │ -00022b70: 7946 696c 654e 616d 6528 293b 2020 2020 yFileName(); │ │ │ │ +00022b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00022b60: 3920 3a20 4469 7231 203d 2074 656d 706f 9 : Dir1 = tempo │ │ │ │ +00022b70: 7261 7279 4669 6c65 4e61 6d65 2829 3b20 raryFileName(); │ │ │ │ 00022b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ba0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022ba0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00022bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ -00022c00: 3a20 6d61 6b65 4469 7265 6374 6f72 7920 : makeDirectory │ │ │ │ -00022c10: 4469 7231 2020 2020 2020 2020 2020 2020 Dir1 │ │ │ │ +00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00022c00: 3130 203a 206d 616b 6544 6972 6563 746f 10 : makeDirecto │ │ │ │ +00022c10: 7279 2044 6972 3120 2020 2020 2020 2020 ry Dir1 │ │ │ │ 00022c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00022c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022c90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -00022ca0: 3a20 7772 6974 6550 6172 616d 6574 6572 : writeParameter │ │ │ │ -00022cb0: 4669 6c65 2873 746f 7265 424d 3246 696c File(storeBM2Fil │ │ │ │ -00022cc0: 6573 2c7b 322c 332c 352c 377d 293b 2020 es,{2,3,5,7}); │ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ce0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00022ca0: 3131 203a 2077 7269 7465 5061 7261 6d65 11 : writeParame │ │ │ │ +00022cb0: 7465 7246 696c 6528 7374 6f72 6542 4d32 terFile(storeBM2 │ │ │ │ +00022cc0: 4669 6c65 732c 7b32 2c33 2c35 2c37 7d29 Files,{2,3,5,7}) │ │ │ │ +00022cd0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00022ce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00022cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022d30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ -00022d40: 3a20 6d6f 7665 4227 4669 6c65 2873 746f : moveB'File(sto │ │ │ │ -00022d50: 7265 424d 3246 696c 6573 2c22 6669 6e61 reBM2Files,"fina │ │ │ │ -00022d60: 6c5f 7061 7261 6d65 7465 7273 222c 2273 l_parameters","s │ │ │ │ -00022d70: 7461 7274 5f70 6172 616d 6574 6572 7322 tart_parameters" │ │ │ │ -00022d80: 2c20 2020 2020 2020 207c 0a7c 2d2d 2d2d , |.|---- │ │ │ │ +00022d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00022d40: 3132 203a 206d 6f76 6542 2746 696c 6528 12 : moveB'File( │ │ │ │ +00022d50: 7374 6f72 6542 4d32 4669 6c65 732c 2266 storeBM2Files,"f │ │ │ │ +00022d60: 696e 616c 5f70 6172 616d 6574 6572 7322 inal_parameters" │ │ │ │ +00022d70: 2c22 7374 6172 745f 7061 7261 6d65 7465 ,"start_paramete │ │ │ │ +00022d80: 7273 222c 2020 2020 2020 2020 7c0a 7c2d rs", |.|- │ │ │ │ 00022d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022dd0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 4d6f 7665 ---------|.|Move │ │ │ │ -00022de0: 546f 4469 7265 6374 6f72 793d 3e44 6972 ToDirectory=>Dir │ │ │ │ -00022df0: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ +00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c4d ------------|.|M │ │ │ │ +00022de0: 6f76 6554 6f44 6972 6563 746f 7279 3d3e oveToDirectory=> │ │ │ │ +00022df0: 4469 7231 2920 2020 2020 2020 2020 2020 Dir1) │ │ │ │ 00022e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022e20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00022e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022e70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ -00022e80: 3a20 6669 6c65 4578 6973 7473 2844 6972 : fileExists(Dir │ │ │ │ -00022e90: 317c 222f 7374 6172 745f 7061 7261 6d65 1|"/start_parame │ │ │ │ -00022ea0: 7465 7273 2229 2020 2020 2020 2020 2020 ters") │ │ │ │ +00022e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00022e80: 3133 203a 2066 696c 6545 7869 7374 7328 13 : fileExists( │ │ │ │ +00022e90: 4469 7231 7c22 2f73 7461 7274 5f70 6172 Dir1|"/start_par │ │ │ │ +00022ea0: 616d 6574 6572 7322 2920 2020 2020 2020 ameters") │ │ │ │ 00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ec0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022ec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f10: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ -00022f20: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ +00022f10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00022f20: 3133 203d 2074 7275 6520 2020 2020 2020 13 = true │ │ │ │ 00022f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022f60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00022f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022fb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d 2d2d ---------+.+---- │ │ │ │ +00022fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d ------------+.+- │ │ │ │ 00022fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023000: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ -00023010: 3a20 6d61 6b65 4469 7265 6374 6f72 7920 : makeDirectory │ │ │ │ -00023020: 2873 746f 7265 424d 3246 696c 6573 7c22 (storeBM2Files|" │ │ │ │ -00023030: 2f44 6972 3222 2920 2020 2020 2020 2020 /Dir2") │ │ │ │ +00023000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00023010: 3134 203a 206d 616b 6544 6972 6563 746f 14 : makeDirecto │ │ │ │ +00023020: 7279 2028 7374 6f72 6542 4d32 4669 6c65 ry (storeBM2File │ │ │ │ +00023030: 737c 222f 4469 7232 2229 2020 2020 2020 s|"/Dir2") │ │ │ │ 00023040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023050: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00023050: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 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 2d2d ---------------- │ │ │ │ 00023090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 ---------+.|i15 │ │ │ │ -000230b0: 3a20 7772 6974 6550 6172 616d 6574 6572 : writeParameter │ │ │ │ -000230c0: 4669 6c65 2873 746f 7265 424d 3246 696c File(storeBM2Fil │ │ │ │ -000230d0: 6573 2c7b 322c 332c 352c 377d 293b 2020 es,{2,3,5,7}); │ │ │ │ -000230e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000230a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000230b0: 3135 203a 2077 7269 7465 5061 7261 6d65 15 : writeParame │ │ │ │ +000230c0: 7465 7246 696c 6528 7374 6f72 6542 4d32 terFile(storeBM2 │ │ │ │ +000230d0: 4669 6c65 732c 7b32 2c33 2c35 2c37 7d29 Files,{2,3,5,7}) │ │ │ │ +000230e0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +000230f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00023100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ -00023150: 3a20 6d6f 7665 4227 4669 6c65 2873 746f : moveB'File(sto │ │ │ │ -00023160: 7265 424d 3246 696c 6573 2c22 6669 6e61 reBM2Files,"fina │ │ │ │ -00023170: 6c5f 7061 7261 6d65 7465 7273 222c 2273 l_parameters","s │ │ │ │ -00023180: 7461 7274 5f70 6172 616d 6574 6572 7322 tart_parameters" │ │ │ │ -00023190: 2c20 2020 2020 2020 207c 0a7c 2d2d 2d2d , |.|---- │ │ │ │ +00023140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00023150: 3136 203a 206d 6f76 6542 2746 696c 6528 16 : moveB'File( │ │ │ │ +00023160: 7374 6f72 6542 4d32 4669 6c65 732c 2266 storeBM2Files,"f │ │ │ │ +00023170: 696e 616c 5f70 6172 616d 6574 6572 7322 inal_parameters" │ │ │ │ +00023180: 2c22 7374 6172 745f 7061 7261 6d65 7465 ,"start_paramete │ │ │ │ +00023190: 7273 222c 2020 2020 2020 2020 7c0a 7c2d rs", |.|- │ │ │ │ 000231a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000231e0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5375 6246 ---------|.|SubF │ │ │ │ -000231f0: 6f6c 6465 723d 3e22 4469 7232 2229 2020 older=>"Dir2") │ │ │ │ -00023200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c53 ------------|.|S │ │ │ │ +000231f0: 7562 466f 6c64 6572 3d3e 2244 6972 3222 ubFolder=>"Dir2" │ │ │ │ +00023200: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00023210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023230: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00023230: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00023240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023280: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 ---------+.|i17 │ │ │ │ -00023290: 3a20 6669 6c65 4578 6973 7473 2873 746f : fileExists(sto │ │ │ │ -000232a0: 7265 424d 3246 696c 6573 7c22 2f44 6972 reBM2Files|"/Dir │ │ │ │ -000232b0: 322f 7374 6172 745f 7061 7261 6d65 7465 2/start_paramete │ │ │ │ -000232c0: 7273 2229 2020 2020 2020 2020 2020 2020 rs") │ │ │ │ -000232d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00023280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00023290: 3137 203a 2066 696c 6545 7869 7374 7328 17 : fileExists( │ │ │ │ +000232a0: 7374 6f72 6542 4d32 4669 6c65 737c 222f storeBM2Files|"/ │ │ │ │ +000232b0: 4469 7232 2f73 7461 7274 5f70 6172 616d Dir2/start_param │ │ │ │ +000232c0: 6574 6572 7322 2920 2020 2020 2020 2020 eters") │ │ │ │ +000232d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000232e0: 2020 2020 2020 2020 2020 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 207c 0a7c 6f31 3720 |.|o17 │ │ │ │ -00023330: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00023330: 3137 203d 2074 7275 6520 2020 2020 2020 17 = true │ │ │ │ 00023340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023370: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00023370: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000233c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 ---------+..Ways │ │ │ │ -000233d0: 2074 6f20 7573 6520 6d6f 7665 4227 4669 to use moveB'Fi │ │ │ │ -000233e0: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ -000233f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00023400: 2022 6d6f 7665 4227 4669 6c65 2853 7472 "moveB'File(Str │ │ │ │ -00023410: 696e 672c 5374 7269 6e67 2c53 7472 696e ing,String,Strin │ │ │ │ -00023420: 6729 220a 0a46 6f72 2074 6865 2070 726f g)"..For the pro │ │ │ │ -00023430: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00023440: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00023450: 6f62 6a65 6374 202a 6e6f 7465 206d 6f76 object *note mov │ │ │ │ -00023460: 6542 2746 696c 653a 206d 6f76 6542 2746 eB'File: moveB'F │ │ │ │ -00023470: 696c 652c 2069 7320 6120 2a6e 6f74 6520 ile, is a *note │ │ │ │ -00023480: 6d65 7468 6f64 2066 756e 6374 696f 6e20 method function │ │ │ │ -00023490: 7769 7468 0a6f 7074 696f 6e73 3a20 284d with.options: (M │ │ │ │ -000234a0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -000234b0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -000234c0: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ +000233c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 ------------+..W │ │ │ │ +000233d0: 6179 7320 746f 2075 7365 206d 6f76 6542 ays to use moveB │ │ │ │ +000233e0: 2746 696c 653a 0a3d 3d3d 3d3d 3d3d 3d3d 'File:.========= │ │ │ │ +000233f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00023400: 2020 2a20 226d 6f76 6542 2746 696c 6528 * "moveB'File( │ │ │ │ +00023410: 5374 7269 6e67 2c53 7472 696e 672c 5374 String,String,St │ │ │ │ +00023420: 7269 6e67 2922 0a0a 466f 7220 7468 6520 ring)"..For the │ │ │ │ +00023430: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00023440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00023450: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00023460: 6d6f 7665 4227 4669 6c65 3a20 6d6f 7665 moveB'File: move │ │ │ │ +00023470: 4227 4669 6c65 2c20 6973 2061 202a 6e6f B'File, is a *no │ │ │ │ +00023480: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00023490: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a on with.options: │ │ │ │ +000234a0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000234b0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +000234c0: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ 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 0a0a 5468 6520 736f --------..The so │ │ │ │ -00023520: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00023530: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00023540: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00023550: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00023560: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ -00023570: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00023580: 4265 7274 696e 692e 6d32 3a0a 3335 3139 Bertini.m2:.3519 │ │ │ │ -00023590: 3a30 2e0a 1f0a 4669 6c65 3a20 4265 7274 :0....File: Bert │ │ │ │ -000235a0: 696e 692e 696e 666f 2c20 4e6f 6465 3a20 ini.info, Node: │ │ │ │ -000235b0: 4e75 6d62 6572 546f 4227 5374 7269 6e67 NumberToB'String │ │ │ │ -000235c0: 2c20 4e65 7874 3a20 5061 7468 4c69 7374 , Next: PathList │ │ │ │ -000235d0: 2c20 5072 6576 3a20 6d6f 7665 4227 4669 , Prev: moveB'Fi │ │ │ │ -000235e0: 6c65 2c20 5570 3a20 546f 700a 0a4e 756d le, Up: Top..Num │ │ │ │ -000235f0: 6265 7254 6f42 2753 7472 696e 6720 2d2d berToB'String -- │ │ │ │ -00023600: 2054 7261 6e73 6c61 7465 7320 6120 6e75 Translates a nu │ │ │ │ -00023610: 6d62 6572 2074 6f20 6120 7374 7269 6e67 mber to a string │ │ │ │ -00023620: 2074 6861 7420 4265 7274 696e 6920 6361 that Bertini ca │ │ │ │ -00023630: 6e20 7265 6164 2e0a 2a2a 2a2a 2a2a 2a2a n read..******** │ │ │ │ +00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00023520: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00023530: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00023540: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00023550: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00023560: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ +00023570: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00023580: 6573 2f42 6572 7469 6e69 2e6d 323a 0a33 es/Bertini.m2:.3 │ │ │ │ +00023590: 3531 393a 302e 0a1f 0a46 696c 653a 2042 519:0....File: B │ │ │ │ +000235a0: 6572 7469 6e69 2e69 6e66 6f2c 204e 6f64 ertini.info, Nod │ │ │ │ +000235b0: 653a 204e 756d 6265 7254 6f42 2753 7472 e: NumberToB'Str │ │ │ │ +000235c0: 696e 672c 204e 6578 743a 2050 6174 684c ing, Next: PathL │ │ │ │ +000235d0: 6973 742c 2050 7265 763a 206d 6f76 6542 ist, Prev: moveB │ │ │ │ +000235e0: 2746 696c 652c 2055 703a 2054 6f70 0a0a 'File, Up: Top.. │ │ │ │ +000235f0: 4e75 6d62 6572 546f 4227 5374 7269 6e67 NumberToB'String │ │ │ │ +00023600: 202d 2d20 5472 616e 736c 6174 6573 2061 -- Translates a │ │ │ │ +00023610: 206e 756d 6265 7220 746f 2061 2073 7472 number to a str │ │ │ │ +00023620: 696e 6720 7468 6174 2042 6572 7469 6e69 ing that Bertini │ │ │ │ +00023630: 2063 616e 2072 6561 642e 0a2a 2a2a 2a2a can read..***** │ │ │ │ 00023640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00023650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00023660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00023670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00023680: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -00023690: 2020 2020 2020 2020 4e75 6d62 6572 546f NumberTo │ │ │ │ -000236a0: 4227 5374 7269 6e67 286e 290a 2020 2a20 B'String(n). * │ │ │ │ -000236b0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -000236c0: 6e2c 2061 202a 6e6f 7465 2074 6869 6e67 n, a *note thing │ │ │ │ -000236d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000236e0: 5468 696e 672c 2c20 6e20 6973 2061 206e Thing,, n is a n │ │ │ │ -000236f0: 756d 6265 722e 0a20 202a 202a 6e6f 7465 umber.. * *note │ │ │ │ -00023700: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -00023710: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00023720: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ -00023730: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ -00023740: 7075 7473 2c3a 0a20 2020 2020 202a 204d puts,:. * M │ │ │ │ -00023750: 3250 7265 6369 7369 6f6e 2028 6d69 7373 2Precision (miss │ │ │ │ -00023760: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -00023770: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ -00023780: 6c74 2076 616c 7565 2035 332c 200a 0a44 lt value 53, ..D │ │ │ │ -00023790: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -000237a0: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e ======..This fun │ │ │ │ -000237b0: 6374 696f 6e20 7461 6b65 7320 6120 6e75 ction takes a nu │ │ │ │ -000237c0: 6d62 6572 2061 7320 616e 2069 6e70 7574 mber as an input │ │ │ │ -000237d0: 2074 6865 6e20 6f75 7470 7574 7320 6120 then outputs a │ │ │ │ -000237e0: 7374 7269 6e67 2074 6f20 7265 7072 6573 string to repres │ │ │ │ -000237f0: 656e 740a 7468 6973 206e 756d 6265 7220 ent.this number │ │ │ │ -00023800: 746f 2042 6572 7469 6e69 2e20 5468 6520 to Bertini. The │ │ │ │ -00023810: 6e75 6d62 6572 7320 6172 6520 636f 6e76 numbers are conv │ │ │ │ -00023820: 6572 7465 6420 746f 2066 6c6f 6174 696e erted to floatin │ │ │ │ -00023830: 6720 706f 696e 7420 746f 0a70 7265 6369 g point to.preci │ │ │ │ -00023840: 7369 6f6e 2064 6574 6572 6d69 6e65 6420 sion determined │ │ │ │ -00023850: 6279 2074 6865 206f 7074 696f 6e20 4d32 by the option M2 │ │ │ │ -00023860: 5072 6563 6973 696f 6e2e 0a0a 2b2d 2d2d Precision...+--- │ │ │ │ +00023680: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00023690: 3a20 0a20 2020 2020 2020 204e 756d 6265 : . Numbe │ │ │ │ +000236a0: 7254 6f42 2753 7472 696e 6728 6e29 0a20 rToB'String(n). │ │ │ │ +000236b0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000236c0: 202a 206e 2c20 6120 2a6e 6f74 6520 7468 * n, a *note th │ │ │ │ +000236d0: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +000236e0: 6f63 2954 6869 6e67 2c2c 206e 2069 7320 oc)Thing,, n is │ │ │ │ +000236f0: 6120 6e75 6d62 6572 2e0a 2020 2a20 2a6e a number.. * *n │ │ │ │ +00023700: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +00023710: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ +00023720: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ +00023730: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00023740: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ +00023750: 2a20 4d32 5072 6563 6973 696f 6e20 286d * M2Precision (m │ │ │ │ +00023760: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +00023770: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +00023780: 6661 756c 7420 7661 6c75 6520 3533 2c20 fault value 53, │ │ │ │ +00023790: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +000237a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 =========..This │ │ │ │ +000237b0: 6675 6e63 7469 6f6e 2074 616b 6573 2061 function takes a │ │ │ │ +000237c0: 206e 756d 6265 7220 6173 2061 6e20 696e number as an in │ │ │ │ +000237d0: 7075 7420 7468 656e 206f 7574 7075 7473 put then outputs │ │ │ │ +000237e0: 2061 2073 7472 696e 6720 746f 2072 6570 a string to rep │ │ │ │ +000237f0: 7265 7365 6e74 0a74 6869 7320 6e75 6d62 resent.this numb │ │ │ │ +00023800: 6572 2074 6f20 4265 7274 696e 692e 2054 er to Bertini. T │ │ │ │ +00023810: 6865 206e 756d 6265 7273 2061 7265 2063 he numbers are c │ │ │ │ +00023820: 6f6e 7665 7274 6564 2074 6f20 666c 6f61 onverted to floa │ │ │ │ +00023830: 7469 6e67 2070 6f69 6e74 2074 6f0a 7072 ting point to.pr │ │ │ │ +00023840: 6563 6973 696f 6e20 6465 7465 726d 696e ecision determin │ │ │ │ +00023850: 6564 2062 7920 7468 6520 6f70 7469 6f6e ed by the option │ │ │ │ +00023860: 204d 3250 7265 6369 7369 6f6e 2e0a 0a2b M2Precision...+ │ │ │ │ 00023870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000238b0: 203a 204e 756d 6265 7254 6f42 2753 7472 : NumberToB'Str │ │ │ │ -000238c0: 696e 6728 322b 352a 6969 2920 2020 2020 ing(2+5*ii) │ │ │ │ +000238a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000238b0: 7c69 3120 3a20 4e75 6d62 6572 546f 4227 |i1 : NumberToB' │ │ │ │ +000238c0: 5374 7269 6e67 2832 2b35 2a69 6929 2020 String(2+5*ii) │ │ │ │ 000238d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000238e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000238f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000238e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000238f0: 0a7c 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 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023930: 6f31 203d 202e 3265 3120 2e35 6531 2020 o1 = .2e1 .5e1 │ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023930: 7c0a 7c6f 3120 3d20 2e32 6531 202e 3565 |.|o1 = .2e1 .5e │ │ │ │ +00023940: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00023950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00023970: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00023960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023970: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00023980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000239a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000239b0: 0a7c 6932 203a 204e 756d 6265 7254 6f42 .|i2 : NumberToB │ │ │ │ -000239c0: 2753 7472 696e 6728 312f 332c 4d32 5072 'String(1/3,M2Pr │ │ │ │ -000239d0: 6563 6973 696f 6e3d 3e31 3629 2020 2020 ecision=>16) │ │ │ │ +000239a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000239b0: 2d2d 2b0a 7c69 3220 3a20 4e75 6d62 6572 --+.|i2 : Number │ │ │ │ +000239c0: 546f 4227 5374 7269 6e67 2831 2f33 2c4d ToB'String(1/3,M │ │ │ │ +000239d0: 3250 7265 6369 7369 6f6e 3d3e 3136 2920 2Precision=>16) │ │ │ │ 000239e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239f0: 7c0a 7c57 6172 6e69 6e67 3a20 7261 7469 |.|Warning: rati │ │ │ │ -00023a00: 6f6e 616c 206e 756d 6265 7273 2077 696c onal numbers wil │ │ │ │ -00023a10: 6c20 6265 2063 6f6e 7665 7274 6564 2074 l be converted t │ │ │ │ -00023a20: 6f20 666c 6f61 7469 6e67 2070 6f69 6e74 o floating point │ │ │ │ -00023a30: 2e7c 0a7c 2020 2020 2020 2020 2020 2020 .|.| │ │ │ │ +000239f0: 2020 207c 0a7c 5761 726e 696e 673a 2072 |.|Warning: r │ │ │ │ +00023a00: 6174 696f 6e61 6c20 6e75 6d62 6572 7320 ational numbers │ │ │ │ +00023a10: 7769 6c6c 2062 6520 636f 6e76 6572 7465 will be converte │ │ │ │ +00023a20: 6420 746f 2066 6c6f 6174 696e 6720 706f d to floating po │ │ │ │ +00023a30: 696e 742e 7c0a 7c20 2020 2020 2020 2020 int.|.| │ │ │ │ 00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a70: 2020 7c0a 7c6f 3220 3d20 2e33 3333 3333 |.|o2 = .33333 │ │ │ │ -00023a80: 3333 3333 3333 3333 3333 3331 6530 202e 333333333331e0 . │ │ │ │ -00023a90: 3065 3020 2020 2020 2020 2020 2020 2020 0e0 │ │ │ │ +00023a70: 2020 2020 207c 0a7c 6f32 203d 202e 3333 |.|o2 = .33 │ │ │ │ +00023a80: 3333 3333 3333 3333 3333 3333 3333 3165 333333333333331e │ │ │ │ +00023a90: 3020 2e30 6530 2020 2020 2020 2020 2020 0 .0e0 │ │ │ │ 00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ab0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00023ab0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023af0: 2d2d 2d2d 2b0a 7c69 3320 3a20 4e75 6d62 ----+.|i3 : Numb │ │ │ │ -00023b00: 6572 546f 4227 5374 7269 6e67 2831 2f33 erToB'String(1/3 │ │ │ │ -00023b10: 2c4d 3250 7265 6369 7369 6f6e 3d3e 3132 ,M2Precision=>12 │ │ │ │ -00023b20: 3829 2020 2020 2020 2020 2020 2020 2020 8) │ │ │ │ -00023b30: 2020 2020 207c 0a7c 5761 726e 696e 673a |.|Warning: │ │ │ │ -00023b40: 2072 6174 696f 6e61 6c20 6e75 6d62 6572 rational number │ │ │ │ -00023b50: 7320 7769 6c6c 2062 6520 636f 6e76 6572 s will be conver │ │ │ │ -00023b60: 7465 6420 746f 2066 6c6f 6174 696e 6720 ted to floating │ │ │ │ -00023b70: 706f 696e 742e 7c0a 7c20 2020 2020 2020 point.|.| │ │ │ │ +00023af0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204e -------+.|i3 : N │ │ │ │ +00023b00: 756d 6265 7254 6f42 2753 7472 696e 6728 umberToB'String( │ │ │ │ +00023b10: 312f 332c 4d32 5072 6563 6973 696f 6e3d 1/3,M2Precision= │ │ │ │ +00023b20: 3e31 3238 2920 2020 2020 2020 2020 2020 >128) │ │ │ │ +00023b30: 2020 2020 2020 2020 7c0a 7c57 6172 6e69 |.|Warni │ │ │ │ +00023b40: 6e67 3a20 7261 7469 6f6e 616c 206e 756d ng: rational num │ │ │ │ +00023b50: 6265 7273 2077 696c 6c20 6265 2063 6f6e bers will be con │ │ │ │ +00023b60: 7665 7274 6564 2074 6f20 666c 6f61 7469 verted to floati │ │ │ │ +00023b70: 6e67 2070 6f69 6e74 2e7c 0a7c 2020 2020 ng point.|.| │ │ │ │ 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 207c 0a7c 6f33 203d 202e |.|o3 = . │ │ │ │ -00023bc0: 3333 3333 3333 3333 3333 3333 3333 3333 3333333333333333 │ │ │ │ +00023bb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +00023bc0: 3d20 2e33 3333 3333 3333 3333 3333 3333 = .3333333333333 │ │ │ │ 00023bd0: 3333 3333 3333 3333 3333 3333 3333 3333 3333333333333333 │ │ │ │ -00023be0: 3333 3333 3333 3338 6530 202e 3065 3020 33333338e0 .0e0 │ │ │ │ -00023bf0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00023be0: 3333 3333 3333 3333 3333 3865 3020 2e30 33333333338e0 .0 │ │ │ │ +00023bf0: 6530 2020 2020 2020 2020 207c 0a2b 2d2d e0 |.+-- │ │ │ │ 00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 ---------+..Ways │ │ │ │ -00023c40: 2074 6f20 7573 6520 4e75 6d62 6572 546f to use NumberTo │ │ │ │ -00023c50: 4227 5374 7269 6e67 3a0a 3d3d 3d3d 3d3d B'String:.====== │ │ │ │ +00023c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 ------------+..W │ │ │ │ +00023c40: 6179 7320 746f 2075 7365 204e 756d 6265 ays to use Numbe │ │ │ │ +00023c50: 7254 6f42 2753 7472 696e 673a 0a3d 3d3d rToB'String:.=== │ │ │ │ 00023c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023c70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4e75 =======.. * "Nu │ │ │ │ -00023c80: 6d62 6572 546f 4227 5374 7269 6e67 2854 mberToB'String(T │ │ │ │ -00023c90: 6869 6e67 2922 0a0a 466f 7220 7468 6520 hing)"..For the │ │ │ │ -00023ca0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00023cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00023cc0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00023cd0: 4e75 6d62 6572 546f 4227 5374 7269 6e67 NumberToB'String │ │ │ │ -00023ce0: 3a20 4e75 6d62 6572 546f 4227 5374 7269 : NumberToB'Stri │ │ │ │ -00023cf0: 6e67 2c20 6973 2061 202a 6e6f 7465 206d ng, is a *note m │ │ │ │ -00023d00: 6574 686f 6420 6675 6e63 7469 6f6e 0a77 ethod function.w │ │ │ │ -00023d10: 6974 6820 6f70 7469 6f6e 733a 2028 4d61 ith options: (Ma │ │ │ │ -00023d20: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00023d30: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ -00023d40: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ +00023c70: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00023c80: 224e 756d 6265 7254 6f42 2753 7472 696e "NumberToB'Strin │ │ │ │ +00023c90: 6728 5468 696e 6729 220a 0a46 6f72 2074 g(Thing)"..For t │ │ │ │ +00023ca0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +00023cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00023cc0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +00023cd0: 7465 204e 756d 6265 7254 6f42 2753 7472 te NumberToB'Str │ │ │ │ +00023ce0: 696e 673a 204e 756d 6265 7254 6f42 2753 ing: NumberToB'S │ │ │ │ +00023cf0: 7472 696e 672c 2069 7320 6120 2a6e 6f74 tring, is a *not │ │ │ │ +00023d00: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00023d10: 6e0a 7769 7468 206f 7074 696f 6e73 3a20 n.with options: │ │ │ │ +00023d20: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00023d30: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00023d40: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ 00023d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023d90: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00023da0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00023db0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00023dc0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00023dd0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00023de0: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ -00023df0: 756c 6179 322f 7061 636b 6167 6573 2f42 ulay2/packages/B │ │ │ │ -00023e00: 6572 7469 6e69 2e6d 323a 0a33 3732 353a ertini.m2:.3725: │ │ │ │ -00023e10: 302e 0a1f 0a46 696c 653a 2042 6572 7469 0....File: Berti │ │ │ │ -00023e20: 6e69 2e69 6e66 6f2c 204e 6f64 653a 2050 ni.info, Node: P │ │ │ │ -00023e30: 6174 684c 6973 742c 204e 6578 743a 2072 athList, Next: r │ │ │ │ -00023e40: 6164 6963 616c 4c69 7374 2c20 5072 6576 adicalList, Prev │ │ │ │ -00023e50: 3a20 4e75 6d62 6572 546f 4227 5374 7269 : NumberToB'Stri │ │ │ │ -00023e60: 6e67 2c20 5570 3a20 546f 700a 0a50 6174 ng, Up: Top..Pat │ │ │ │ -00023e70: 684c 6973 740a 2a2a 2a2a 2a2a 2a2a 0a0a hList.********.. │ │ │ │ -00023e80: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ -00023e90: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ -00023ea0: 206e 616d 6564 2050 6174 684c 6973 743a named PathList: │ │ │ │ -00023eb0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00023d90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00023da0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00023db0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00023dc0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00023dd0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00023de0: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ +00023df0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00023e00: 732f 4265 7274 696e 692e 6d32 3a0a 3337 s/Bertini.m2:.37 │ │ │ │ +00023e10: 3235 3a30 2e0a 1f0a 4669 6c65 3a20 4265 25:0....File: Be │ │ │ │ +00023e20: 7274 696e 692e 696e 666f 2c20 4e6f 6465 rtini.info, Node │ │ │ │ +00023e30: 3a20 5061 7468 4c69 7374 2c20 4e65 7874 : PathList, Next │ │ │ │ +00023e40: 3a20 7261 6469 6361 6c4c 6973 742c 2050 : radicalList, P │ │ │ │ +00023e50: 7265 763a 204e 756d 6265 7254 6f42 2753 rev: NumberToB'S │ │ │ │ +00023e60: 7472 696e 672c 2055 703a 2054 6f70 0a0a tring, Up: Top.. │ │ │ │ +00023e70: 5061 7468 4c69 7374 0a2a 2a2a 2a2a 2a2a PathList.******* │ │ │ │ +00023e80: 2a0a 0a46 756e 6374 696f 6e73 2077 6974 *..Functions wit │ │ │ │ +00023e90: 6820 6f70 7469 6f6e 616c 2061 7267 756d h optional argum │ │ │ │ +00023ea0: 656e 7420 6e61 6d65 6420 5061 7468 4c69 ent named PathLi │ │ │ │ +00023eb0: 7374 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d st:.============ │ │ │ │ 00023ec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00023ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023ee0: 3d0a 0a20 202a 2069 6d70 6f72 744d 6169 =.. * importMai │ │ │ │ -00023ef0: 6e44 6174 6146 696c 6528 2e2e 2e2c 5061 nDataFile(...,Pa │ │ │ │ -00023f00: 7468 4c69 7374 3d3e 2e2e 2e29 2028 6d69 thList=>...) (mi │ │ │ │ -00023f10: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -00023f20: 696f 6e29 0a0a 466f 7220 7468 6520 7072 ion)..For the pr │ │ │ │ -00023f30: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00023f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00023f50: 206f 626a 6563 7420 5061 7468 4c69 7374 object PathList │ │ │ │ -00023f60: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -00023f70: 6e74 6174 696f 6e29 2069 7320 6120 2a6e ntation) is a *n │ │ │ │ -00023f80: 6f74 6520 7379 6d62 6f6c 3a0a 284d 6163 ote symbol:.(Mac │ │ │ │ -00023f90: 6175 6c61 7932 446f 6329 5379 6d62 6f6c aulay2Doc)Symbol │ │ │ │ -00023fa0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +00023ee0: 3d3d 3d3d 0a0a 2020 2a20 696d 706f 7274 ====.. * import │ │ │ │ +00023ef0: 4d61 696e 4461 7461 4669 6c65 282e 2e2e MainDataFile(... │ │ │ │ +00023f00: 2c50 6174 684c 6973 743d 3e2e 2e2e 2920 ,PathList=>...) │ │ │ │ +00023f10: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +00023f20: 7461 7469 6f6e 290a 0a46 6f72 2074 6865 tation)..For the │ │ │ │ +00023f30: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +00023f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00023f50: 5468 6520 6f62 6a65 6374 2050 6174 684c The object PathL │ │ │ │ +00023f60: 6973 7420 286d 6973 7369 6e67 2064 6f63 ist (missing doc │ │ │ │ +00023f70: 756d 656e 7461 7469 6f6e 2920 6973 2061 umentation) is a │ │ │ │ +00023f80: 202a 6e6f 7465 2073 796d 626f 6c3a 0a28 *note symbol:.( │ │ │ │ +00023f90: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ +00023fa0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ff0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00024000: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00024010: 2069 7320 696e 2063 7572 7265 6e74 5374 is in currentSt │ │ │ │ -00024020: 7269 6e67 3a31 3a33 3832 2e0a 1f0a 4669 ring:1:382....Fi │ │ │ │ -00024030: 6c65 3a20 4265 7274 696e 692e 696e 666f le: Bertini.info │ │ │ │ -00024040: 2c20 4e6f 6465 3a20 7261 6469 6361 6c4c , Node: radicalL │ │ │ │ -00024050: 6973 742c 204e 6578 743a 2073 746f 7265 ist, Next: store │ │ │ │ -00024060: 424d 3246 696c 6573 2c20 5072 6576 3a20 BM2Files, Prev: │ │ │ │ -00024070: 5061 7468 4c69 7374 2c20 5570 3a20 546f PathList, Up: To │ │ │ │ -00024080: 700a 0a72 6164 6963 616c 4c69 7374 202d p..radicalList - │ │ │ │ -00024090: 2d20 4120 7375 7070 6f72 7420 6675 6e63 - A support func │ │ │ │ -000240a0: 7469 6f6e 2074 6861 7420 7265 6d6f 7665 tion that remove │ │ │ │ -000240b0: 7320 6d75 6c74 6970 6c69 6369 7469 6573 s multiplicities │ │ │ │ -000240c0: 206f 6620 6e75 6d62 6572 7320 696e 2061 of numbers in a │ │ │ │ -000240d0: 206c 6973 7420 7570 2074 6f20 6120 746f list up to a to │ │ │ │ -000240e0: 6c65 7261 6e63 652e 0a2a 2a2a 2a2a 2a2a lerance..******* │ │ │ │ +00023ff0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00024000: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00024010: 656e 7420 6973 2069 6e20 6375 7272 656e ent is in curren │ │ │ │ +00024020: 7453 7472 696e 673a 313a 3338 322e 0a1f tString:1:382... │ │ │ │ +00024030: 0a46 696c 653a 2042 6572 7469 6e69 2e69 .File: Bertini.i │ │ │ │ +00024040: 6e66 6f2c 204e 6f64 653a 2072 6164 6963 nfo, Node: radic │ │ │ │ +00024050: 616c 4c69 7374 2c20 4e65 7874 3a20 7374 alList, Next: st │ │ │ │ +00024060: 6f72 6542 4d32 4669 6c65 732c 2050 7265 oreBM2Files, Pre │ │ │ │ +00024070: 763a 2050 6174 684c 6973 742c 2055 703a v: PathList, Up: │ │ │ │ +00024080: 2054 6f70 0a0a 7261 6469 6361 6c4c 6973 Top..radicalLis │ │ │ │ +00024090: 7420 2d2d 2041 2073 7570 706f 7274 2066 t -- A support f │ │ │ │ +000240a0: 756e 6374 696f 6e20 7468 6174 2072 656d unction that rem │ │ │ │ +000240b0: 6f76 6573 206d 756c 7469 706c 6963 6974 oves multiplicit │ │ │ │ +000240c0: 6965 7320 6f66 206e 756d 6265 7273 2069 ies of numbers i │ │ │ │ +000240d0: 6e20 6120 6c69 7374 2075 7020 746f 2061 n a list up to a │ │ │ │ +000240e0: 2074 6f6c 6572 616e 6365 2e0a 2a2a 2a2a tolerance..**** │ │ │ │ 000240f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00024150: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00024160: 2020 2020 7261 6469 6361 6c4c 6973 7428 radicalList( │ │ │ │ -00024170: 4c69 7374 2c4e 756d 6265 7229 0a20 2020 List,Number). │ │ │ │ -00024180: 2020 2020 2072 6164 6963 616c 4c69 7374 radicalList │ │ │ │ -00024190: 284c 6973 7429 0a20 202a 2049 6e70 7574 (List). * Input │ │ │ │ -000241a0: 733a 0a20 2020 2020 202a 204c 2c20 6120 s:. * L, a │ │ │ │ -000241b0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -000241c0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -000241d0: 2041 206c 6973 7420 6f66 2063 6f6d 706c A list of compl │ │ │ │ -000241e0: 6578 206f 7220 7265 616c 0a20 2020 2020 ex or real. │ │ │ │ -000241f0: 2020 206e 756d 6265 7273 2e0a 2020 2020 numbers.. │ │ │ │ -00024200: 2020 2a20 4e2c 2061 202a 6e6f 7465 206e * N, a *note n │ │ │ │ -00024210: 756d 6265 723a 2028 4d61 6361 756c 6179 umber: (Macaulay │ │ │ │ -00024220: 3244 6f63 294e 756d 6265 722c 2c20 4120 2Doc)Number,, A │ │ │ │ -00024230: 736d 616c 6c20 7265 616c 206e 756d 6265 small real numbe │ │ │ │ -00024240: 722e 0a0a 4465 7363 7269 7074 696f 6e0a r...Description. │ │ │ │ -00024250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00024260: 7320 6f75 7470 7574 7320 6120 7375 626c s outputs a subl │ │ │ │ -00024270: 6973 7420 6f66 2063 6f6d 706c 6578 206f ist of complex o │ │ │ │ -00024280: 7220 7265 616c 206e 756d 6265 7273 2074 r real numbers t │ │ │ │ -00024290: 6861 7420 616c 6c20 6861 7665 2064 6973 hat all have dis │ │ │ │ -000242a0: 7469 6e63 7420 6e6f 726d 730a 7570 2074 tinct norms.up t │ │ │ │ -000242b0: 6f20 7468 6520 746f 6c65 7261 6e63 6520 o the tolerance │ │ │ │ -000242c0: 4e20 2864 6566 6175 6c74 2069 7320 3165 N (default is 1e │ │ │ │ -000242d0: 2d31 3029 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d -10)...+-------- │ │ │ │ +00024140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024150: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00024160: 2020 2020 2020 2072 6164 6963 616c 4c69 radicalLi │ │ │ │ +00024170: 7374 284c 6973 742c 4e75 6d62 6572 290a st(List,Number). │ │ │ │ +00024180: 2020 2020 2020 2020 7261 6469 6361 6c4c radicalL │ │ │ │ +00024190: 6973 7428 4c69 7374 290a 2020 2a20 496e ist(List). * In │ │ │ │ +000241a0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ +000241b0: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ +000241c0: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ +000241d0: 742c 2c20 4120 6c69 7374 206f 6620 636f t,, A list of co │ │ │ │ +000241e0: 6d70 6c65 7820 6f72 2072 6561 6c0a 2020 mplex or real. │ │ │ │ +000241f0: 2020 2020 2020 6e75 6d62 6572 732e 0a20 numbers.. │ │ │ │ +00024200: 2020 2020 202a 204e 2c20 6120 2a6e 6f74 * N, a *not │ │ │ │ +00024210: 6520 6e75 6d62 6572 3a20 284d 6163 6175 e number: (Macau │ │ │ │ +00024220: 6c61 7932 446f 6329 4e75 6d62 6572 2c2c lay2Doc)Number,, │ │ │ │ +00024230: 2041 2073 6d61 6c6c 2072 6561 6c20 6e75 A small real nu │ │ │ │ +00024240: 6d62 6572 2e0a 0a44 6573 6372 6970 7469 mber...Descripti │ │ │ │ +00024250: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00024260: 5468 6973 206f 7574 7075 7473 2061 2073 This outputs a s │ │ │ │ +00024270: 7562 6c69 7374 206f 6620 636f 6d70 6c65 ublist of comple │ │ │ │ +00024280: 7820 6f72 2072 6561 6c20 6e75 6d62 6572 x or real number │ │ │ │ +00024290: 7320 7468 6174 2061 6c6c 2068 6176 6520 s that all have │ │ │ │ +000242a0: 6469 7374 696e 6374 206e 6f72 6d73 0a75 distinct norms.u │ │ │ │ +000242b0: 7020 746f 2074 6865 2074 6f6c 6572 616e p to the toleran │ │ │ │ +000242c0: 6365 204e 2028 6465 6661 756c 7420 6973 ce N (default is │ │ │ │ +000242d0: 2031 652d 3130 292e 0a0a 2b2d 2d2d 2d2d 1e-10)...+----- │ │ │ │ 000242e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00024300: 6931 203a 2072 6164 6963 616c 4c69 7374 i1 : radicalList │ │ │ │ -00024310: 287b 322e 3030 302c 312e 3939 397d 2920 ({2.000,1.999}) │ │ │ │ -00024320: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000242f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024300: 2b0a 7c69 3120 3a20 7261 6469 6361 6c4c +.|i1 : radicalL │ │ │ │ +00024310: 6973 7428 7b32 2e30 3030 2c31 2e39 3939 ist({2.000,1.999 │ │ │ │ +00024320: 7d29 2020 2020 2020 7c0a 7c20 2020 2020 }) |.| │ │ │ │ 00024330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024340: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024350: 6f31 203d 207b 322c 2031 2e39 3939 7d20 o1 = {2, 1.999} │ │ │ │ -00024360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024370: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00024340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024350: 7c0a 7c6f 3120 3d20 7b32 2c20 312e 3939 |.|o1 = {2, 1.99 │ │ │ │ +00024360: 397d 2020 2020 2020 2020 2020 2020 2020 9} │ │ │ │ +00024370: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00024380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024390: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000243a0: 6f31 203a 204c 6973 7420 2020 2020 2020 o1 : List │ │ │ │ +00024390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000243a0: 7c0a 7c6f 3120 3a20 4c69 7374 2020 2020 |.|o1 : List │ │ │ │ 000243b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000243c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000243c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000243d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -000243f0: 6932 203a 2072 6164 6963 616c 4c69 7374 i2 : radicalList │ │ │ │ -00024400: 287b 322e 3030 302c 312e 3939 397d 2c31 ({2.000,1.999},1 │ │ │ │ -00024410: 652d 3130 297c 0a7c 2020 2020 2020 2020 e-10)|.| │ │ │ │ +000243e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000243f0: 2b0a 7c69 3220 3a20 7261 6469 6361 6c4c +.|i2 : radicalL │ │ │ │ +00024400: 6973 7428 7b32 2e30 3030 2c31 2e39 3939 ist({2.000,1.999 │ │ │ │ +00024410: 7d2c 3165 2d31 3029 7c0a 7c20 2020 2020 },1e-10)|.| │ │ │ │ 00024420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024430: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024440: 6f32 203d 207b 322c 2031 2e39 3939 7d20 o2 = {2, 1.999} │ │ │ │ -00024450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024460: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00024430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024440: 7c0a 7c6f 3220 3d20 7b32 2c20 312e 3939 |.|o2 = {2, 1.99 │ │ │ │ +00024450: 397d 2020 2020 2020 2020 2020 2020 2020 9} │ │ │ │ +00024460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00024470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024480: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024490: 6f32 203a 204c 6973 7420 2020 2020 2020 o2 : List │ │ │ │ +00024480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024490: 7c0a 7c6f 3220 3a20 4c69 7374 2020 2020 |.|o2 : List │ │ │ │ 000244a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000244b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000244b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000244c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -000244e0: 6933 203a 2072 6164 6963 616c 4c69 7374 i3 : radicalList │ │ │ │ -000244f0: 287b 322e 3030 302c 312e 3939 397d 2c31 ({2.000,1.999},1 │ │ │ │ -00024500: 652d 3229 207c 0a7c 2020 2020 2020 2020 e-2) |.| │ │ │ │ +000244d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000244e0: 2b0a 7c69 3320 3a20 7261 6469 6361 6c4c +.|i3 : radicalL │ │ │ │ +000244f0: 6973 7428 7b32 2e30 3030 2c31 2e39 3939 ist({2.000,1.999 │ │ │ │ +00024500: 7d2c 3165 2d32 2920 7c0a 7c20 2020 2020 },1e-2) |.| │ │ │ │ 00024510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024530: 6f33 203d 207b 327d 2020 2020 2020 2020 o3 = {2} │ │ │ │ +00024520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024530: 7c0a 7c6f 3320 3d20 7b32 7d20 2020 2020 |.|o3 = {2} │ │ │ │ 00024540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024550: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00024550: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00024560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024580: 6f33 203a 204c 6973 7420 2020 2020 2020 o3 : List │ │ │ │ +00024570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024580: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020 |.|o3 : List │ │ │ │ 00024590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000245a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000245a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000245b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -000245d0: 5761 7973 2074 6f20 7573 6520 7261 6469 Ways to use radi │ │ │ │ -000245e0: 6361 6c4c 6973 743a 0a3d 3d3d 3d3d 3d3d calList:.======= │ │ │ │ +000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245d0: 2b0a 0a57 6179 7320 746f 2075 7365 2072 +..Ways to use r │ │ │ │ +000245e0: 6164 6963 616c 4c69 7374 3a0a 3d3d 3d3d adicalList:.==== │ │ │ │ 000245f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00024600: 3d0a 0a20 202a 2022 7261 6469 6361 6c4c =.. * "radicalL │ │ │ │ -00024610: 6973 7428 4c69 7374 2922 0a20 202a 2022 ist(List)". * " │ │ │ │ -00024620: 7261 6469 6361 6c4c 6973 7428 4c69 7374 radicalList(List │ │ │ │ -00024630: 2c4e 756d 6265 7229 220a 0a46 6f72 2074 ,Number)"..For t │ │ │ │ -00024640: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00024650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00024660: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00024670: 7465 2072 6164 6963 616c 4c69 7374 3a20 te radicalList: │ │ │ │ -00024680: 7261 6469 6361 6c4c 6973 742c 2069 7320 radicalList, is │ │ │ │ -00024690: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -000246a0: 756e 6374 696f 6e20 7769 7468 0a6f 7074 unction with.opt │ │ │ │ -000246b0: 696f 6e73 3a20 284d 6163 6175 6c61 7932 ions: (Macaulay2 │ │ │ │ -000246c0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -000246d0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ -000246e0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00024600: 3d3d 3d3d 0a0a 2020 2a20 2272 6164 6963 ====.. * "radic │ │ │ │ +00024610: 616c 4c69 7374 284c 6973 7429 220a 2020 alList(List)". │ │ │ │ +00024620: 2a20 2272 6164 6963 616c 4c69 7374 284c * "radicalList(L │ │ │ │ +00024630: 6973 742c 4e75 6d62 6572 2922 0a0a 466f ist,Number)"..Fo │ │ │ │ +00024640: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00024650: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00024660: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00024670: 2a6e 6f74 6520 7261 6469 6361 6c4c 6973 *note radicalLis │ │ │ │ +00024680: 743a 2072 6164 6963 616c 4c69 7374 2c20 t: radicalList, │ │ │ │ +00024690: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +000246a0: 6420 6675 6e63 7469 6f6e 2077 6974 680a d function with. │ │ │ │ +000246b0: 6f70 7469 6f6e 733a 2028 4d61 6361 756c options: (Macaul │ │ │ │ +000246c0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +000246d0: 6374 696f 6e57 6974 684f 7074 696f 6e73 ctionWithOptions │ │ │ │ +000246e0: 2c2e 0a0a 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024730: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00024740: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00024750: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00024760: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00024770: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ -00024780: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00024790: 6163 6b61 6765 732f 4265 7274 696e 692e ackages/Bertini. │ │ │ │ -000247a0: 6d32 3a0a 3337 3032 3a30 2e0a 1f0a 4669 m2:.3702:0....Fi │ │ │ │ -000247b0: 6c65 3a20 4265 7274 696e 692e 696e 666f le: Bertini.info │ │ │ │ -000247c0: 2c20 4e6f 6465 3a20 7374 6f72 6542 4d32 , Node: storeBM2 │ │ │ │ -000247d0: 4669 6c65 732c 204e 6578 743a 2073 7562 Files, Next: sub │ │ │ │ -000247e0: 506f 696e 742c 2050 7265 763a 2072 6164 Point, Prev: rad │ │ │ │ -000247f0: 6963 616c 4c69 7374 2c20 5570 3a20 546f icalList, Up: To │ │ │ │ -00024800: 700a 0a73 746f 7265 424d 3246 696c 6573 p..storeBM2Files │ │ │ │ -00024810: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a .*************.. │ │ │ │ -00024820: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00024830: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00024840: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00024850: 7420 7374 6f72 6542 4d32 4669 6c65 7320 t storeBM2Files │ │ │ │ -00024860: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -00024870: 7461 7469 6f6e 2920 6973 2061 202a 6e6f tation) is a *no │ │ │ │ -00024880: 7465 2073 7472 696e 673a 0a28 4d61 6361 te string:.(Maca │ │ │ │ -00024890: 756c 6179 3244 6f63 2953 7472 696e 672c ulay2Doc)String, │ │ │ │ -000248a0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +00024730: 2d2d 2d0a 0a54 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6e6f 6d69 616c 206f , A polynomial o │ │ │ │ -00024a80: 7220 6120 6d61 7472 6978 2e0a 2020 2020 r a matrix.. │ │ │ │ -00024a90: 2020 2a20 762c 2061 202a 6e6f 7465 206c * v, a *note l │ │ │ │ -00024aa0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -00024ab0: 6f63 294c 6973 742c 2c20 4c69 7374 206f oc)List,, List o │ │ │ │ -00024ac0: 6620 7661 7269 6162 6c65 7320 7468 6174 f variables that │ │ │ │ -00024ad0: 2077 6520 7769 6c6c 2062 650a 2020 2020 we will be. │ │ │ │ -00024ae0: 2020 2020 6576 616c 7561 7465 6420 6174 evaluated at │ │ │ │ -00024af0: 2074 6865 2070 6f69 6e74 2e0a 2020 2020 the point.. │ │ │ │ -00024b00: 2020 2a20 702c 2061 202a 6e6f 7465 2074 * p, a *note t │ │ │ │ -00024b10: 6869 6e67 3a20 284d 6163 6175 6c61 7932 hing: (Macaulay2 │ │ │ │ -00024b20: 446f 6329 5468 696e 672c 2c20 4120 706f Doc)Thing,, A po │ │ │ │ -00024b30: 696e 7420 6f72 2061 206c 6973 7420 6f66 int or a list of │ │ │ │ -00024b40: 0a20 2020 2020 2020 2063 6f6f 7264 696e . coordin │ │ 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7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -00024c30: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -00024c40: 6661 6c73 652c 200a 2020 2020 2020 2a20 false, . * │ │ │ │ -00024c50: 5375 6249 6e74 6f43 4320 286d 6973 7369 SubIntoCC (missi │ │ │ │ -00024c60: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -00024c70: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -00024c80: 7420 7661 6c75 6520 6661 6c73 652c 200a t value false, . │ │ │ │ -00024c90: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -00024ca0: 3d3d 3d3d 3d3d 3d3d 0a0a 4576 616c 7561 ========..Evalua │ │ │ │ -00024cb0: 7465 2066 2061 7420 6120 706f 696e 742e te f at a point. │ │ │ │ -00024cc0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00024a10: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00024a20: 0a20 2020 2020 2020 2073 7562 506f 696e . subPoin │ │ │ │ +00024a30: 7428 662c 762c 7029 0a20 202a 2049 6e70 t(f,v,p). * Inp │ │ │ │ +00024a40: 7574 733a 0a20 2020 2020 202a 2066 2c20 uts:. * f, │ │ │ │ +00024a50: 6120 2a6e 6f74 6520 7468 696e 673a 2028 a *note thing: ( │ │ │ │ +00024a60: 4d61 6361 756c 6179 3244 6f63 2954 6869 Macaulay2Doc)Thi │ │ │ │ +00024a70: 6e67 2c2c 2041 2070 6f6c 796e 6f6d 6961 ng,, A polynomia │ │ │ │ +00024a80: 6c20 6f72 2061 206d 6174 7269 782e 0a20 l or a matrix.. │ │ │ │ +00024a90: 2020 2020 202a 2076 2c20 6120 2a6e 6f74 * v, a *not │ │ │ │ +00024aa0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00024ab0: 7932 446f 6329 4c69 7374 2c2c 204c 6973 y2Doc)List,, Lis │ │ │ │ +00024ac0: 7420 6f66 2076 6172 6961 626c 6573 2074 t of variables t │ │ │ │ +00024ad0: 6861 7420 7765 2077 696c 6c20 6265 0a20 hat we will be. │ │ │ │ +00024ae0: 2020 2020 2020 2065 7661 6c75 6174 6564 evaluated │ │ │ │ +00024af0: 2061 7420 7468 6520 706f 696e 742e 0a20 at the point.. │ │ │ │ +00024b00: 2020 2020 202a 2070 2c20 6120 2a6e 6f74 * p, a *not │ │ │ │ +00024b10: 6520 7468 696e 673a 2028 4d61 6361 756c e thing: (Macaul │ │ │ │ +00024b20: 6179 3244 6f63 2954 6869 6e67 2c2c 2041 ay2Doc)Thing,, A │ │ │ │ +00024b30: 2070 6f69 6e74 206f 7220 6120 6c69 7374 point or a list │ │ │ │ +00024b40: 206f 660a 2020 2020 2020 2020 636f 6f72 of. coor │ │ │ │ +00024b50: 6469 6e61 7465 7320 6f72 2061 206d 6174 dinates or a mat │ │ │ │ +00024b60: 7269 782e 0a20 202a 202a 6e6f 7465 204f rix.. * *note O │ │ │ │ +00024b70: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +00024b80: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +00024b90: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +00024ba0: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +00024bb0: 7473 2c3a 0a20 2020 2020 202a 204d 3250 ts,:. * M2P │ │ │ │ +00024bc0: 7265 6369 7369 6f6e 2028 6d69 7373 696e recision (missin │ │ │ │ +00024bd0: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +00024be0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00024bf0: 2076 616c 7565 2035 332c 200a 2020 2020 value 53, . │ │ │ │ +00024c00: 2020 2a20 5370 6563 6966 7956 6172 6961 * SpecifyVaria │ │ │ │ +00024c10: 626c 6573 2028 6d69 7373 696e 6720 646f bles (missing do │ │ │ │ +00024c20: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ +00024c30: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +00024c40: 7565 2066 616c 7365 2c20 0a20 2020 2020 ue false, . │ │ │ │ +00024c50: 202a 2053 7562 496e 746f 4343 2028 6d69 * SubIntoCC (mi │ │ │ │ +00024c60: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +00024c70: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +00024c80: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ +00024c90: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +00024ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a45 7661 ===========..Eva │ │ │ │ +00024cb0: 6c75 6174 6520 6620 6174 2061 2070 6f69 luate f at a 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2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e20: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00024e20: 2020 2020 2020 2020 2020 207c 0a2b 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 2b0a 7c69 3220 3a20 663d 7a2a ----+.|i2 : f=z* │ │ │ │ -00024e70: 782b 7920 2020 2020 2020 2020 2020 2020 x+y │ │ │ │ +00024e60: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 -------+.|i2 : f │ │ │ │ +00024e70: 3d7a 2a78 2b79 2020 2020 2020 2020 2020 =z*x+y │ │ │ │ 00024e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024ea0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00024ee0: 3220 3d20 782a 7a20 2b20 7920 2020 2020 2 = x*z + y │ │ │ │ +00024ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00024ee0: 0a7c 6f32 203d 2078 2a7a 202b 2079 2020 .|o2 = x*z + y │ │ │ │ 00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00024f10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00024f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f50: 2020 2020 7c0a 7c6f 3220 3a20 5220 2020 |.|o2 : R │ │ │ │ +00024f50: 2020 2020 2020 207c 0a7c 6f32 203a 2052 |.|o2 : R │ │ │ │ 00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f80: 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ 00025020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025040: 2020 2020 7c0a 7c6f 3320 3d20 2e31 7a20 |.|o3 = .1z │ │ │ │ -00025050: 2b20 2e32 2020 2020 2020 2020 2020 2020 + .2 │ │ │ │ +00025040: 2020 2020 2020 207c 0a7c 6f33 203d 202e |.|o3 = . │ │ │ │ +00025050: 317a 202b 202e 3220 2020 2020 2020 2020 1z + .2 │ │ │ │ 00025060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025080: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025080: 2020 207c 0a7c 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 7c0a 7c6f |.|o │ │ │ │ -000250c0: 3320 3a20 5220 2020 2020 2020 2020 2020 3 : R │ │ │ │ +000250b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000250c0: 0a7c 6f33 203a 2052 2020 2020 2020 2020 .|o3 : R │ │ │ │ 000250d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000250f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00025100: 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3420 3a20 7375 6250 ----+.|i4 : subP │ │ │ │ -00025140: 6f69 6e74 2866 2c7b 782c 792c 7a7d 2c7b oint(f,{x,y,z},{ │ │ │ │ -00025150: 2e31 2c2e 322c 2e33 7d2c 5370 6563 6966 .1,.2,.3},Specif │ │ │ │ -00025160: 7956 6172 6961 626c 6573 3d3e 7b79 7d29 yVariables=>{y}) │ │ │ │ -00025170: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2073 -------+.|i4 : s │ │ │ │ +00025140: 7562 506f 696e 7428 662c 7b78 2c79 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2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000252c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000252d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252e0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 523d ------+.|i5 : R= │ │ │ │ -000252f0: 4343 5f32 3030 5b78 2c79 2c7a 5d20 2020 CC_200[x,y,z] │ │ │ │ +000252e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +000252f0: 2052 3d43 435f 3230 305b 782c 792c 7a5d R=CC_200[x,y,z] │ │ │ │ 00025300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025330: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025330: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00025340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025370: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00025380: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +00025370: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025380: 6f35 203d 2052 2020 2020 2020 2020 2020 o5 = R │ │ │ │ 00025390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000253c0: 2020 2020 2020 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00025410: 7c6f 3520 3a20 506f 6c79 6e6f 6d69 616c |o5 : Polynomial │ │ │ │ -00025420: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025410: 207c 0a7c 6f35 203a 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---------------- │ │ │ │ -00025650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00025660: 7c69 3720 3a20 7375 6250 6f69 6e74 2866 |i7 : subPoint(f │ │ │ │ -00025670: 2c7b 782c 792c 7a7d 2c7b 2e31 2c2e 322c ,{x,y,z},{.1,.2, │ │ │ │ -00025680: 2e33 7d2c 5375 6249 6e74 6f43 433d 3e74 .3},SubIntoCC=>t │ │ │ │ -00025690: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ -000256a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00025650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025660: 2d2b 0a7c 6937 203a 2073 7562 506f 696e -+.|i7 : subPoin │ │ │ │ +00025670: 7428 662c 7b78 2c79 2c7a 7d2c 7b2e 312c t(f,{x,y,z},{.1, │ │ │ │ +00025680: 2e32 2c2e 337d 2c53 7562 496e 746f 4343 .2,.3},SubIntoCC │ │ │ │ +00025690: 3d3e 7472 7565 2920 2020 2020 2020 2020 =>true) │ │ │ │ +000256a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 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 7c0a 7c6f 3720 3d20 2e32 3320 2020 |.|o7 = .23 │ │ │ │ +000256f0: 2020 2020 207c 0a7c 6f37 203d 202e 3233 |.|o7 = .23 │ │ │ │ 00025700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00025740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00025740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00025750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025780: 2020 2020 2020 7c0a 7c6f 3720 3a20 4343 |.|o7 : CC │ │ │ │ -00025790: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ -000257a0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ +00025780: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +00025790: 2043 4320 286f 6620 7072 6563 6973 696f CC (of precisio │ │ │ │ +000257a0: 6e20 3533 2920 2020 2020 2020 2020 2020 n 53) │ │ │ │ 000257b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000257c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000257d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000257e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000257f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025810: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ -00025820: 3a20 7375 6250 6f69 6e74 2866 2c7b 782c : subPoint(f,{x, │ │ │ │ -00025830: 792c 7a7d 2c7b 2e31 3233 3435 3637 3839 y,z},{.123456789 │ │ │ │ -00025840: 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+00025870: 2020 2020 2020 2030 2c31 7d2c 5375 6249 0,1},SubI │ │ │ │ +00025880: 6e74 6f43 433d 3e74 7275 652c 4d32 5072 ntoCC=>true,M2Pr │ │ │ │ +00025890: 6563 6973 696f 6e3d 3e32 3030 2920 2020 ecision=>200) │ │ │ │ +000258a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000258b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000258d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000258e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258f0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20 |.|o8 = │ │ │ │ -00025900: 2e31 3233 3435 3637 3839 3031 3233 3435 .123456789012345 │ │ │ │ -00025910: 3637 3839 3031 3233 3435 3637 3839 3031 6789012345678901 │ │ │ │ -00025920: 3233 3435 3637 3839 2020 2020 2020 2020 23456789 │ │ │ │ +000258f0: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ +00025900: 203d 202e 3132 3334 3536 3738 3930 3132 = .123456789012 │ │ │ │ +00025910: 3334 3536 3738 3930 3132 3334 3536 3738 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6250 Ways to use subP │ │ │ │ -00025aa0: 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d oint:.========== │ │ │ │ -00025ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00025ac0: 2022 7375 6250 6f69 6e74 2854 6869 6e67 "subPoint(Thing │ │ │ │ -00025ad0: 2c4c 6973 742c 5468 696e 6729 220a 0a46 ,List,Thing)"..F │ │ │ │ -00025ae0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00025af0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00025b00: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00025b10: 202a 6e6f 7465 2073 7562 506f 696e 743a *note subPoint: │ │ │ │ -00025b20: 2073 7562 506f 696e 742c 2069 7320 6120 subPoint, is a │ │ │ │ -00025b30: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ -00025b40: 6374 696f 6e20 7769 7468 206f 7074 696f ction with optio │ │ │ │ -00025b50: 6e73 3a0a 284d 6163 6175 6c61 7932 446f ns:.(Macaulay2Do │ │ │ │ -00025b60: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -00025b70: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ -00025b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a20: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +00025a30: 3d3d 3d0a 0a57 6865 6e20 5375 6249 6e74 ===..When SubInt │ │ │ │ +00025a40: 6f43 4320 6973 2073 6574 2074 6f20 7472 oCC is set to tr │ │ │ │ +00025a50: 7565 2074 6865 6e20 756e 7365 7420 7661 ue then unset va │ │ │ │ +00025a60: 7269 6162 6c65 7320 7769 6c6c 2062 6520 riables will be │ │ │ │ +00025a70: 7365 7420 746f 207a 6572 6f20 6f72 0a75 set to zero or.u │ │ │ │ +00025a80: 6e65 7870 6563 7465 6420 7661 6c75 6573 nexpected values │ │ │ │ +00025a90: 2e0a 0a57 6179 7320 746f 2075 7365 2073 ...Ways to use s │ │ │ │ +00025aa0: 7562 506f 696e 743a 0a3d 3d3d 3d3d 3d3d ubPoint:.======= │ │ │ │ +00025ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00025ac0: 2020 2a20 2273 7562 506f 696e 7428 5468 * "subPoint(Th │ │ │ │ +00025ad0: 696e 672c 4c69 7374 2c54 6869 6e67 2922 ing,List,Thing)" │ │ │ │ +00025ae0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +00025af0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +00025b00: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +00025b10: 6563 7420 2a6e 6f74 6520 7375 6250 6f69 ect *note subPoi │ │ │ │ +00025b20: 6e74 3a20 7375 6250 6f69 6e74 2c20 6973 nt: subPoint, is │ │ │ │ +00025b30: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00025b40: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ +00025b50: 7469 6f6e 733a 0a28 4d61 6361 756c 6179 tions:.(Macaulay │ │ │ │ +00025b60: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +00025b70: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +00025b80: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 00025b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00025bd0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00025be0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00025bf0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00025c00: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00025c10: 6c61 7932 2d31 2e32 352e 3036 2b64 732f lay2-1.25.06+ds/ │ │ │ │ -00025c20: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00025c30: 6b61 6765 732f 4265 7274 696e 692e 6d32 kages/Bertini.m2 │ │ │ │ -00025c40: 3a0a 3337 3838 3a30 2e0a 1f0a 4669 6c65 :.3788:0....File │ │ │ │ -00025c50: 3a20 4265 7274 696e 692e 696e 666f 2c20 : Bertini.info, │ │ │ │ -00025c60: 4e6f 6465 3a20 546f 7044 6972 6563 746f Node: TopDirecto │ │ │ │ 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│ │ │ │ +00025d30: 2020 2020 2020 6265 7274 696e 6950 6172 bertiniPar │ │ │ │ +00025d40: 616d 6574 6572 486f 6d6f 746f 7079 282e ameterHomotopy(. │ │ │ │ +00025d50: 2e2e 2c54 6f70 4469 7265 6374 6f72 793d ..,TopDirectory= │ │ │ │ +00025d60: 3e53 7472 696e 6729 0a20 2020 2020 2020 >String). │ │ │ │ +00025d70: 2062 6572 7469 6e69 5a65 726f 4469 6d53 bertiniZeroDimS │ │ │ │ +00025d80: 6f6c 7665 282e 2e2e 2c54 6f70 4469 7265 olve(...,TopDire │ │ │ │ +00025d90: 6374 6f72 793d 3e53 7472 696e 6729 0a20 ctory=>String). │ │ │ │ +00025da0: 2020 2020 2020 2062 6572 7469 6e69 5573 bertiniUs │ │ │ │ +00025db0: 6572 486f 6d6f 746f 7079 282e 2e2e 2c54 erHomotopy(...,T │ │ │ │ +00025dc0: 6f70 4469 7265 6374 6f72 793d 3e53 7472 opDirectory=>Str │ │ │ │ +00025dd0: 696e 6729 0a0a 4465 7363 7269 7074 696f ing)..Descriptio │ │ │ │ +00025de0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 n.===========..T │ │ │ │ +00025df0: 6869 7320 6f70 7469 6f6e 2073 7065 6369 his option speci │ │ │ │ +00025e00: 6669 6573 2061 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ay2Doc)Symbol,.. │ │ │ │ -00025fb0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00025e90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00025ea0: 2262 6572 7469 6e69 5061 7261 6d65 7465 "bertiniParamete │ │ │ │ +00025eb0: 7248 6f6d 6f74 6f70 7928 2e2e 2e2c 546f rHomotopy(...,To │ │ │ │ +00025ec0: 7044 6972 6563 746f 7279 3d3e 2e2e 2e29 pDirectory=>...) │ │ │ │ +00025ed0: 220a 2020 2a20 2262 6572 7469 6e69 5573 ". * "bertiniUs │ │ │ │ +00025ee0: 6572 486f 6d6f 746f 7079 282e 2e2e 2c54 erHomotopy(...,T │ │ │ │ +00025ef0: 6f70 4469 7265 6374 6f72 793d 3e2e 2e2e opDirectory=>... │ │ │ │ +00025f00: 2922 0a20 202a 2022 6265 7274 696e 695a )". * "bertiniZ │ │ │ │ +00025f10: 6572 6f44 696d 536f 6c76 6528 2e2e 2e2c eroDimSolve(..., │ │ │ │ +00025f20: 546f 7044 6972 6563 746f 7279 3d3e 2e2e TopDirectory=>.. │ │ │ │ +00025f30: 2e29 220a 0a46 6f72 2074 6865 2070 726f .)"..For the pro │ │ │ │ +00025f40: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00025f50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00025f60: 6f62 6a65 6374 202a 6e6f 7465 2054 6f70 object *note Top │ │ │ │ +00025f70: 4469 7265 6374 6f72 793a 2054 6f70 4469 Directory: TopDi │ │ │ │ +00025f80: 7265 6374 6f72 792c 2069 7320 6120 2a6e rectory, is a *n │ │ │ │ +00025f90: 6f74 6520 7379 6d62 6f6c 3a0a 284d 6163 ote symbol:.(Mac │ │ │ │ +00025fa0: 6175 6c61 7932 446f 6329 5379 6d62 6f6c aulay2Doc)Symbol │ │ │ │ +00025fb0: 2c2e 0a0a 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026000: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00026010: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00026020: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00026030: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00026040: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ -00026050: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00026060: 6163 6b61 6765 732f 4265 7274 696e 692e ackages/Bertini. │ │ │ │ -00026070: 6d32 3a0a 3236 3932 3a30 2e0a 1f0a 4669 m2:.2692:0....Fi │ │ │ │ -00026080: 6c65 3a20 4265 7274 696e 692e 696e 666f le: Bertini.info │ │ │ │ -00026090: 2c20 4e6f 6465 3a20 5573 6552 6567 656e , Node: UseRegen │ │ │ │ -000260a0: 6572 6174 696f 6e2c 204e 6578 743a 2076 eration, Next: v │ │ │ │ -000260b0: 616c 7565 424d 322c 2050 7265 763a 2054 alueBM2, Prev: T │ │ │ │ -000260c0: 6f70 4469 7265 6374 6f72 792c 2055 703a opDirectory, Up: │ │ │ │ -000260d0: 2054 6f70 0a0a 5573 6552 6567 656e 6572 Top..UseRegener │ │ │ │ -000260e0: 6174 696f 6e20 2d2d 2061 6e20 6f70 7469 ation -- an opti │ │ │ │ -000260f0: 6f6e 2073 7065 6369 6679 696e 6720 7768 on specifying wh │ │ │ │ -00026100: 656e 2074 6f20 7573 6520 7265 6765 6e65 en to use regene │ │ │ │ -00026110: 7261 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a ration.********* │ │ │ │ +00026000: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00026010: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00026020: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00026030: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00026040: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ +00026050: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00026060: 322f 7061 636b 6167 6573 2f42 6572 7469 2/packages/Berti │ │ │ │ +00026070: 6e69 2e6d 323a 0a32 3639 323a 302e 0a1f ni.m2:.2692:0... │ │ │ │ +00026080: 0a46 696c 653a 2042 6572 7469 6e69 2e69 .File: Bertini.i │ │ │ │ +00026090: 6e66 6f2c 204e 6f64 653a 2055 7365 5265 nfo, Node: UseRe │ │ │ │ +000260a0: 6765 6e65 7261 7469 6f6e 2c20 4e65 7874 generation, Next │ │ │ │ +000260b0: 3a20 7661 6c75 6542 4d32 2c20 5072 6576 : valueBM2, Prev │ │ │ │ +000260c0: 3a20 546f 7044 6972 6563 746f 7279 2c20 : TopDirectory, │ │ │ │ +000260d0: 5570 3a20 546f 700a 0a55 7365 5265 6765 Up: Top..UseRege │ │ │ │ +000260e0: 6e65 7261 7469 6f6e 202d 2d20 616e 206f neration -- an o │ │ │ │ +000260f0: 7074 696f 6e20 7370 6563 6966 7969 6e67 ption specifying │ │ │ │ +00026100: 2077 6865 6e20 746f 2075 7365 2072 6567 when to use reg │ │ │ │ +00026110: 656e 6572 6174 696f 6e0a 2a2a 2a2a 2a2a eneration.****** │ │ │ │ 00026120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00026130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00026140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00026150: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -00026160: 6765 3a20 0a20 2020 2020 2020 2062 6572 ge: . ber │ │ │ │ -00026170: 7469 6e69 5061 7261 6d65 7465 7248 6f6d tiniParameterHom │ │ │ │ -00026180: 6f74 6f70 7928 2e2e 2e2c 546f 7044 6972 otopy(...,TopDir │ │ │ │ -00026190: 6563 746f 7279 3d3e 5374 7269 6e67 290a ectory=>String). │ │ │ │ -000261a0: 2020 2020 2020 2020 6265 7274 696e 695a bertiniZ │ │ │ │ -000261b0: 6572 6f44 696d 536f 6c76 6528 2e2e 2e2c eroDimSolve(..., │ │ │ │ -000261c0: 546f 7044 6972 6563 746f 7279 3d3e 5374 TopDirectory=>St │ │ │ │ -000261d0: 7269 6e67 290a 2020 2020 2020 2020 6265 ring). be │ │ │ │ -000261e0: 7274 696e 6955 7365 7248 6f6d 6f74 6f70 rtiniUserHomotop │ │ │ │ -000261f0: 7928 2e2e 2e2c 546f 7044 6972 6563 746f y(...,TopDirecto │ │ │ │ -00026200: 7279 3d3e 5374 7269 6e67 290a 0a44 6573 ry=>String)..Des │ │ │ │ -00026210: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00026220: 3d3d 3d3d 0a0a 5468 6973 206f 7074 696f ====..This optio │ │ │ │ -00026230: 6e20 6973 2073 6574 2074 6f20 3120 746f n is set to 1 to │ │ │ │ -00026240: 2068 6176 6520 4265 7274 696e 6920 7573 have Bertini us │ │ │ │ -00026250: 6520 7265 6765 6e65 7261 7469 6f6e 2077 e regeneration w │ │ │ │ -00026260: 6865 6e20 736f 6c76 696e 6720 610a 706f hen solving a.po │ │ │ │ -00026270: 6c79 6e6f 6d69 616c 2073 7973 7465 6d2e lynomial system. │ │ │ │ -00026280: 0a0a 0a0a 4675 6e63 7469 6f6e 7320 7769 ....Functions wi │ │ │ │ -00026290: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -000262a0: 6d65 6e74 206e 616d 6564 2055 7365 5265 ment named UseRe │ │ │ │ -000262b0: 6765 6e65 7261 7469 6f6e 3a0a 3d3d 3d3d generation:.==== │ │ │ │ +00026150: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00026160: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +00026170: 6265 7274 696e 6950 6172 616d 6574 6572 bertiniParameter │ │ │ │ +00026180: 486f 6d6f 746f 7079 282e 2e2e 2c54 6f70 Homotopy(...,Top │ │ │ │ +00026190: 4469 7265 6374 6f72 793d 3e53 7472 696e Directory=>Strin │ │ │ │ +000261a0: 6729 0a20 2020 2020 2020 2062 6572 7469 g). berti │ │ │ │ +000261b0: 6e69 5a65 726f 4469 6d53 6f6c 7665 282e niZeroDimSolve(. │ │ │ │ +000261c0: 2e2e 2c54 6f70 4469 7265 6374 6f72 793d ..,TopDirectory= │ │ │ │ +000261d0: 3e53 7472 696e 6729 0a20 2020 2020 2020 >String). │ │ │ │ +000261e0: 2062 6572 7469 6e69 5573 6572 486f 6d6f bertiniUserHomo │ │ │ │ +000261f0: 746f 7079 282e 2e2e 2c54 6f70 4469 7265 topy(...,TopDire │ │ │ │ +00026200: 6374 6f72 793d 3e53 7472 696e 6729 0a0a ctory=>String).. │ │ │ │ +00026210: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00026220: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6f70 =======..This op │ │ │ │ +00026230: 7469 6f6e 2069 7320 7365 7420 746f 2031 tion is set to 1 │ │ │ │ +00026240: 2074 6f20 6861 7665 2042 6572 7469 6e69 to have Bertini │ │ │ │ +00026250: 2075 7365 2072 6567 656e 6572 6174 696f use regeneratio │ │ │ │ +00026260: 6e20 7768 656e 2073 6f6c 7669 6e67 2061 n when solving a │ │ │ │ +00026270: 0a70 6f6c 796e 6f6d 6961 6c20 7379 7374 .polynomial syst │ │ │ │ +00026280: 656d 2e0a 0a0a 0a46 756e 6374 696f 6e73 em.....Functions │ │ │ │ +00026290: 2077 6974 6820 6f70 7469 6f6e 616c 2061 with optional a │ │ │ │ +000262a0: 7267 756d 656e 7420 6e61 6d65 6420 5573 rgument named Us │ │ │ │ +000262b0: 6552 6567 656e 6572 6174 696f 6e3a 0a3d eRegeneration:.= │ │ │ │ 000262c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000262d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000262e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000262f0: 3d3d 3d0a 0a20 202a 2062 6572 7469 6e69 ===.. * bertini │ │ │ │ -00026300: 5a65 726f 4469 6d53 6f6c 7665 282e 2e2e ZeroDimSolve(... │ │ │ │ -00026310: 2c55 7365 5265 6765 6e65 7261 7469 6f6e ,UseRegeneration │ │ │ │ -00026320: 3d3e 2e2e 2e29 2028 6d69 7373 696e 6720 =>...) 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(missi │ │ │ │ +00026330: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00026340: 290a 0a46 6f72 2074 6865 2070 726f 6772 )..For the progr │ │ │ │ +00026350: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00026360: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00026370: 6a65 6374 202a 6e6f 7465 2055 7365 5265 ject *note UseRe │ │ │ │ +00026380: 6765 6e65 7261 7469 6f6e 3a20 5573 6552 generation: UseR │ │ │ │ +00026390: 6567 656e 6572 6174 696f 6e2c 2069 7320 egeneration, is │ │ │ │ +000263a0: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a0a a *note symbol:. │ │ │ │ +000263b0: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +000263c0: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ 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 ---------------- │ │ │ │ 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763a 2055 7365 5265 6765 6e65 7261 7469 v: UseRegenerati │ │ │ │ -000264e0: 6f6e 2c20 5570 3a20 546f 700a 0a76 616c on, Up: Top..val │ │ │ │ -000264f0: 7565 424d 3220 2d2d 2054 6869 7320 6675 ueBM2 -- This fu │ │ │ │ -00026500: 6e63 7469 6f6e 206d 616b 6573 2061 206e nction makes a n │ │ │ │ -00026510: 756d 6265 7220 696e 2043 4320 6672 6f6d umber in CC from │ │ │ │ -00026520: 2061 2073 7472 696e 672e 0a2a 2a2a 2a2a a string..***** │ │ │ │ +00026410: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00026420: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00026430: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00026440: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00026450: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00026460: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ +00026470: 756c 6179 322f 7061 636b 6167 6573 2f42 ulay2/packages/B │ │ │ │ +00026480: 6572 7469 6e69 2e6d 323a 0a32 3730 373a ertini.m2:.2707: │ │ │ │ +00026490: 302e 0a1f 0a46 696c 653a 2042 6572 7469 0....File: Berti │ │ │ │ +000264a0: 6e69 2e69 6e66 6f2c 204e 6f64 653a 2076 ni.info, Node: v │ │ │ │ +000264b0: 616c 7565 424d 322c 204e 6578 743a 2056 alueBM2, Next: V │ │ │ │ +000264c0: 6172 6961 626c 6520 6772 6f75 7073 2c20 ariable groups, │ │ │ │ +000264d0: 5072 6576 3a20 5573 6552 6567 656e 6572 Prev: UseRegener │ │ │ │ +000264e0: 6174 696f 6e2c 2055 703a 2054 6f70 0a0a ation, Up: Top.. │ │ │ │ +000264f0: 7661 6c75 6542 4d32 202d 2d20 5468 6973 valueBM2 -- This │ │ │ │ +00026500: 2066 756e 6374 696f 6e20 6d61 6b65 7320 function makes │ │ │ │ +00026510: 6120 6e75 6d62 6572 2069 6e20 4343 2066 a number in CC f │ │ │ │ +00026520: 726f 6d20 6120 7374 7269 6e67 2e0a 2a2a rom a string..** │ │ │ │ 00026530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00026540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00026550: 2a2a 2a2a 2a2a 2a2a 2a2a 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-00026630: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ -00026640: 3a0a 2020 2020 2020 2a20 4d32 5072 6563 :. * M2Prec │ │ │ │ -00026650: 6973 696f 6e20 286d 6973 7369 6e67 2064 ision (missing d │ │ │ │ -00026660: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ -00026670: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -00026680: 6c75 6520 3533 2c20 0a0a 4465 7363 7269 lue 53, ..Descri │ │ │ │ -00026690: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -000266a0: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e =..This function │ │ │ │ -000266b0: 2074 616b 6520 6120 7374 7269 6e67 2072 take a string r │ │ │ │ -000266c0: 6570 7265 7365 6e74 696e 6720 6120 636f epresenting a co │ │ │ │ -000266d0: 6f72 6469 6e61 7465 2069 6e20 6120 4265 ordinate in a Be │ │ │ │ -000266e0: 7274 696e 6920 736f 6c75 7469 6f6e 730a rtini solutions. │ │ │ │ -000266f0: 6669 6c65 206f 7220 7061 7261 6d65 7465 file or paramete │ │ │ │ -00026700: 7220 6669 6c65 2061 6e64 206d 616b 6573 r file and makes │ │ │ │ -00026710: 2061 206e 756d 6265 7220 696e 2043 432e a number in CC. │ │ │ │ -00026720: 2057 6520 6361 6e20 6164 6a75 7374 2074 We can adjust t │ │ │ │ -00026730: 6865 2070 7265 6369 7369 6f6e 0a75 7369 he precision.usi │ │ │ │ -00026740: 6e67 2074 6865 204d 3250 7265 6369 7369 ng the M2Precisi │ │ │ │ -00026750: 6f6e 206f 7074 696f 6e2e 2046 7261 6374 on option. Fract │ │ │ │ -00026760: 696f 6e73 2073 686f 756c 6420 6e6f 7420 ions should not │ │ │ │ -00026770: 6265 2069 6e20 7468 6520 7374 7269 6e67 be in the string │ │ │ │ -00026780: 2073 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d s...+---------- │ │ │ │ +00026560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00026570: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00026580: 2076 616c 7565 424d 3228 7329 0a20 202a valueBM2(s). * │ │ │ │ +00026590: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +000265a0: 2073 2c20 6120 2a6e 6f74 6520 7374 7269 s, a *note stri │ │ │ │ +000265b0: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ +000265c0: 6329 5374 7269 6e67 2c2c 2041 2073 7472 c)String,, A str │ │ │ │ +000265d0: 696e 6720 7468 6174 2067 6976 6573 2061 ing that gives a │ │ │ │ +000265e0: 0a20 2020 2020 2020 2063 6f6f 7264 696e . coordin │ │ │ │ +000265f0: 6174 652e 0a20 202a 202a 6e6f 7465 204f ate.. * *note O │ │ │ │ +00026600: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +00026610: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +00026620: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +00026630: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +00026640: 7473 2c3a 0a20 2020 2020 202a 204d 3250 ts,:. * M2P │ │ │ │ +00026650: 7265 6369 7369 6f6e 2028 6d69 7373 696e recision (missin │ │ │ │ +00026660: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +00026670: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00026680: 2076 616c 7565 2035 332c 200a 0a44 6573 value 53, ..Des │ │ │ │ +00026690: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +000266a0: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374 ====..This funct │ │ │ │ +000266b0: 696f 6e20 7461 6b65 2061 2073 7472 696e ion take a strin │ │ │ │ +000266c0: 6720 7265 7072 6573 656e 7469 6e67 2061 g representing a │ │ │ │ +000266d0: 2063 6f6f 7264 696e 6174 6520 696e 2061 coordinate in a │ │ │ │ +000266e0: 2042 6572 7469 6e69 2073 6f6c 7574 696f Bertini solutio │ │ │ │ +000266f0: 6e73 0a66 696c 6520 6f72 2070 6172 616d ns.file or param │ │ │ │ +00026700: 6574 6572 2066 696c 6520 616e 6420 6d61 eter file and ma │ │ │ │ +00026710: 6b65 7320 6120 6e75 6d62 6572 2069 6e20 kes a number in │ │ │ │ +00026720: 4343 2e20 5765 2063 616e 2061 646a 7573 CC. We can adjus │ │ │ │ +00026730: 7420 7468 6520 7072 6563 6973 696f 6e0a t the precision. │ │ │ │ +00026740: 7573 696e 6720 7468 6520 4d32 5072 6563 using the M2Prec │ │ │ │ +00026750: 6973 696f 6e20 6f70 7469 6f6e 2e20 4672 ision option. Fr │ │ │ │ +00026760: 6163 7469 6f6e 7320 7368 6f75 6c64 206e actions should n │ │ │ │ +00026770: 6f74 2062 6520 696e 2074 6865 2073 7472 ot be in the str │ │ │ │ +00026780: 696e 6720 732e 0a0a 2b2d 2d2d 2d2d 2d2d ing s...+------- │ │ │ │ 00026790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000267a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000267b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000267c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000267d0: 2d2d 2d2b 0a7c 6931 203a 2076 616c 7565 ---+.|i1 : value │ │ │ │ -000267e0: 424d 3228 2231 2e32 3265 2d32 2034 652d BM2("1.22e-2 4e- │ │ │ │ -000267f0: 3522 2920 2020 2020 2020 2020 2020 2020 5") │ │ │ │ +000267d0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7661 ------+.|i1 : va │ │ │ │ +000267e0: 6c75 6542 4d32 2822 312e 3232 652d 3220 lueBM2("1.22e-2 │ │ │ │ +000267f0: 3465 2d35 2229 2020 2020 2020 2020 2020 4e-5") │ │ │ │ 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 |.| │ │ │ │ +00026820: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 │ │ │ │ -00026870: 2020 207c 0a7c 6f31 203d 202e 3031 3232 |.|o1 = .0122 │ │ │ │ -00026880: 2b2e 3030 3030 342a 6969 2020 2020 2020 +.00004*ii │ │ │ │ +00026870: 2020 2020 2020 7c0a 7c6f 3120 3d20 2e30 |.|o1 = .0 │ │ │ │ +00026880: 3132 322b 2e30 3030 3034 2a69 6920 2020 122+.00004*ii │ │ │ │ 00026890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000268a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000268b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000268c0: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 │ │ │ │ -00026910: 2020 207c 0a7c 6f31 203a 2043 4320 286f |.|o1 : CC (o │ │ │ │ -00026920: 6620 7072 6563 6973 696f 6e20 3533 2920 f precision 53) │ │ │ │ -00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026910: 2020 2020 2020 7c0a 7c6f 3120 3a20 4343 |.|o1 : CC │ │ │ │ +00026920: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ +00026930: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 00026940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026960: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026960: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00026970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000269a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000269b0: 2d2d 2d2b 0a7c 6932 203a 2076 616c 7565 ---+.|i2 : value │ │ │ │ -000269c0: 424d 3228 2231 2e32 3220 3465 2d35 2229 BM2("1.22 4e-5") │ │ │ │ -000269d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000269b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7661 ------+.|i2 : va │ │ │ │ +000269c0: 6c75 6542 4d32 2822 312e 3232 2034 652d lueBM2("1.22 4e- │ │ │ │ +000269d0: 3522 2920 2020 2020 2020 2020 2020 2020 5") │ │ │ │ 000269e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000269f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026a00: 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ 00026a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a50: 2020 207c 0a7c 6f32 203d 2031 2e32 322b |.|o2 = 1.22+ │ │ │ │ -00026a60: 2e30 3030 3034 2a69 6920 2020 2020 2020 .00004*ii │ │ │ │ +00026a50: 2020 2020 2020 7c0a 7c6f 3220 3d20 312e |.|o2 = 1. │ │ │ │ +00026a60: 3232 2b2e 3030 3030 342a 6969 2020 2020 22+.00004*ii │ │ │ │ 00026a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026aa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026aa0: 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026af0: 2020 207c 0a7c 6f32 203a 2043 4320 286f |.|o2 : CC (o │ │ │ │ -00026b00: 6620 7072 6563 6973 696f 6e20 3533 2920 f precision 53) │ │ │ │ -00026b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026af0: 2020 2020 2020 7c0a 7c6f 3220 3a20 4343 |.|o2 : CC │ │ │ │ +00026b00: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ +00026b10: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 00026b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026b40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00026b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b90: 2d2d 2d2b 0a7c 6933 203a 2076 616c 7565 ---+.|i3 : value │ │ │ │ -00026ba0: 424d 3228 2231 2e32 3220 3422 2920 2020 BM2("1.22 4") │ │ │ │ +00026b90: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7661 ------+.|i3 : va │ │ │ │ +00026ba0: 6c75 6542 4d32 2822 312e 3232 2034 2229 lueBM2("1.22 4") │ │ │ │ 00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026be0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c30: 2020 207c 0a7c 6f33 203d 2031 2e32 322b |.|o3 = 1.22+ │ │ │ │ -00026c40: 342a 6969 2020 2020 2020 2020 2020 2020 4*ii │ │ │ │ +00026c30: 2020 2020 2020 7c0a 7c6f 3320 3d20 312e |.|o3 = 1. │ │ │ │ +00026c40: 3232 2b34 2a69 6920 2020 2020 2020 2020 22+4*ii │ │ │ │ 00026c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00026c90: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f33 203a 2043 4320 286f |.|o3 : CC (o │ │ │ │ -00026ce0: 6620 7072 6563 6973 696f 6e20 3533 2920 f precision 53) │ │ │ │ -00026cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026cd0: 2020 2020 2020 7c0a 7c6f 3320 3a20 4343 |.|o3 : CC │ │ │ │ +00026ce0: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ +00026cf0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 00026d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026d20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00026d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026d70: 2d2d 2d2b 0a7c 6934 203a 2076 616c 7565 ---+.|i4 : value │ │ │ │ -00026d80: 424d 3228 2231 2e32 3265 2b32 2034 2022 BM2("1.22e+2 4 " │ │ │ │ -00026d90: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00026d70: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7661 ------+.|i4 : va │ │ │ │ +00026d80: 6c75 6542 4d32 2822 312e 3232 652b 3220 lueBM2("1.22e+2 │ │ │ │ +00026d90: 3420 2229 2020 2020 2020 2020 2020 2020 4 ") │ │ │ │ 00026da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e10: 2020 207c 0a7c 6f34 203d 2031 3232 2b34 |.|o4 = 122+4 │ │ │ │ -00026e20: 2a69 6920 2020 2020 2020 2020 2020 2020 *ii │ │ │ │ +00026e10: 2020 2020 2020 7c0a 7c6f 3420 3d20 3132 |.|o4 = 12 │ │ │ │ +00026e20: 322b 342a 6969 2020 2020 2020 2020 2020 2+4*ii │ │ │ │ 00026e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026e60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00026e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026eb0: 2020 207c 0a7c 6f34 203a 2043 4320 286f |.|o4 : CC (o │ │ │ │ -00026ec0: 6620 7072 6563 6973 696f 6e20 3533 2920 f precision 53) │ │ │ │ -00026ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026eb0: 2020 2020 2020 7c0a 7c6f 3420 3a20 4343 |.|o4 : CC │ │ │ │ +00026ec0: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ +00026ed0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 00026ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026f00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026f00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00026f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026f50: 2d2d 2d2b 0a7c 6935 203a 206e 313d 7661 ---+.|i5 : n1=va │ │ │ │ -00026f60: 6c75 6542 4d32 2822 312e 3131 222c 4d32 lueBM2("1.11",M2 │ │ │ │ -00026f70: 5072 6563 6973 696f 6e3d 3e35 3229 2020 Precision=>52) │ │ │ │ -00026f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026f50: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6e31 ------+.|i5 : n1 │ │ │ │ +00026f60: 3d76 616c 7565 424d 3228 2231 2e31 3122 =valueBM2("1.11" │ │ │ │ +00026f70: 2c4d 3250 7265 6369 7369 6f6e 3d3e 3532 ,M2Precision=>52 │ │ │ │ +00026f80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00026f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026fa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 207c 0a7c 6f35 203d 2031 2e31 3120 |.|o5 = 1.11 │ │ │ │ -00027000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026ff0: 2020 2020 2020 7c0a 7c6f 3520 3d20 312e |.|o5 = 1. │ │ │ │ +00027000: 3131 2020 2020 2020 2020 2020 2020 2020 11 │ │ │ │ 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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027040: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 207c 0a7c 6f35 203a 2052 5220 286f |.|o5 : RR (o │ │ │ │ -000270a0: 6620 7072 6563 6973 696f 6e20 3532 2920 f precision 52) │ │ │ │ -000270b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027090: 2020 2020 2020 7c0a 7c6f 3520 3a20 5252 |.|o5 : RR │ │ │ │ +000270a0: 2028 6f66 2070 7265 6369 7369 6f6e 2035 (of precision 5 │ │ │ │ +000270b0: 3229 2020 2020 2020 2020 2020 2020 2020 2) │ │ │ │ 000270c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000270d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000270e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000270e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000270f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027130: 2d2d 2d2b 0a7c 6936 203a 206e 323d 7661 ---+.|i6 : n2=va │ │ │ │ -00027140: 6c75 6542 4d32 2822 312e 3131 222c 4d32 lueBM2("1.11",M2 │ │ │ │ -00027150: 5072 6563 6973 696f 6e3d 3e33 3030 2920 Precision=>300) │ │ │ │ -00027160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027130: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 6e32 ------+.|i6 : n2 │ │ │ │ +00027140: 3d76 616c 7565 424d 3228 2231 2e31 3122 =valueBM2("1.11" │ │ │ │ +00027150: 2c4d 3250 7265 6369 7369 6f6e 3d3e 3330 ,M2Precision=>30 │ │ │ │ +00027160: 3029 2020 2020 2020 2020 2020 2020 2020 0) │ │ │ │ 00027170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027180: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027180: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 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3939 3939 3939 3939 1099999999999999 │ │ │ │ +000273d0: 7035 3265 3120 2020 2020 2020 2020 2020 p52e1 │ │ │ │ 000273e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000273f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027400: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027400: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00027410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027450: 2d2d 2d2b 0a7c 6938 203a 2074 6f45 7874 ---+.|i8 : toExt │ │ │ │ -00027460: 6572 6e61 6c53 7472 696e 6720 6e32 2020 ernalString n2 │ │ │ │ -00027470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027450: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 746f ------+.|i8 : to │ │ │ │ +00027460: 4578 7465 726e 616c 5374 7269 6e67 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0000000000000000 │ │ │ │ 00027530: 3030 3030 3030 3030 3030 3030 3030 3030 0000000000000000 │ │ │ │ -00027540: 3030 307c 0a7c 2020 2020 2030 3030 3030 000|.| 00000 │ │ │ │ -00027550: 3030 3030 3030 3030 3030 3030 3030 3033 0000000000000003 │ │ │ │ -00027560: 7033 3030 6531 2020 2020 2020 2020 2020 p300e1 │ │ │ │ +00027540: 3030 3030 3030 7c0a 7c20 2020 2020 3030 000000|.| 00 │ │ │ │ +00027550: 3030 3030 3030 3030 3030 3030 3030 3030 0000000000000000 │ │ │ │ +00027560: 3030 3370 3330 3065 3120 2020 2020 2020 003p300e1 │ │ │ │ 00027570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027590: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027590: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000275b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000275c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000275d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275e0: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573 ---+..Ways to us │ │ │ │ -000275f0: 6520 7661 6c75 6542 4d32 3a0a 3d3d 3d3d e valueBM2:.==== │ │ │ │ +000275e0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ +000275f0: 2075 7365 2076 616c 7565 424d 323a 0a3d use valueBM2:.= │ │ │ │ 00027600: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027610: 3d0a 0a20 202a 2022 7661 6c75 6542 4d32 =.. * "valueBM2 │ │ │ │ -00027620: 2853 7472 696e 6729 220a 0a46 6f72 2074 (String)"..For t │ │ │ │ -00027630: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00027640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027650: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00027660: 7465 2076 616c 7565 424d 323a 2076 616c te valueBM2: val │ │ │ │ -00027670: 7565 424d 322c 2069 7320 6120 2a6e 6f74 ueBM2, is a *not │ │ │ │ -00027680: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -00027690: 6e20 7769 7468 206f 7074 696f 6e73 3a0a n with options:. │ │ │ │ -000276a0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -000276b0: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -000276c0: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00027610: 3d3d 3d3d 0a0a 2020 2a20 2276 616c 7565 ====.. * "value │ │ │ │ +00027620: 424d 3228 5374 7269 6e67 2922 0a0a 466f BM2(String)"..Fo │ │ │ │ +00027630: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00027640: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00027650: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00027660: 2a6e 6f74 6520 7661 6c75 6542 4d32 3a20 *note valueBM2: │ │ │ │ +00027670: 7661 6c75 6542 4d32 2c20 6973 2061 202a valueBM2, is a * │ │ │ │ +00027680: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +00027690: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e tion with option │ │ │ │ +000276a0: 733a 0a28 4d61 6361 756c 6179 3244 6f63 s:.(Macaulay2Doc │ │ │ │ +000276b0: 294d 6574 686f 6446 756e 6374 696f 6e57 )MethodFunctionW │ │ │ │ +000276c0: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d ithOptions,...-- │ │ │ │ 000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000276f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027710: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00027720: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00027730: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00027740: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00027750: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00027760: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ -00027770: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00027780: 732f 4265 7274 696e 692e 6d32 3a0a 3337 s/Bertini.m2:.37 │ │ │ │ -00027790: 3533 3a30 2e0a 1f0a 4669 6c65 3a20 4265 53:0....File: Be │ │ │ │ -000277a0: 7274 696e 692e 696e 666f 2c20 4e6f 6465 rtini.info, Node │ │ │ │ -000277b0: 3a20 5661 7269 6162 6c65 2067 726f 7570 : Variable group │ │ │ │ -000277c0: 732c 204e 6578 743a 2077 7269 7465 5374 s, Next: writeSt │ │ │ │ -000277d0: 6172 7446 696c 652c 2050 7265 763a 2076 artFile, Prev: v │ │ │ │ -000277e0: 616c 7565 424d 322c 2055 703a 2054 6f70 alueBM2, Up: Top │ │ │ │ -000277f0: 0a0a 5661 7269 6162 6c65 2067 726f 7570 ..Variable group │ │ │ │ -00027800: 7320 2d2d 2061 6e20 6f70 7469 6f6e 2074 s -- an option t │ │ │ │ -00027810: 6f20 6772 6f75 7020 7661 7269 6162 6c65 o group variable │ │ │ │ -00027820: 7320 616e 6420 7573 6520 6d75 6c74 6968 s and use multih │ │ │ │ -00027830: 6f6d 6f67 656e 656f 7573 2068 6f6d 6f74 omogeneous homot │ │ │ │ -00027840: 6f70 6965 730a 2a2a 2a2a 2a2a 2a2a 2a2a opies.********** │ │ │ │ +00027710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00027720: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +00027730: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +00027740: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +00027750: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +00027760: 6179 322d 312e 3235 2e30 362b 6473 2f4d ay2-1.25.06+ds/M │ │ │ │ +00027770: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +00027780: 6167 6573 2f42 6572 7469 6e69 2e6d 323a ages/Bertini.m2: │ │ │ │ +00027790: 0a33 3735 333a 302e 0a1f 0a46 696c 653a .3753:0....File: │ │ │ │ +000277a0: 2042 6572 7469 6e69 2e69 6e66 6f2c 204e Bertini.info, N │ │ │ │ +000277b0: 6f64 653a 2056 6172 6961 626c 6520 6772 ode: Variable gr │ │ │ │ +000277c0: 6f75 7073 2c20 4e65 7874 3a20 7772 6974 oups, Next: writ │ │ │ │ +000277d0: 6553 7461 7274 4669 6c65 2c20 5072 6576 eStartFile, Prev │ │ │ │ +000277e0: 3a20 7661 6c75 6542 4d32 2c20 5570 3a20 : valueBM2, Up: │ │ │ │ +000277f0: 546f 700a 0a56 6172 6961 626c 6520 6772 Top..Variable gr │ │ │ │ +00027800: 6f75 7073 202d 2d20 616e 206f 7074 696f oups -- an optio │ │ │ │ +00027810: 6e20 746f 2067 726f 7570 2076 6172 6961 n to group varia │ │ │ │ +00027820: 626c 6573 2061 6e64 2075 7365 206d 756c bles and use mul │ │ │ │ +00027830: 7469 686f 6d6f 6765 6e65 6f75 7320 686f tihomogeneous ho │ │ │ │ +00027840: 6d6f 746f 7069 6573 0a2a 2a2a 2a2a 2a2a motopies.******* │ │ │ │ 00027850: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027860: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027890: 2a2a 2a2a 2a2a 2a2a 2a0a 0a47 726f 7570 *********..Group │ │ │ │ -000278a0: 696e 6720 7468 6520 7661 7269 6162 6c65 ing the variable │ │ │ │ -000278b0: 7320 6861 7320 4265 7274 696e 6920 736f s has Bertini so │ │ │ │ -000278c0: 6c76 6520 7a65 726f 2064 696d 656e 7369 lve zero dimensi │ │ │ │ -000278d0: 6f6e 616c 2073 7973 7465 6d73 2075 7369 onal systems usi │ │ │ │ -000278e0: 6e67 0a6d 756c 7469 686f 6d6f 6765 6e65 ng.multihomogene │ │ │ │ -000278f0: 6f75 7320 686f 6d6f 746f 7069 6573 2e0a ous homotopies.. │ │ │ │ -00027900: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00027890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4772 ************..Gr │ │ │ │ +000278a0: 6f75 7069 6e67 2074 6865 2076 6172 6961 ouping the varia │ │ │ │ +000278b0: 626c 6573 2068 6173 2042 6572 7469 6e69 bles has Bertini │ │ │ │ +000278c0: 2073 6f6c 7665 207a 6572 6f20 6469 6d65 solve zero dime │ │ │ │ +000278d0: 6e73 696f 6e61 6c20 7379 7374 656d 7320 nsional systems │ │ │ │ +000278e0: 7573 696e 670a 6d75 6c74 6968 6f6d 6f67 using.multihomog │ │ │ │ +000278f0: 656e 656f 7573 2068 6f6d 6f74 6f70 6965 eneous homotopie │ │ │ │ +00027900: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ 00027910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027940: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2043 ---+.|i1 : R = C │ │ │ │ -00027950: 435b 782c 795d 3b20 2020 2020 2020 2020 C[x,y]; │ │ │ │ +00027940: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ +00027950: 3d20 4343 5b78 2c79 5d3b 2020 2020 2020 = CC[x,y]; │ │ │ │ 00027960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027980: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00027980: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -000279d0: 203a 2046 3120 3d20 7b78 2a79 2b31 2c32 : F1 = {x*y+1,2 │ │ │ │ -000279e0: 2a78 2a79 2b33 2a78 2b34 2a79 2b35 7d3b *x*y+3*x+4*y+5}; │ │ │ │ -000279f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000279c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000279d0: 7c69 3220 3a20 4631 203d 207b 782a 792b |i2 : F1 = {x*y+ │ │ │ │ +000279e0: 312c 322a 782a 792b 332a 782b 342a 792b 1,2*x*y+3*x+4*y+ │ │ │ │ +000279f0: 357d 3b20 2020 2020 2020 2020 2020 2020 5}; │ │ │ │ +00027a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00027a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027a50: 2d2d 2d2b 0a7c 6933 203a 2062 6572 7469 ---+.|i3 : berti │ │ │ │ -00027a60: 6e69 5a65 726f 4469 6d53 6f6c 7665 2846 niZeroDimSolve(F │ │ │ │ -00027a70: 312c 2041 6666 5661 7269 6162 6c65 4772 1, AffVariableGr │ │ │ │ -00027a80: 6f75 703d 3e7b 7b78 7d2c 7b79 7d7d 293b oup=>{{x},{y}}); │ │ │ │ -00027a90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00027a50: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6265 ------+.|i3 : be │ │ │ │ +00027a60: 7274 696e 695a 6572 6f44 696d 536f 6c76 rtiniZeroDimSolv │ │ │ │ +00027a70: 6528 4631 2c20 4166 6656 6172 6961 626c e(F1, AffVariabl │ │ │ │ +00027a80: 6547 726f 7570 3d3e 7b7b 787d 2c7b 797d eGroup=>{{x},{y} │ │ │ │ +00027a90: 7d29 3b20 2020 2020 2020 7c0a 2b2d 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 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -00027ae0: 203a 2068 5220 3d43 435b 7830 2c78 312c : hR =CC[x0,x1, │ │ │ │ -00027af0: 7930 2c79 315d 2020 2020 2020 2020 2020 y0,y1] │ │ │ │ +00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00027ae0: 7c69 3420 3a20 6852 203d 4343 5b78 302c |i4 : hR =CC[x0, │ │ │ │ +00027af0: 7831 2c79 302c 7931 5d20 2020 2020 2020 x1,y0,y1] │ │ │ │ 00027b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027b20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00027b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b60: 2020 207c 0a7c 6f34 203d 2068 5220 2020 |.|o4 = hR │ │ │ │ +00027b60: 2020 2020 2020 7c0a 7c6f 3420 3d20 6852 |.|o4 = hR │ │ │ │ 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 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00027ba0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027be0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -00027bf0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -00027c00: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00027be0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027bf0: 7c6f 3420 3a20 506f 6c79 6e6f 6d69 616c |o4 : Polynomial │ │ │ │ +00027c00: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00027c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027c30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00027c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027c30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00027c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c70: 2d2d 2d2b 0a7c 6935 203a 2046 3220 3d20 ---+.|i5 : F2 = │ │ │ │ -00027c80: 7b78 312a 7931 2b78 302a 7930 2c32 2a78 {x1*y1+x0*y0,2*x │ │ │ │ -00027c90: 312a 7931 2b33 2a78 312a 7930 2b34 2a78 1*y1+3*x1*y0+4*x │ │ │ │ -00027ca0: 302a 7931 2b35 2a78 302a 7930 7d3b 2020 0*y1+5*x0*y0}; │ │ │ │ -00027cb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00027c70: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 4632 ------+.|i5 : F2 │ │ │ │ +00027c80: 203d 207b 7831 2a79 312b 7830 2a79 302c = {x1*y1+x0*y0, │ │ │ │ +00027c90: 322a 7831 2a79 312b 332a 7831 2a79 302b 2*x1*y1+3*x1*y0+ │ │ │ │ +00027ca0: 342a 7830 2a79 312b 352a 7830 2a79 307d 4*x0*y1+5*x0*y0} │ │ │ │ +00027cb0: 3b20 2020 2020 2020 2020 7c0a 2b2d 2d2d ; |.+--- │ │ │ │ 00027cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -00027d00: 203a 2062 6572 7469 6e69 5a65 726f 4469 : bertiniZeroDi │ │ │ │ -00027d10: 6d53 6f6c 7665 2846 322c 486f 6d56 6172 mSolve(F2,HomVar │ │ │ │ -00027d20: 6961 626c 6547 726f 7570 3d3e 7b7b 7830 iableGroup=>{{x0 │ │ │ │ -00027d30: 2c78 317d 2c7b 7930 2c79 317d 7d29 3b7c ,x1},{y0,y1}});| │ │ │ │ -00027d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00027d00: 7c69 3620 3a20 6265 7274 696e 695a 6572 |i6 : bertiniZer │ │ │ │ +00027d10: 6f44 696d 536f 6c76 6528 4632 2c48 6f6d oDimSolve(F2,Hom │ │ │ │ +00027d20: 5661 7269 6162 6c65 4772 6f75 703d 3e7b VariableGroup=>{ │ │ │ │ +00027d30: 7b78 302c 7831 7d2c 7b79 302c 7931 7d7d {x0,x1},{y0,y1}} │ │ │ │ +00027d40: 293b 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d );|.+----------- │ │ │ │ 00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d80: 2d2d 2d2b 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ---+.----------- │ │ │ │ +00027d80: 2d2d 2d2d 2d2d 2b0a 2d2d 2d2d 2d2d 2d2d ------+.-------- │ │ │ │ 00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027dd0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00027de0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00027df0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00027e00: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00027e10: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -00027e20: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00027e30: 7932 2f70 6163 6b61 6765 732f 4265 7274 y2/packages/Bert │ │ │ │ -00027e40: 696e 692e 6d32 3a0a 3332 3732 3a30 2e0a ini.m2:.3272:0.. │ │ │ │ -00027e50: 1f0a 4669 6c65 3a20 4265 7274 696e 692e ..File: Bertini. │ │ │ │ -00027e60: 696e 666f 2c20 4e6f 6465 3a20 7772 6974 info, Node: writ │ │ │ │ -00027e70: 6553 7461 7274 4669 6c65 2c20 5072 6576 eStartFile, Prev │ │ │ │ -00027e80: 3a20 5661 7269 6162 6c65 2067 726f 7570 : Variable group │ │ │ │ -00027e90: 732c 2055 703a 2054 6f70 0a0a 7772 6974 s, Up: Top..writ │ │ │ │ -00027ea0: 6553 7461 7274 4669 6c65 202d 2d20 5772 eStartFile -- Wr │ │ │ │ -00027eb0: 6974 6573 2074 6865 206c 6973 7420 6f66 ites the list of │ │ │ │ -00027ec0: 206c 6973 7420 6f66 2063 6f6f 7264 696e list of coordin │ │ │ │ -00027ed0: 6174 6573 2074 6f20 6120 6669 6c65 2074 ates to a file t │ │ │ │ -00027ee0: 6861 7420 4265 7274 696e 6920 6361 6e20 hat Bertini can │ │ │ │ -00027ef0: 7265 6164 2e0a 2a2a 2a2a 2a2a 2a2a 2a2a read..********** │ │ │ │ +00027dd0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00027de0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00027df0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00027e00: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00027e10: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00027e20: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ +00027e30: 756c 6179 322f 7061 636b 6167 6573 2f42 ulay2/packages/B │ │ │ │ +00027e40: 6572 7469 6e69 2e6d 323a 0a33 3237 323a ertini.m2:.3272: │ │ │ │ +00027e50: 302e 0a1f 0a46 696c 653a 2042 6572 7469 0....File: Berti │ │ │ │ +00027e60: 6e69 2e69 6e66 6f2c 204e 6f64 653a 2077 ni.info, Node: w │ │ │ │ +00027e70: 7269 7465 5374 6172 7446 696c 652c 2050 riteStartFile, P │ │ │ │ +00027e80: 7265 763a 2056 6172 6961 626c 6520 6772 rev: Variable gr │ │ │ │ +00027e90: 6f75 7073 2c20 5570 3a20 546f 700a 0a77 oups, Up: Top..w │ │ │ │ +00027ea0: 7269 7465 5374 6172 7446 696c 6520 2d2d riteStartFile -- │ │ │ │ +00027eb0: 2057 7269 7465 7320 7468 6520 6c69 7374 Writes the list │ │ │ │ +00027ec0: 206f 6620 6c69 7374 206f 6620 636f 6f72 of list of coor │ │ │ │ +00027ed0: 6469 6e61 7465 7320 746f 2061 2066 696c dinates to a fil │ │ │ │ +00027ee0: 6520 7468 6174 2042 6572 7469 6e69 2063 e that Bertini c │ │ │ │ +00027ef0: 616e 2072 6561 642e 0a2a 2a2a 2a2a 2a2a an read..******* │ │ │ │ 00027f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00027f50: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00027f60: 2020 2020 2077 7269 7465 5374 6172 7446 writeStartF │ │ │ │ -00027f70: 696c 6528 732c 7629 0a20 202a 2049 6e70 ile(s,v). * Inp │ │ │ │ -00027f80: 7574 733a 0a20 2020 2020 202a 2073 2c20 uts:. * s, │ │ │ │ -00027f90: 6120 2a6e 6f74 6520 7374 7269 6e67 3a20 a *note string: │ │ │ │ -00027fa0: 284d 6163 6175 6c61 7932 446f 6329 5374 (Macaulay2Doc)St │ │ │ │ -00027fb0: 7269 6e67 2c2c 2054 6865 2064 6972 6563 ring,, The direc │ │ │ │ -00027fc0: 746f 7279 2077 6865 7265 2074 6865 0a20 tory where the. │ │ │ │ -00027fd0: 2020 2020 2020 2042 6572 7469 6e69 2066 Bertini f │ │ │ │ -00027fe0: 696c 6520 7769 6c6c 2062 6520 7772 6974 ile will be writ │ │ │ │ -00027ff0: 7465 6e2e 0a20 2020 2020 202a 2076 2c20 ten.. * v, │ │ │ │ -00028000: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -00028010: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -00028020: 2c2c 2041 206c 6973 7420 6f66 206c 6973 ,, A list of lis │ │ │ │ -00028030: 7420 6e75 6d62 6572 7320 7468 6174 2077 t numbers that w │ │ │ │ -00028040: 696c 6c0a 2020 2020 2020 2020 6265 2077 ill. be w │ │ │ │ -00028050: 7269 7474 656e 2074 6f20 7468 6520 6669 ritten to the fi │ │ │ │ -00028060: 6c65 2e0a 2020 2a20 2a6e 6f74 6520 4f70 le.. * *note Op │ │ │ │ -00028070: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -00028080: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -00028090: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -000280a0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -000280b0: 732c 3a0a 2020 2020 2020 2a20 4d32 5072 s,:. * M2Pr │ │ │ │ -000280c0: 6563 6973 696f 6e20 286d 6973 7369 6e67 ecision (missing │ │ │ │ -000280d0: 2064 6f63 756d 656e 7461 7469 6f6e 2920 documentation) │ │ │ │ -000280e0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -000280f0: 7661 6c75 6520 3533 2c20 0a20 2020 2020 value 53, . │ │ │ │ -00028100: 202a 204e 616d 6553 7461 7274 4669 6c65 * NameStartFile │ │ │ │ -00028110: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -00028120: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ -00028130: 2064 6566 6175 6c74 2076 616c 7565 2022 default value " │ │ │ │ -00028140: 7374 6172 7422 2c20 0a20 2020 2020 202a start", . * │ │ │ │ -00028150: 2053 746f 7261 6765 466f 6c64 6572 2028 StorageFolder ( │ │ │ │ -00028160: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -00028170: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -00028180: 6566 6175 6c74 2076 616c 7565 206e 756c efault value nul │ │ │ │ -00028190: 6c2c 200a 0a44 6573 6372 6970 7469 6f6e l, ..Description │ │ │ │ -000281a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -000281b0: 6973 2066 756e 6374 696f 6e20 6361 6e20 is function can │ │ │ │ -000281c0: 6265 2075 7365 6420 746f 2077 7269 7465 be used to write │ │ │ │ -000281d0: 2022 7374 6172 7422 2066 696c 6573 2061 "start" files a │ │ │ │ -000281e0: 6e64 2061 6e79 206f 7468 6572 2073 6f6c nd any other sol │ │ │ │ -000281f0: 7574 696f 6e20 6669 6c65 0a75 7369 6e67 ution file.using │ │ │ │ -00028200: 2074 6865 206f 7074 696f 6e20 4e61 6d65 the option Name │ │ │ │ -00028210: 5374 6172 7446 696c 653d 3e22 416e 794e StartFile=>"AnyN │ │ │ │ -00028220: 616d 6559 6f75 5761 6e74 222e 0a0a 2b2d ameYouWant"...+- │ │ │ │ -00028230: 2d2d 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Bertin │ │ │ │ +00027fe0: 6920 6669 6c65 2077 696c 6c20 6265 2077 i file will be w │ │ │ │ +00027ff0: 7269 7474 656e 2e0a 2020 2020 2020 2a20 ritten.. * │ │ │ │ +00028000: 762c 2061 202a 6e6f 7465 206c 6973 743a v, a *note list: │ │ │ │ +00028010: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ +00028020: 6973 742c 2c20 4120 6c69 7374 206f 6620 ist,, A list of │ │ │ │ +00028030: 6c69 7374 206e 756d 6265 7273 2074 6861 list numbers tha │ │ │ │ +00028040: 7420 7769 6c6c 0a20 2020 2020 2020 2062 t will. b │ │ │ │ +00028050: 6520 7772 6974 7465 6e20 746f 2074 6865 e written to the │ │ │ │ +00028060: 2066 696c 652e 0a20 202a 202a 6e6f 7465 file.. * *note │ │ │ │ +00028070: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ +00028080: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00028090: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ +000280a0: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ +000280b0: 7075 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---------------- │ │ │ │ -00028260: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2063 -------+.|i1 : c │ │ │ │ -00028270: 6f6f 7264 696e 6174 6573 4f66 5477 6f50 oordinatesOfTwoP │ │ │ │ -00028280: 6e74 733d 7b7b 312c 307d 2c7b 332c 347d nts={{1,0},{3,4} │ │ │ │ -00028290: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -000282a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028260: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00028270: 3a20 636f 6f72 6469 6e61 7465 734f 6654 : coordinatesOfT │ │ │ │ +00028280: 776f 506e 7473 3d7b 7b31 2c30 7d2c 7b33 woPnts={{1,0},{3 │ │ │ │ +00028290: 2c34 7d7d 2020 2020 2020 2020 2020 2020 ,4}} │ │ │ │ +000282a0: 2020 2020 207c 0a7c 2020 2020 2020 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 207c 0a7c |.| │ │ │ │ -000282e0: 6f31 203d 207b 7b31 2c20 307d 2c20 7b33 o1 = {{1, 0}, {3 │ │ │ │ -000282f0: 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UseReg │ │ │ │ +00028a20: 656e 6572 6174 696f 6e7f 3135 3537 3735 eneration.155775 │ │ │ │ +00028a30: 0a4e 6f64 653a 2076 616c 7565 424d 327f .Node: valueBM2. │ │ │ │ +00028a40: 3135 3638 3139 0a4e 6f64 653a 2056 6172 156819.Node: Var │ │ │ │ +00028a50: 6961 626c 6520 6772 6f75 7073 7f31 3631 iable groups.161 │ │ │ │ +00028a60: 3638 390a 4e6f 6465 3a20 7772 6974 6553 689.Node: writeS │ │ │ │ +00028a70: 7461 7274 4669 6c65 7f31 3633 3431 310a tartFile.163411. │ │ │ │ +00028a80: 1f0a 456e 6420 5461 6720 5461 626c 650a ..End Tag Table. │ │ ├── ./usr/share/info/BettiCharacters.info.gz │ │ │ ├── BettiCharacters.info │ │ │ │ @@ -12973,15 +12973,15 @@ │ │ │ │ 00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ae0: 2d2b 0a7c 6939 203a 2065 6c61 7073 6564 -+.|i9 : elapsed │ │ │ │ 00032af0: 5469 6d65 2063 203d 2063 6861 7261 6374 Time c = charact │ │ │ │ 00032b00: 6572 2041 2020 2020 2020 2020 2020 2020 er A │ │ │ │ 00032b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032b30: 207c 0a7c 202d 2d20 2e34 3436 3139 3273 |.| -- .446192s │ │ │ │ +00032b30: 207c 0a7c 202d 2d20 2e33 3934 3830 3573 |.| -- .394805s │ │ │ │ 00032b40: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 00032b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00032b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032ba0: 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 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ 00037680: 2065 6c61 7073 6564 5469 6d65 2063 3d63 elapsedTime c=c │ │ │ │ 00037690: 6861 7261 6374 6572 2041 2020 2020 2020 haracter A │ │ │ │ 000376a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000376b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000376c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000376d0: 2e35 3936 3133 3273 2065 6c61 7073 6564 .596132s elapsed │ │ │ │ +000376d0: 2e34 3234 3235 3573 2065 6c61 7073 6564 .424255s elapsed │ │ │ │ 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 207c 0a7c 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 │ │ │ │ @@ -15614,15 +15614,15 @@ │ │ │ │ 0003cfd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 ----------+.|i20 │ │ │ │ 0003d010: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 0003d020: 3120 3d20 6368 6172 6163 7465 7220 4131 1 = character A1 │ │ │ │ 0003d030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003d040: 0a7c 202d 2d20 2e37 3439 3635 3273 2065 .| -- .749652s e │ │ │ │ +0003d040: 0a7c 202d 2d20 2e37 3135 3534 3973 2065 .| -- .715549s e │ │ │ │ 0003d050: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0003d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d070: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0003d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0a0: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ 0003d0b0: 3d20 4368 6172 6163 7465 7220 6f76 6572 = Character over │ │ │ │ @@ -15654,16 +15654,16 @@ │ │ │ │ 0003d250: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d280: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2065 ------+.|i21 : e │ │ │ │ 0003d290: 6c61 7073 6564 5469 6d65 2061 3220 3d20 lapsedTime a2 = │ │ │ │ 0003d2a0: 6368 6172 6163 7465 7220 4132 2020 2020 character A2 │ │ │ │ 0003d2b0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0003d2c0: 2d20 3333 2e38 3439 3773 2065 6c61 7073 - 33.8497s elaps │ │ │ │ -0003d2d0: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ +0003d2c0: 2d20 3238 2e37 7320 656c 6170 7365 6420 - 28.7s elapsed │ │ │ │ +0003d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d320: 2020 2020 207c 0a7c 6f32 3120 3d20 4368 |.|o21 = Ch │ │ │ │ 0003d330: 6172 6163 7465 7220 6f76 6572 2052 2020 aracter over R │ │ │ │ 0003d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -16112,16 +16112,16 @@ │ │ │ │ 0003eef0: 6374 696f 6e4f 6e47 7261 6465 644d 6f64 ctionOnGradedMod │ │ │ │ 0003ef00: 756c 6520 2020 2020 2020 2020 2020 7c0a ule |. │ │ │ │ 0003ef10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ef30: 2d2d 2d2d 2d2d 2b0a 7c69 3332 203a 2065 ------+.|i32 : e │ │ │ │ 0003ef40: 6c61 7073 6564 5469 6d65 2062 203d 2063 lapsedTime b = c │ │ │ │ 0003ef50: 6861 7261 6374 6572 2842 2c32 3129 7c0a haracter(B,21)|. │ │ │ │ -0003ef60: 7c20 2d2d 2031 352e 3037 3873 2065 6c61 | -- 15.078s ela │ │ │ │ -0003ef70: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ +0003ef60: 7c20 2d2d 2031 322e 3634 3332 7320 656c | -- 12.6432s el │ │ │ │ +0003ef70: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 0003ef80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003efa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0003efb0: 7c6f 3332 203d 2043 6861 7261 6374 6572 |o32 = Character │ │ │ │ 0003efc0: 206f 7665 7220 5220 2020 2020 2020 2020 over R │ │ │ │ 0003efd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Bruns.info.gz │ │ │ ├── Bruns.info │ │ │ │ @@ -1095,17 +1095,17 @@ │ │ │ │ 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 2e31 3335 3933 -- used 0.13593 │ │ │ │ -000044e0: 3373 2028 6370 7529 3b20 302e 3133 3730 3s (cpu); 0.1370 │ │ │ │ -000044f0: 3137 7320 2874 6872 6561 6429 3b20 3073 17s (thread); 0s │ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e31 3734 3233 -- used 0.17423 │ │ │ │ +000044e0: 3873 2028 6370 7529 3b20 302e 3137 3539 8s (cpu); 0.1759 │ │ │ │ +000044f0: 3635 7320 2874 6872 6561 6429 3b20 3073 65s (thread); 0s │ │ │ │ 00004500: 2028 6763 2920 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 |.| │ │ ├── ./usr/share/info/CellularResolutions.info.gz │ │ │ ├── CellularResolutions.info │ │ │ │ @@ -2297,63 +2297,63 @@ │ │ │ │ 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1 1 ||.| │ │ │ │ 000090d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000090e0: 2020 7c20 3020 3020 3020 3020 3120 3020 | 0 0 0 0 1 0 │ │ │ │ -000090f0: 3120 3020 3020 3120 3020 3020 3120 3020 1 0 0 1 0 0 1 0 │ │ │ │ -00009100: 3020 3120 3120 7c7c 0a7c 2020 2020 2020 0 1 1 ||.| │ │ │ │ +000090f0: 3020 3020 3120 3020 3120 3120 3120 3120 0 0 1 0 1 1 1 1 │ │ │ │ +00009100: 3120 3120 3120 7c7c 0a7c 2020 2020 2020 1 1 1 ||.| │ │ │ │ 00009110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009120: 7c20 3020 3020 3020 3020 3020 3120 3020 | 0 0 0 0 0 1 0 │ │ │ │ -00009130: 3020 3020 3020 3020 3020 3020 3120 3120 0 0 0 0 0 0 1 1 │ │ │ │ +00009130: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ 00009140: 3020 3020 7c7c 0a7c 2020 2020 2020 2020 0 0 ||.| │ │ │ │ 00009150: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ 00009160: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009170: 3020 3020 3020 3020 3020 3020 3020 3020 0 0 0 0 0 0 0 0 │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b970: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000b980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b990: 2020 3520 2020 3420 2020 2033 2032 2020 5 4 3 2 │ │ │ │ -0000b9a0: 2032 2033 2020 2020 2034 2020 2035 2020 2 3 4 5 │ │ │ │ +0000b990: 2020 2020 3420 2020 3520 2020 3520 2020 4 5 5 │ │ │ │ +0000b9a0: 3420 2020 2033 2032 2020 2032 2033 2020 4 3 2 2 3 │ │ │ │ 0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 } │ │ │ │ +0000b9e0: 7b78 2a79 202c 2078 202c 2078 202c 2078 {x*y , x , x , x │ │ │ │ +0000b9f0: 2079 2c20 7820 7920 2c20 7820 7920 7d20 y, x y , x y } │ │ │ │ 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 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 |.| │ │ │ │ +0000ba30: 2020 3520 2020 2035 2032 2020 2035 2033 5 5 2 5 3 │ │ │ │ +0000ba40: 2020 2035 2034 2020 2034 2032 2020 2034 5 4 4 2 4 │ │ │ │ +0000ba50: 2034 2020 2035 2020 2020 3320 3320 2020 4 5 3 3 │ │ │ │ +0000ba60: 3520 3220 2020 2020 207c 0a7c 2020 2020 5 2 |.| │ │ │ │ 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 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 , |.| │ │ │ │ +0000ba80: 7b78 2079 2c20 7820 7920 2c20 7820 7920 {x y, x y , x y │ │ │ │ +0000ba90: 2c20 7820 7920 2c20 7820 7920 2c20 7820 , x y , x y , x │ │ │ │ +0000baa0: 7920 2c20 7820 792c 2078 2079 202c 2078 y , x y, x y , x │ │ │ │ +0000bab0: 2079 202c 2020 2020 207c 0a7c 2020 2020 y , |.| │ │ │ │ 0000bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ │ │ @@ -3007,25 +3007,25 @@ │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ 0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ -0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 792c |.|x y, │ │ │ │ -0000bcf0: 2078 2079 202c 2078 2079 207d 2020 2020 x y , x y } │ │ │ │ +0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 7920 |.|x y │ │ │ │ +0000bcf0: 2c20 7820 7920 2c20 7820 7920 7d20 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 ---------------- │ │ │ │ @@ -5234,24 +5234,24 @@ │ │ │ │ 00014710: 2066 6163 6550 6f73 6574 2043 2020 2020 facePoset C │ │ │ │ 00014720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014730: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 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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 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 │ │ │ │ +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 │ │ │ │ 00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018bb0: 2020 2020 207c 0a7c 6f35 203d 207b 7820 |.|o5 = {x │ │ │ │ -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} │ │ │ │ +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} │ │ │ │ 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 │ │ │ │ @@ -8276,21 +8276,21 @@ │ │ │ │ 00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020540: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00020550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ 00020630: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00020640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8570,18 +8570,18 @@ │ │ │ │ 00021790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ 000217b0: 203a 2063 656c 6c73 2831 2c44 292f 6365 : cells(1,D)/ce │ │ │ │ 000217c0: 6c6c 4c61 6265 6c20 2020 2020 2020 2020 llLabel │ │ │ │ 000217d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000217e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000217f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00021800: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00021800: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 00021810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021820: 0a7c 6f31 3220 3d20 7b61 2a62 2c20 622a .|o12 = {a*b, b* │ │ │ │ -00021830: 6320 7d20 2020 2020 2020 2020 2020 2020 c } │ │ │ │ +00021820: 0a7c 6f31 3220 3d20 7b62 2a63 202c 2061 .|o12 = {b*c , a │ │ │ │ +00021830: 2a62 7d20 2020 2020 2020 2020 2020 2020 *b} │ │ │ │ 00021840: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021860: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00021870: 6f31 3220 3a20 4c69 7374 2020 2020 2020 o12 : List │ │ │ │ 00021880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 000218a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/ChainComplexExtras.info.gz │ │ │ ├── ChainComplexExtras.info │ │ │ │ @@ -4819,17 +4819,17 @@ │ │ │ │ 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 3331 3937 3173 2028 6370 7529 3b20 0.31971s (cpu); │ │ │ │ -00012da0: 302e 3235 3534 3938 7320 2874 6872 6561 0.255498s (threa │ │ │ │ -00012db0: 6429 3b20 3073 2028 6763 2920 7c0a 2b2d d); 0s (gc) |.+- │ │ │ │ +00012d90: 302e 3335 3234 3831 7320 2863 7075 293b 0.352481s (cpu); │ │ │ │ +00012da0: 2030 2e32 3736 3334 3773 2028 7468 7265 0.276347s (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 │ │ │ │ 00012e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ @@ -6579,31 +6579,31 @@ │ │ │ │ 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 3434 3131 -- used 0.094411 │ │ │ │ -00019ba0: 3473 2028 6370 7529 3b20 302e 3039 3635 4s (cpu); 0.0965 │ │ │ │ -00019bb0: 3439 3173 2028 7468 7265 6164 293b 2030 491s (thread); 0 │ │ │ │ -00019bc0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3130 3737 3637 -- used 0.107767 │ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3036 3330 s (cpu); 0.10630 │ │ │ │ +00019bb0: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ +00019bc0: 2867 6329 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 3036 3831 -- used 0.210681 │ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3431 3934 s (cpu); 0.14194 │ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3234 3532 3039 -- used 0.245209 │ │ │ │ +00019c90: 7320 2863 7075 293b 2030 2e31 3635 3732 s (cpu); 0.16572 │ │ │ │ 00019ca0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (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 ---------------- │ │ ├── ./usr/share/info/CharacteristicClasses.info.gz │ │ │ ├── CharacteristicClasses.info │ │ │ │ @@ -1170,17 +1170,17 @@ │ │ │ │ 00004910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004920: 2d2d 2b0a 7c69 3320 3a20 7469 6d65 2043 --+.|i3 : time C │ │ │ │ 00004930: 534d 2055 2020 2020 2020 2020 2020 2020 SM U │ │ │ │ 00004940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004970: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004980: 3233 3238 3136 7320 2863 7075 293b 2030 232816s (cpu); 0 │ │ │ │ -00004990: 2e31 3533 3335 7320 2874 6872 6561 6429 .15335s (thread) │ │ │ │ -000049a0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00004980: 3233 3233 3039 7320 2863 7075 293b 2030 232309s (cpu); 0 │ │ │ │ +00004990: 2e31 3630 3330 3573 2028 7468 7265 6164 .160305s (thread │ │ │ │ +000049a0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 000049b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000049d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004a10: 2020 7c0a 7c20 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ @@ -1255,17 +1255,17 @@ │ │ │ │ 00004e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004e70: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2043 --+.|i4 : time C │ │ │ │ 00004e80: 534d 2855 2c43 6865 636b 536d 6f6f 7468 SM(U,CheckSmooth │ │ │ │ 00004e90: 3d3e 6661 6c73 6529 2020 2020 2020 2020 =>false) │ │ │ │ 00004ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ec0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004ed0: 3331 3234 7320 2863 7075 293b 2030 2e32 3124s (cpu); 0.2 │ │ │ │ -00004ee0: 3637 3230 3273 2028 7468 7265 6164 293b 67202s (thread); │ │ │ │ -00004ef0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00004ed0: 3337 3633 3438 7320 2863 7075 293b 2030 376348s (cpu); 0 │ │ │ │ +00004ee0: 2e33 3036 3236 3673 2028 7468 7265 6164 .306266s (thread │ │ │ │ +00004ef0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00004f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00004f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 │ │ │ │ 00004f60: 2020 7c0a 7c20 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ @@ -4296,17 +4296,17 @@ │ │ │ │ 00010c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c90: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2043 --+.|i5 : time C │ │ │ │ 00010ca0: 534d 2849 2c43 6f6d 704d 6574 686f 643d SM(I,CompMethod= │ │ │ │ 00010cb0: 3e50 726f 6a65 6374 6976 6544 6567 7265 >ProjectiveDegre │ │ │ │ 00010cc0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 00010cd0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00010ce0: 2030 2e39 3837 3637 3173 2028 6370 7529 0.987671s (cpu) │ │ │ │ -00010cf0: 3b20 302e 3431 3030 3431 7320 2874 6872 ; 0.410041s (thr │ │ │ │ -00010d00: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00010ce0: 2031 2e35 3732 3336 7320 2863 7075 293b 1.57236s (cpu); │ │ │ │ +00010cf0: 2030 2e34 3634 3731 3173 2028 7468 7265 0.464711s (thre │ │ │ │ +00010d00: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00010d10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00010d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00010d60: 2020 2020 2035 2020 2020 2020 3420 2020 5 4 │ │ │ │ 00010d70: 2020 2033 2020 2020 2020 3220 2020 2020 3 2 │ │ │ │ @@ -4354,17 +4354,17 @@ │ │ │ │ 00011010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00011040: 3620 3a20 7469 6d65 2043 534d 2849 2c43 6 : time CSM(I,C │ │ │ │ 00011050: 6f6d 704d 6574 686f 643d 3e50 6e52 6573 ompMethod=>PnRes │ │ │ │ 00011060: 6964 7561 6c29 2020 2020 2020 2020 2020 idual) │ │ │ │ 00011070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011080: 0a7c 202d 2d20 7573 6564 2032 2e31 3138 .| -- used 2.118 │ │ │ │ -00011090: 3336 7320 2863 7075 293b 2031 2e38 3134 36s (cpu); 1.814 │ │ │ │ -000110a0: 3432 7320 2874 6872 6561 6429 3b20 3073 42s (thread); 0s │ │ │ │ +00011080: 0a7c 202d 2d20 7573 6564 2032 2e31 3735 .| -- used 2.175 │ │ │ │ +00011090: 3831 7320 2863 7075 293b 2032 2e30 3033 81s (cpu); 2.003 │ │ │ │ +000110a0: 3835 7320 2874 6872 6561 6429 3b20 3073 85s (thread); 0s │ │ │ │ 000110b0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000110c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000110d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011100: 2020 2020 207c 0a7c 2020 2020 2020 2035 |.| 5 │ │ │ │ 00011110: 2020 2020 2020 3420 2020 2020 2033 2020 4 3 │ │ │ │ @@ -4442,17 +4442,17 @@ │ │ │ │ 00011590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000115b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 000115c0: 3020 3a20 7469 6d65 2043 534d 284b 2c43 0 : time CSM(K,C │ │ │ │ 000115d0: 6f6d 704d 6574 686f 643d 3e50 726f 6a65 ompMethod=>Proje │ │ │ │ 000115e0: 6374 6976 6544 6567 7265 6529 2020 2020 ctiveDegree) │ │ │ │ 000115f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00011600: 7c20 2d2d 2075 7365 6420 302e 3236 3933 | -- used 0.2693 │ │ │ │ -00011610: 3032 7320 2863 7075 293b 2030 2e31 3833 02s (cpu); 0.183 │ │ │ │ -00011620: 3233 3673 2028 7468 7265 6164 293b 2030 236s (thread); 0 │ │ │ │ +00011600: 7c20 2d2d 2075 7365 6420 302e 3237 3532 | -- used 0.2752 │ │ │ │ +00011610: 3833 7320 2863 7075 293b 2030 2e32 3033 83s (cpu); 0.203 │ │ │ │ +00011620: 3333 3973 2028 7468 7265 6164 293b 2030 339s (thread); 0 │ │ │ │ 00011630: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00011640: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011680: 2020 2020 7c0a 7c20 2020 2020 2020 2033 |.| 3 │ │ │ │ 00011690: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -4501,16 +4501,16 @@ │ │ │ │ 00011940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011960: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469 -----+.|i11 : ti │ │ │ │ 00011970: 6d65 2043 534d 284b 2c43 6f6d 704d 6574 me CSM(K,CompMet │ │ │ │ 00011980: 686f 643d 3e50 6e52 6573 6964 7561 6c29 hod=>PnResidual) │ │ │ │ 00011990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000119a0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000119b0: 7365 6420 302e 3038 3238 3130 3373 2028 sed 0.0828103s ( │ │ │ │ -000119c0: 6370 7529 3b20 302e 3038 3533 3938 3573 cpu); 0.0853985s │ │ │ │ +000119b0: 7365 6420 302e 3039 3939 3230 3873 2028 sed 0.0999208s ( │ │ │ │ +000119c0: 6370 7529 3b20 302e 3039 3937 3033 3973 cpu); 0.0997039s │ │ │ │ 000119d0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000119e0: 6329 2020 2020 2020 2020 207c 0a7c 2020 c) |.| │ │ │ │ 000119f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00011a30: 7c20 2020 2020 2020 2033 2020 2020 2032 | 3 2 │ │ │ │ @@ -5498,16 +5498,16 @@ │ │ │ │ 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 2d2b ---------------+ │ │ │ │ 000157c0: 0a7c 6931 3520 3a20 7469 6d65 2063 736d .|i15 : time csm │ │ │ │ 000157d0: 4b3d 4353 4d28 412c 4b29 2020 2020 2020 K=CSM(A,K) │ │ │ │ 000157e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000157f0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00015800: 6420 312e 3232 3038 3573 2028 6370 7529 d 1.22085s (cpu) │ │ │ │ -00015810: 3b20 302e 3436 3935 3934 7320 2874 6872 ; 0.469594s (thr │ │ │ │ +00015800: 6420 312e 3332 3133 3673 2028 6370 7529 d 1.32136s (cpu) │ │ │ │ +00015810: 3b20 302e 3432 3130 3136 7320 2874 6872 ; 0.421016s (thr │ │ │ │ 00015820: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ 00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015860: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ 00015870: 2032 2020 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ 00015880: 2032 2020 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ @@ -5675,17 +5675,17 @@ │ │ │ │ 000162a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000162b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000162c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000162d0: 0a7c 6932 3220 3a20 7469 6d65 2043 534d .|i22 : time CSM │ │ │ │ 000162e0: 2841 2c4b 2c6d 2920 2020 2020 2020 2020 (A,K,m) │ │ │ │ 000162f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016300: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00016310: 202d 2d20 7573 6564 2030 2e32 3930 3331 -- used 0.29031 │ │ │ │ -00016320: 3873 2028 6370 7529 3b20 302e 3038 3331 8s (cpu); 0.0831 │ │ │ │ -00016330: 3136 3573 2028 7468 7265 6164 293b 2030 165s (thread); 0 │ │ │ │ +00016310: 202d 2d20 7573 6564 2030 2e31 3039 3232 -- used 0.10922 │ │ │ │ +00016320: 3973 2028 6370 7529 3b20 302e 3037 3030 9s (cpu); 0.0700 │ │ │ │ +00016330: 3830 3573 2028 7468 7265 6164 293b 2030 805s (thread); 0 │ │ │ │ 00016340: 7320 2867 6329 2020 2020 207c 0a7c 2020 s (gc) |.| │ │ │ │ 00016350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016380: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00016390: 2020 2020 3220 3220 2020 2020 3220 2020 2 2 2 │ │ │ │ 000163a0: 2020 2020 2020 3220 2020 2032 2020 2020 2 2 │ │ │ │ @@ -6645,18 +6645,18 @@ │ │ │ │ 00019f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019f60: 3420 3a20 7469 6d65 2045 756c 6572 2849 4 : time Euler(I │ │ │ │ 00019f70: 2c49 6e70 7574 4973 536d 6f6f 7468 3d3e ,InputIsSmooth=> │ │ │ │ 00019f80: 7472 7565 2920 2020 2020 2020 2020 2020 true) │ │ │ │ 00019f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019fb0: 2d2d 2075 7365 6420 302e 3033 3939 3135 -- used 0.039915 │ │ │ │ -00019fc0: 3373 2028 6370 7529 3b20 302e 3033 3834 3s (cpu); 0.0384 │ │ │ │ -00019fd0: 3639 3173 2028 7468 7265 6164 293b 2030 691s (thread); 0 │ │ │ │ -00019fe0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +00019fb0: 2d2d 2075 7365 6420 302e 3132 3534 3533 -- used 0.125453 │ │ │ │ +00019fc0: 7320 2863 7075 293b 2030 2e30 3538 3131 s (cpu); 0.05811 │ │ │ │ +00019fd0: 3331 7320 2874 6872 6561 6429 3b20 3073 31s (thread); 0s │ │ │ │ +00019fe0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00019ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a040: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0001a050: 3420 3d20 3420 2020 2020 2020 2020 2020 4 = 4 │ │ │ │ @@ -6670,17 +6670,17 @@ │ │ │ │ 0001a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0001a0f0: 3520 3a20 7469 6d65 2045 756c 6572 2049 5 : time Euler I │ │ │ │ 0001a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a130: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a140: 2d2d 2075 7365 6420 302e 3337 3434 3832 -- used 0.374482 │ │ │ │ -0001a150: 7320 2863 7075 293b 2030 2e31 3735 3334 s (cpu); 0.17534 │ │ │ │ -0001a160: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ +0001a140: 2d2d 2075 7365 6420 302e 3335 3336 3133 -- used 0.353613 │ │ │ │ +0001a150: 7320 2863 7075 293b 2030 2e31 3932 3638 s (cpu); 0.19268 │ │ │ │ +0001a160: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ 0001a170: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001a180: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -6874,17 +6874,17 @@ │ │ │ │ 0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001adb0: 2b0a 7c69 3130 203a 2074 696d 6520 4575 +.|i10 : time Eu │ │ │ │ 0001adc0: 6c65 7228 4a2c 4d65 7468 6f64 3d3e 4469 ler(J,Method=>Di │ │ │ │ 0001add0: 7265 6374 436f 6d70 6c65 7465 496e 7429 rectCompleteInt) │ │ │ │ 0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001adf0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -0001ae00: 302e 3235 3630 3737 7320 2863 7075 293b 0.256077s (cpu); │ │ │ │ -0001ae10: 2030 2e30 3738 3937 3733 7320 2874 6872 0.0789773s (thr │ │ │ │ -0001ae20: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0001ae00: 302e 3236 3530 3835 7320 2863 7075 293b 0.265085s (cpu); │ │ │ │ +0001ae10: 2030 2e31 3132 3430 3773 2028 7468 7265 0.112407s (thre │ │ │ │ +0001ae20: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0001ae30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0001ae80: 3130 203d 2032 2020 2020 2020 2020 2020 10 = 2 │ │ │ │ 0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6895,18 +6895,18 @@ │ │ │ │ 0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001af00: 2d2d 2d2d 2b0a 7c69 3131 203a 2074 696d ----+.|i11 : tim │ │ │ │ 0001af10: 6520 4575 6c65 7228 4a2c 4d65 7468 6f64 e Euler(J,Method │ │ │ │ 0001af20: 3d3e 4469 7265 6374 436f 6d70 6c65 7465 =>DirectComplete │ │ │ │ 0001af30: 496e 742c 496e 6473 4f66 536d 6f6f 7468 Int,IndsOfSmooth │ │ │ │ 0001af40: 3d3e 7b30 2c31 7d29 7c0a 7c20 2d2d 2075 =>{0,1})|.| -- u │ │ │ │ -0001af50: 7365 6420 302e 3238 3530 3231 7320 2863 sed 0.285021s (c │ │ │ │ -0001af60: 7075 293b 2030 2e30 3831 3732 3838 7320 pu); 0.0817288s │ │ │ │ -0001af70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0001af80: 2920 2020 2020 2020 2020 2020 7c0a 7c20 ) |.| │ │ │ │ +0001af50: 7365 6420 302e 3332 3533 3836 7320 2863 sed 0.325386s (c │ │ │ │ +0001af60: 7075 293b 2030 2e31 3137 3730 3673 2028 pu); 0.117706s ( │ │ │ │ +0001af70: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0001af80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001afd0: 7c0a 7c6f 3131 203d 2032 2020 2020 2020 |.|o11 = 2 │ │ │ │ 0001afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7514,17 +7514,17 @@ │ │ │ │ 0001d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0001d5c0: 3320 3a20 7469 6d65 2043 534d 2849 2c4d 3 : time CSM(I,M │ │ │ │ 0001d5d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d ethod=>DirectCom │ │ │ │ 0001d5e0: 706c 6574 496e 7429 2020 2020 2020 2020 pletInt) │ │ │ │ 0001d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d600: 2020 207c 0a7c 202d 2d20 7573 6564 2033 |.| -- used 3 │ │ │ │ -0001d610: 2e34 3733 3832 7320 2863 7075 293b 2031 .47382s (cpu); 1 │ │ │ │ -0001d620: 2e30 3731 3673 2028 7468 7265 6164 293b .0716s (thread); │ │ │ │ +0001d600: 2020 207c 0a7c 202d 2d20 7573 6564 2036 |.| -- used 6 │ │ │ │ +0001d610: 2e32 3930 3337 7320 2863 7075 293b 2031 .29037s (cpu); 1 │ │ │ │ +0001d620: 2e34 3536 3273 2028 7468 7265 6164 293b .4562s (thread); │ │ │ │ 0001d630: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001d640: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d690: 207c 0a7c 2020 2020 2020 2032 2032 2020 |.| 2 2 │ │ │ │ @@ -7577,17 +7577,17 @@ │ │ │ │ 0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0001d9a0: 7c69 3420 3a20 7469 6d65 2043 534d 2849 |i4 : time CSM(I │ │ │ │ 0001d9b0: 2c4d 6574 686f 643d 3e44 6972 6563 7443 ,Method=>DirectC │ │ │ │ 0001d9c0: 6f6d 706c 6574 496e 742c 496e 6473 4f66 ompletInt,IndsOf │ │ │ │ 0001d9d0: 536d 6f6f 7468 3d3e 7b31 2c32 7d29 2020 Smooth=>{1,2}) │ │ │ │ 0001d9e0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001d9f0: 2033 2e37 3934 3034 7320 2863 7075 293b 3.79404s (cpu); │ │ │ │ -0001da00: 2031 2e32 3535 3333 7320 2874 6872 6561 1.25533s (threa │ │ │ │ -0001da10: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0001d9f0: 2035 2e35 3731 3173 2028 6370 7529 3b20 5.5711s (cpu); │ │ │ │ +0001da00: 312e 3335 3934 3773 2028 7468 7265 6164 1.35947s (thread │ │ │ │ +0001da10: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0001da20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da70: 2020 207c 0a7c 2020 2020 2020 2032 2032 |.| 2 2 │ │ │ │ 0001da80: 2020 2020 2032 2020 2020 2020 2020 2032 2 2 │ │ │ │ @@ -7727,16 +7727,16 @@ │ │ │ │ 0001e2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001e310: 0a7c 6933 203a 2074 696d 6520 4353 4d20 .|i3 : time CSM │ │ │ │ 0001e320: 4920 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ 0001e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e340: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0001e350: 7573 6564 2030 2e39 3530 3834 3173 2028 used 0.950841s ( │ │ │ │ -0001e360: 6370 7529 3b20 302e 3438 3539 3473 2028 cpu); 0.48594s ( │ │ │ │ +0001e350: 7573 6564 2031 2e33 3332 3433 7320 2863 used 1.33243s (c │ │ │ │ +0001e360: 7075 293b 2030 2e35 3332 3132 3273 2028 pu); 0.532122s ( │ │ │ │ 0001e370: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 0001e380: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001e3c0: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ 0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7778,16 +7778,16 @@ │ │ │ │ 0001e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 0001e640: 203a 2074 696d 6520 4353 4d28 492c 496e : time CSM(I,In │ │ │ │ 0001e650: 7075 7449 7353 6d6f 6f74 683d 3e74 7275 putIsSmooth=>tru │ │ │ │ 0001e660: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 0001e670: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001e680: 2030 2e30 3633 3331 3536 7320 2863 7075 0.0633156s (cpu │ │ │ │ -0001e690: 293b 2030 2e30 3337 3438 3333 7320 2874 ); 0.0374833s (t │ │ │ │ +0001e680: 2030 2e30 3934 3738 3036 7320 2863 7075 0.0947806s (cpu │ │ │ │ +0001e690: 293b 2030 2e30 3433 3539 3635 7320 2874 ); 0.0435965s (t │ │ │ │ 0001e6a0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ 0001e6b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e6e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0001e6f0: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7833,4197 +7833,4199 @@ │ │ │ │ 0001e980: 7175 6976 616c 656e 746c 792c 2075 7365 quivalently, use │ │ │ │ 0001e990: 2074 6865 2063 6f6d 6d61 6e64 202a 6e6f the command *no │ │ │ │ 0001e9a0: 7465 2043 6865 726e 3a20 4368 6572 6e2c te Chern: Chern, │ │ │ │ 0001e9b0: 2069 6e73 7465 6164 0a69 6e20 7468 6973 instead.in this │ │ │ │ 0001e9c0: 2063 6173 652e 0a0a 2b2d 2d2d 2d2d 2d2d case...+------- │ │ │ │ 0001e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ea00: 7c69 3520 3a20 7469 6d65 2043 6865 726e |i5 : time Chern │ │ │ │ -0001ea10: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ea00: 2b0a 7c69 3520 3a20 7469 6d65 2043 6865 +.|i5 : time Che │ │ │ │ +0001ea10: 726e 2049 2020 2020 2020 2020 2020 2020 rn I │ │ │ │ 0001ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ea30: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0001ea40: 6420 302e 3132 3230 3873 2028 6370 7529 d 0.12208s (cpu) │ │ │ │ -0001ea50: 3b20 302e 3033 3834 3136 3573 2028 7468 ; 0.0384165s (th │ │ │ │ -0001ea60: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ -0001ea70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001ea30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ +0001ea40: 2075 7365 6420 302e 3039 3435 3332 3373 used 0.0945323s │ │ │ │ +0001ea50: 2028 6370 7529 3b20 302e 3033 3839 3933 (cpu); 0.038993 │ │ │ │ +0001ea60: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +0001ea70: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ 0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eaa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001eab0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0001eaa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001eab0: 7c20 2020 2020 2020 3320 2020 2020 2020 | 3 │ │ │ │ 0001eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001eae0: 7c6f 3520 3d20 3468 2020 2020 2020 2020 |o5 = 4h │ │ │ │ -0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eae0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ +0001eaf0: 3468 2020 2020 2020 2020 2020 2020 2020 4h │ │ │ │ 0001eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001eb20: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eb20: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ 0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001eb50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb80: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ -0001eb90: 5b68 205d 2020 2020 2020 2020 2020 2020 [h ] │ │ │ │ -0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ebc0: 7c20 2020 2020 2020 2020 3120 2020 2020 | 1 │ │ │ │ -0001ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eb90: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ +0001eba0: 5b68 205d 2020 2020 2020 2020 2020 2020 [h ] │ │ │ │ +0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ebd0: 7c0a 7c20 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ 0001ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebf0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d |.|o5 : -- │ │ │ │ -0001ec00: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ec30: 7c20 2020 2020 2020 2035 2020 2020 2020 | 5 │ │ │ │ -0001ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ec00: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +0001ec10: 3a20 2d2d 2d2d 2d2d 2020 2020 2020 2020 : ------ │ │ │ │ +0001ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ec40: 2020 2020 7c0a 7c20 2020 2020 2020 2035 |.| 5 │ │ │ │ 0001ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ec70: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ -0001ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001eca0: 7c20 2020 2020 2020 2031 2020 2020 2020 | 1 │ │ │ │ -0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ecd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -0001ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ed10: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ -0001ed20: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ -0001ed30: 7420 6e61 6d65 6420 496e 7075 7449 7353 t named InputIsS │ │ │ │ -0001ed40: 6d6f 6f74 683a 0a3d 3d3d 3d3d 3d3d 3d3d mooth:.========= │ │ │ │ -0001ed50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001ed60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001ed70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0001ed80: 2a20 2243 534d 282e 2e2e 2c49 6e70 7574 * "CSM(...,Input │ │ │ │ -0001ed90: 4973 536d 6f6f 7468 3d3e 2e2e 2e29 2220 IsSmooth=>...)" │ │ │ │ -0001eda0: 2d2d 2073 6565 202a 6e6f 7465 2043 534d -- see *note CSM │ │ │ │ -0001edb0: 3a20 4353 4d2c 202d 2d20 5468 650a 2020 : CSM, -- The. │ │ │ │ -0001edc0: 2020 4368 6572 6e2d 5363 6877 6172 747a Chern-Schwartz │ │ │ │ -0001edd0: 2d4d 6163 5068 6572 736f 6e20 636c 6173 -MacPherson clas │ │ │ │ -0001ede0: 730a 2020 2a20 4575 6c65 7228 2e2e 2e2c s. * Euler(..., │ │ │ │ -0001edf0: 496e 7075 7449 7353 6d6f 6f74 683d 3e2e InputIsSmooth=>. │ │ │ │ -0001ee00: 2e2e 2920 286d 6973 7369 6e67 2064 6f63 ..) (missing doc │ │ │ │ -0001ee10: 756d 656e 7461 7469 6f6e 290a 0a46 6f72 umentation)..For │ │ │ │ -0001ee20: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0001ee30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001ee40: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0001ee50: 6e6f 7465 2049 6e70 7574 4973 536d 6f6f note InputIsSmoo │ │ │ │ -0001ee60: 7468 3a20 496e 7075 7449 7353 6d6f 6f74 th: InputIsSmoot │ │ │ │ -0001ee70: 682c 2069 7320 6120 2a6e 6f74 6520 7379 h, is a *note sy │ │ │ │ -0001ee80: 6d62 6f6c 3a0a 284d 6163 6175 6c61 7932 mbol:.(Macaulay2 │ │ │ │ -0001ee90: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d Doc)Symbol,...-- │ │ │ │ -0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ec70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001ec80: 7c20 2020 2020 2020 6820 2020 2020 2020 | h │ │ │ │ +0001ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ecb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001ecc0: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ecf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ │ │ +0001ed30: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +0001ed40: 7469 6f6e 616c 2061 7267 756d 656e 7420 tional argument │ │ │ │ +0001ed50: 6e61 6d65 6420 496e 7075 7449 7353 6d6f named InputIsSmo │ │ │ │ +0001ed60: 6f74 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d oth:.=========== │ 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2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +0001ee50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001ee60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +0001ee70: 7465 2049 6e70 7574 4973 536d 6f6f 7468 te InputIsSmooth │ │ │ │ +0001ee80: 3a20 496e 7075 7449 7353 6d6f 6f74 682c : InputIsSmooth, │ │ │ │ +0001ee90: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +0001eea0: 6f6c 3a0a 284d 6163 6175 6c61 7932 446f ol:.(Macaulay2Do │ │ │ │ +0001eeb0: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ 0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0001eef0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0001ef00: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0001ef10: 0a2f 6275 696c 642f 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---------------- │ │ │ │ +0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +0001ef10: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0001ef20: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0001ef30: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0001ef40: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0001ef50: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ +0001ef60: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0001ef70: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ +0001ef80: 6963 436c 6173 7365 732e 6d32 3a32 3436 icClasses.m2:246 │ │ │ │ +0001ef90: 333a 302e 0a1f 0a46 696c 653a 2043 6861 3:0....File: Cha │ │ │ │ +0001efa0: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ +0001efb0: 6573 2e69 6e66 6f2c 204e 6f64 653a 2069 es.info, Node: i │ │ │ │ +0001efc0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75 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*note Boolean │ │ │ │ -0001f250: 7661 6c75 653a 2028 4d61 6361 756c 6179 value: (Macaulay │ │ │ │ -0001f260: 3244 6f63 2942 6f6f 6c65 616e 2c2c 200a 2Doc)Boolean,, . │ │ │ │ -0001f270: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -0001f280: 3d3d 3d3d 3d3d 3d3d 0a0a 5465 7374 7320 ========..Tests │ │ │ │ -0001f290: 6966 2074 6865 2069 6e70 7574 2049 6465 if the input Ide │ │ │ │ -0001f2a0: 616c 206f 7220 5269 6e67 456c 656d 656e al or RingElemen │ │ │ │ -0001f2b0: 7420 6973 2048 6f6d 6f67 656e 656f 7573 t is Homogeneous │ │ │ │ -0001f2c0: 2077 6974 6820 7265 7370 6563 7420 746f with respect to │ │ │ │ -0001f2d0: 2074 6865 0a67 7261 6469 6e67 206f 6e20 the.grading on │ │ │ │ -0001f2e0: 7468 6520 7269 6e67 2e20 486f 6d6f 6765 the ring. Homoge │ │ │ │ -0001f2f0: 6e65 6f75 7320 696e 7075 7420 6973 2072 neous input is r │ │ │ │ -0001f300: 6571 7569 7265 6420 666f 7220 616c 6c20 equired for all │ │ │ │ -0001f310: 6d65 7468 6f64 7320 746f 2063 6f6d 7075 methods to compu │ │ │ │ -0001f320: 7465 0a63 6861 7261 6374 6572 6973 7469 te.characteristi │ │ │ │ -0001f330: 6320 636c 6173 7365 732e 0a0a 5468 6973 c classes...This │ │ │ │ -0001f340: 206d 6574 686f 6420 776f 726b 7320 666f method works fo │ │ │ │ -0001f350: 7220 6964 6561 6c73 2069 6e20 7468 6520 r ideals in the │ │ │ │ -0001f360: 6772 6164 6564 2063 6f6f 7264 696e 6174 graded coordinat │ │ │ │ -0001f370: 6520 7269 6e67 7320 6f66 2074 6f72 6963 e rings of toric │ │ │ │ -0001f380: 2076 6172 6965 7469 6573 2c0a 616e 6420 varieties,.and │ │ │ │ -0001f390: 6865 6e63 6520 666f 7220 7072 6f64 7563 hence for produc │ │ │ │ -0001f3a0: 7473 206f 6620 7072 6f6a 6563 7469 7665 ts of projective │ │ │ │ -0001f3b0: 2073 7061 6365 732e 2054 6865 7365 2063 spaces. 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Homogene │ │ │ │ +0001f310: 6f75 7320 696e 7075 7420 6973 2072 6571 ous input is req │ │ │ │ +0001f320: 7569 7265 6420 666f 7220 616c 6c20 6d65 uired for all me │ │ │ │ +0001f330: 7468 6f64 7320 746f 2063 6f6d 7075 7465 thods to compute │ │ │ │ +0001f340: 0a63 6861 7261 6374 6572 6973 7469 6320 .characteristic │ │ │ │ +0001f350: 636c 6173 7365 732e 0a0a 5468 6973 206d classes...This m │ │ │ │ +0001f360: 6574 686f 6420 776f 726b 7320 666f 7220 ethod works for │ │ │ │ +0001f370: 6964 6561 6c73 2069 6e20 7468 6520 6772 ideals in the gr │ │ │ │ +0001f380: 6164 6564 2063 6f6f 7264 696e 6174 6520 aded coordinate │ │ │ │ +0001f390: 7269 6e67 7320 6f66 2074 6f72 6963 2076 rings of toric v │ │ │ │ +0001f3a0: 6172 6965 7469 6573 2c0a 616e 6420 6865 arieties,.and he │ │ │ │ +0001f3b0: 6e63 6520 666f 7220 7072 6f64 7563 7473 nce for products │ │ │ │ +0001f3c0: 206f 6620 7072 6f6a 6563 7469 7665 2073 of projective s │ │ │ │ +0001f3d0: 7061 6365 732e 2054 6865 7365 2063 616e paces. These can │ │ │ │ +0001f3e0: 2062 6520 6372 6561 7465 6420 6469 7265 be created dire │ │ │ │ +0001f3f0: 6374 6c79 2c20 6f72 0a75 7369 6e67 206d ctly, or.using m │ │ │ │ +0001f400: 6574 686f 6473 2074 6865 202a 6e6f 7465 ethods the *note │ │ │ │ +0001f410: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ +0001f420: 696e 673a 204d 756c 7469 5072 6f6a 436f ing: MultiProjCo │ │ │ │ +0001f430: 6f72 6452 696e 672c 206d 6574 686f 6420 ordRing, method │ │ │ │ +0001f440: 6f66 2074 6869 730a 7061 636b 6167 652c of this.package, │ │ │ │ +0001f450: 206f 7220 7769 7468 206d 6574 686f 6473 or with methods │ │ │ │ +0001f460: 2066 726f 6d20 7468 6520 4e6f 726d 616c from the Normal │ │ │ │ +0001f470: 546f 7269 6356 6172 6965 7469 6573 2050 ToricVarieties P │ │ │ │ +0001f480: 6163 6b61 6765 2e0a 0a2b 2d2d 2d2d 2d2d ackage...+------ │ │ │ │ 0001f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0001f4b0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f : R=MultiProjCo │ │ │ │ -0001f4c0: 6f72 6452 696e 6728 7b31 2c32 2c31 7d29 ordRing({1,2,1}) │ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f4c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +0001f4d0: 2052 3d4d 756c 7469 5072 6f6a 436f 6f72 R=MultiProjCoor │ │ │ │ +0001f4e0: 6452 696e 6728 7b31 2c32 2c31 7d29 2020 dRing({1,2,1}) │ │ │ │ 0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f520: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f530: 0a7c 6f31 203d 2052 2020 2020 2020 2020 .|o1 = R │ │ │ │ -0001f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f550: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ 0001f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f570: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f580: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5b0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -0001f5c0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ -0001f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f5d0: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ +0001f5e0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f610: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0001f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f630: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2078 -------+.|i2 : x │ │ │ │ -0001f640: 3d67 656e 7328 5229 2020 2020 2020 2020 =gens(R) │ │ │ │ -0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f670: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f650: 2d2d 2d2d 2d2b 0a7c 6932 203a 2078 3d67 -----+.|i2 : x=g │ │ │ │ +0001f660: 656e 7328 5229 2020 2020 2020 2020 2020 ens(R) │ │ │ │ +0001f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0001f6c0: 203d 207b 7820 2c20 7820 2c20 7820 2c20 = {x , x , x , │ │ │ │ -0001f6d0: 7820 2c20 7820 2c20 7820 2c20 7820 7d20 x , x , x , x } │ │ │ │ -0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f700: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ -0001f710: 2020 2033 2020 2034 2020 2035 2020 2036 3 4 5 6 │ │ │ │ -0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001f750: 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 2020 │ │ │ │ +0001f6d0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +0001f6e0: 207b 7820 2c20 7820 2c20 7820 2c20 7820 {x , x , x , x │ │ │ │ +0001f6f0: 2c20 7820 2c20 7820 2c20 7820 7d20 2020 , x , x , x } │ │ │ │ +0001f700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f720: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ +0001f730: 2033 2020 2034 2020 2035 2020 2036 2020 3 4 5 6 │ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f780: 207c 0a7c 6f32 203a 204c 6973 7420 2020 |.|o2 : List │ │ │ │ -0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7a0: 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 207c | │ │ │ │ +0001f7a0: 0a7c 6f32 203a 204c 6973 7420 2020 2020 .|o2 : List │ │ │ │ 0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f7e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f800: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 3d69 -----+.|i3 : I=i │ │ │ │ -0001f810: 6465 616c 2878 5f30 5e32 2a78 5f33 2d78 deal(x_0^2*x_3-x │ │ │ │ -0001f820: 5f31 2a78 5f30 2a78 5f34 2c78 5f36 5e33 _1*x_0*x_4,x_6^3 │ │ │ │ -0001f830: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001f840: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f820: 2d2d 2d2b 0a7c 6933 203a 2049 3d69 6465 ---+.|i3 : I=ide │ │ │ │ +0001f830: 616c 2878 5f30 5e32 2a78 5f33 2d78 5f31 al(x_0^2*x_3-x_1 │ │ │ │ +0001f840: 2a78 5f30 2a78 5f34 2c78 5f36 5e33 2920 *x_0*x_4,x_6^3) │ │ │ │ 0001f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f860: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f880: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001f890: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0001f8a0: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ -0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0001f8d0: 203d 2069 6465 616c 2028 7820 7820 202d = ideal (x x - │ │ │ │ -0001f8e0: 2078 2078 2078 202c 2078 2029 2020 2020 x x x , x ) │ │ │ │ -0001f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f900: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f910: 2020 2020 2020 2020 2020 2020 2030 2033 0 3 │ │ │ │ -0001f920: 2020 2020 3020 3120 3420 2020 3620 2020 0 1 4 6 │ │ │ │ -0001f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f8a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f8b0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0001f8c0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f8e0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +0001f8f0: 2069 6465 616c 2028 7820 7820 202d 2078 ideal (x x - x │ │ │ │ +0001f900: 2078 2078 202c 2078 2029 2020 2020 2020 x x , x ) │ │ │ │ +0001f910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f920: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f930: 2020 2020 2020 2020 2020 2030 2033 2020 0 3 │ │ │ │ +0001f940: 2020 3020 3120 3420 2020 3620 2020 2020 0 1 4 6 │ │ │ │ +0001f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f960: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f990: 207c 0a7c 6f33 203a 2049 6465 616c 206f |.|o3 : Ideal o │ │ │ │ -0001f9a0: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0001f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f9d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f9b0: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ +0001f9c0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fa10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2069 734d -----+.|i4 : isM │ │ │ │ -0001fa20: 756c 7469 486f 6d6f 6765 6e65 6f75 7320 ultiHomogeneous │ │ │ │ -0001fa30: 4920 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ -0001fa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fa50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fa30: 2d2d 2d2b 0a7c 6934 203a 2069 734d 756c ---+.|i4 : isMul │ │ │ │ +0001fa40: 7469 486f 6d6f 6765 6e65 6f75 7320 4920 tiHomogeneous I │ │ │ │ +0001fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fa70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fa90: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ -0001faa0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ -0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0001fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fab0: 2020 2020 2020 207c 0a7c 6f34 203d 2074 |.|o4 = t │ │ │ │ +0001fac0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001faf0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0001fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0001fb20: 6935 203a 2069 734d 756c 7469 486f 6d6f i5 : isMultiHomo │ │ │ │ -0001fb30: 6765 6e65 6f75 7320 6964 6561 6c28 785f geneous ideal(x_ │ │ │ │ -0001fb40: 302a 785f 332d 785f 312a 785f 302a 785f 0*x_3-x_1*x_0*x_ │ │ │ │ -0001fb50: 342c 785f 365e 3329 2020 2020 2020 207c 4,x_6^3) | │ │ │ │ -0001fb60: 0a7c 496e 7075 7420 7465 726d 2062 656c .|Input term bel │ │ │ │ -0001fb70: 6f77 2069 7320 6e6f 7420 686f 6d6f 6765 ow is not homoge │ │ │ │ -0001fb80: 6e65 6f75 7320 7769 7468 2072 6573 7065 neous with respe │ │ │ │ -0001fb90: 6374 2074 6f20 7468 6520 6772 6164 696e ct to the gradin │ │ │ │ -0001fba0: 677c 0a7c 2d20 7820 7820 7820 202b 2078 g|.|- x x x + x │ │ │ │ -0001fbb0: 2078 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ -0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +0001fb40: 203a 2069 734d 756c 7469 486f 6d6f 6765 : isMultiHomoge │ │ │ │ +0001fb50: 6e65 6f75 7320 6964 6561 6c28 785f 302a neous ideal(x_0* │ │ │ │ +0001fb60: 785f 332d 785f 312a 785f 302a 785f 342c x_3-x_1*x_0*x_4, │ │ │ │ +0001fb70: 785f 365e 3329 2020 2020 2020 207c 0a7c x_6^3) |.| │ │ │ │ +0001fb80: 496e 7075 7420 7465 726d 2062 656c 6f77 Input term below │ │ │ │ +0001fb90: 2069 7320 6e6f 7420 686f 6d6f 6765 6e65 is not homogene │ │ │ │ +0001fba0: 6f75 7320 7769 7468 2072 6573 7065 6374 ous with respect │ │ │ │ +0001fbb0: 2074 6f20 7468 6520 6772 6164 696e 677c to the grading| │ │ │ │ +0001fbc0: 0a7c 2d20 7820 7820 7820 202b 2078 2078 .|- x x x + x x │ │ │ │ 0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fbe0: 2020 207c 0a7c 2020 2030 2031 2034 2020 |.| 0 1 4 │ │ │ │ -0001fbf0: 2020 3020 3320 2020 2020 2020 2020 2020 0 3 │ │ │ │ -0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fc00: 207c 0a7c 2020 2030 2031 2034 2020 2020 |.| 0 1 4 │ │ │ │ +0001fc10: 3020 3320 2020 2020 2020 2020 2020 2020 0 3 │ │ │ │ +0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fc40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc60: 2020 2020 2020 207c 0a7c 6f35 203d 2066 |.|o5 = f │ │ │ │ -0001fc70: 616c 7365 2020 2020 2020 2020 2020 2020 alse │ │ │ │ -0001fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fca0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0001fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fc80: 2020 2020 207c 0a7c 6f35 203d 2066 616c |.|o5 = fal │ │ │ │ +0001fc90: 7365 2020 2020 2020 2020 2020 2020 2020 se │ │ │ │ +0001fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fcc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e6f -----------+..No │ │ │ │ -0001fcf0: 7465 2074 6861 7420 666f 7220 616e 2069 te that for an i │ │ │ │ -0001fd00: 6465 616c 2074 6f20 6265 206d 756c 7469 deal to be multi │ │ │ │ -0001fd10: 2d68 6f6d 6f67 656e 656f 7573 2074 6865 -homogeneous the │ │ │ │ -0001fd20: 2064 6567 7265 6520 7665 6374 6f72 206f degree vector o │ │ │ │ -0001fd30: 6620 616c 6c0a 6d6f 6e6f 6d69 616c 7320 f all.monomials │ │ │ │ -0001fd40: 696e 2061 2067 6976 656e 2067 656e 6572 in a given gener │ │ │ │ -0001fd50: 6174 6f72 206d 7573 7420 6265 2074 6865 ator must be the │ │ │ │ -0001fd60: 2073 616d 652e 0a0a 5761 7973 2074 6f20 same...Ways to │ │ │ │ -0001fd70: 7573 6520 6973 4d75 6c74 6948 6f6d 6f67 use isMultiHomog │ │ │ │ -0001fd80: 656e 656f 7573 3a0a 3d3d 3d3d 3d3d 3d3d eneous:.======== │ │ │ │ -0001fd90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001fda0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6973 =======.. * "is │ │ │ │ -0001fdb0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -0001fdc0: 2849 6465 616c 2922 0a20 202a 2022 6973 (Ideal)". * "is │ │ │ │ -0001fdd0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -0001fde0: 2852 696e 6745 6c65 6d65 6e74 2922 0a0a (RingElement)".. │ │ │ │ -0001fdf0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0001fe00: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0001fe10: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0001fe20: 7420 2a6e 6f74 6520 6973 4d75 6c74 6948 t *note isMultiH │ │ │ │ -0001fe30: 6f6d 6f67 656e 656f 7573 3a20 6973 4d75 omogeneous: isMu │ │ │ │ -0001fe40: 6c74 6948 6f6d 6f67 656e 656f 7573 2c20 ltiHomogeneous, │ │ │ │ -0001fe50: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0001fe60: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ -0001fe70: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0001fe80: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ -0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fd00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e6f 7465 ---------+..Note │ │ │ │ +0001fd10: 2074 6861 7420 666f 7220 616e 2069 6465 that for an ide │ │ │ │ +0001fd20: 616c 2074 6f20 6265 206d 756c 7469 2d68 al to be multi-h │ │ │ │ +0001fd30: 6f6d 6f67 656e 656f 7573 2074 6865 2064 omogeneous the d │ │ │ │ +0001fd40: 6567 7265 6520 7665 6374 6f72 206f 6620 egree vector of │ │ │ │ +0001fd50: 616c 6c0a 6d6f 6e6f 6d69 616c 7320 696e all.monomials in │ │ │ │ +0001fd60: 2061 2067 6976 656e 2067 656e 6572 6174 a given generat │ │ │ │ +0001fd70: 6f72 206d 7573 7420 6265 2074 6865 2073 or must be the s │ │ │ │ +0001fd80: 616d 652e 0a0a 5761 7973 2074 6f20 7573 ame...Ways to us │ │ │ │ +0001fd90: 6520 6973 4d75 6c74 6948 6f6d 6f67 656e e isMultiHomogen │ │ │ │ +0001fda0: 656f 7573 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d eous:.========== │ │ │ │ +0001fdb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001fdc0: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 4d75 =====.. * "isMu │ │ │ │ +0001fdd0: 6c74 6948 6f6d 6f67 656e 656f 7573 2849 ltiHomogeneous(I │ │ │ │ +0001fde0: 6465 616c 2922 0a20 202a 2022 6973 4d75 deal)". * "isMu │ │ │ │ +0001fdf0: 6c74 6948 6f6d 6f67 656e 656f 7573 2852 ltiHomogeneous(R │ │ │ │ +0001fe00: 696e 6745 6c65 6d65 6e74 2922 0a0a 466f ingElement)"..Fo │ │ │ │ +0001fe10: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +0001fe20: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0001fe30: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +0001fe40: 2a6e 6f74 6520 6973 4d75 6c74 6948 6f6d *note isMultiHom │ │ │ │ +0001fe50: 6f67 656e 656f 7573 3a20 6973 4d75 6c74 ogeneous: isMult │ │ │ │ +0001fe60: 6948 6f6d 6f67 656e 656f 7573 2c20 6973 iHomogeneous, is │ │ │ │ +0001fe70: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ +0001fe80: 6675 6e63 7469 6f6e 3a20 284d 6163 6175 function: (Macau │ │ │ │ +0001fe90: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +0001fea0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ 0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0001fee0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0001fef0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0001ff00: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0001ff10: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0001ff20: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -0001ff30: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0001ff40: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ -0001ff50: 6963 436c 6173 7365 732e 6d32 3a31 3937 icClasses.m2:197 │ │ │ │ -0001ff60: 353a 302e 0a1f 0a46 696c 653a 2043 6861 5:0....File: Cha │ │ │ │ -0001ff70: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -0001ff80: 6573 2e69 6e66 6f2c 204e 6f64 653a 204d es.info, Node: M │ │ │ │ -0001ff90: 6574 686f 642c 204e 6578 743a 204d 756c ethod, Next: Mul │ │ │ │ -0001ffa0: 7469 5072 6f6a 436f 6f72 6452 696e 672c tiProjCoordRing, │ │ │ │ -0001ffb0: 2050 7265 763a 2069 734d 756c 7469 486f Prev: isMultiHo │ │ │ │ -0001ffc0: 6d6f 6765 6e65 6f75 732c 2055 703a 2054 mogeneous, Up: T │ │ │ │ -0001ffd0: 6f70 0a0a 4d65 7468 6f64 0a2a 2a2a 2a2a op..Method.***** │ │ │ │ -0001ffe0: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ -0001fff0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00020000: 6f70 7469 6f6e 204d 6574 686f 6420 6973 option Method is │ │ │ │ -00020010: 206f 6e6c 7920 7573 6564 2062 7920 7468 only used by th │ │ │ │ -00020020: 6520 636f 6d6d 616e 6473 202a 6e6f 7465 e commands *note │ │ │ │ -00020030: 2043 534d 3a20 4353 4d2c 2061 6e64 202a CSM: CSM, and * │ │ │ │ -00020040: 6e6f 7465 2045 756c 6572 3a0a 4575 6c65 note Euler:.Eule │ │ │ │ -00020050: 722c 2061 6e64 206f 6e6c 7920 696e 2063 r, and only in c │ │ │ │ -00020060: 6f6d 6269 6e61 7469 6f6e 2077 6974 6820 ombination with │ │ │ │ -00020070: 2a6e 6f74 6520 436f 6d70 4d65 7468 6f64 *note CompMethod │ │ │ │ -00020080: 3a0a 436f 6d70 4d65 7468 6f64 2c3d 3e50 :.CompMethod,=>P │ │ │ │ -00020090: 726f 6a65 6374 6976 6544 6567 7265 652e rojectiveDegree. │ │ │ │ -000200a0: 2054 6865 204d 6574 686f 6420 496e 636c The Method Incl │ │ │ │ -000200b0: 7573 696f 6e45 7863 6c75 7369 6f6e 2077 usionExclusion w │ │ │ │ -000200c0: 696c 6c20 616c 7761 7973 2062 650a 7573 ill always be.us │ │ │ │ -000200d0: 6564 2077 6974 6820 2a6e 6f74 6520 436f ed with *note Co │ │ │ │ -000200e0: 6d70 4d65 7468 6f64 3a20 436f 6d70 4d65 mpMethod: CompMe │ │ │ │ -000200f0: 7468 6f64 2c20 506e 5265 7369 6475 616c thod, PnResidual │ │ │ │ -00020100: 206f 7220 6265 7274 696e 692e 2057 6865 or bertini. Whe │ │ │ │ -00020110: 6e20 7468 6520 696e 7075 740a 6964 6561 n the input.idea │ │ │ │ -00020120: 6c20 6973 2061 2063 6f6d 706c 6574 6520 l is a complete │ │ │ │ -00020130: 696e 7465 7273 6563 7469 6f6e 206f 6e65 intersection one │ │ │ │ -00020140: 206d 6179 2c20 706f 7465 6e74 6961 6c6c may, potentiall │ │ │ │ -00020150: 792c 2073 7065 6564 2075 7020 7468 6520 y, speed up the │ │ │ │ -00020160: 636f 6d70 7574 6174 696f 6e0a 6279 2073 computation.by s │ │ │ │ -00020170: 6574 7469 6e67 204d 6574 686f 643d 3e20 etting Method=> │ │ │ │ -00020180: 4469 7265 6374 436f 6d70 6c65 7465 496e DirectCompleteIn │ │ │ │ -00020190: 742e 2054 6865 206f 7074 696f 6e20 4d65 t. The option Me │ │ │ │ -000201a0: 7468 6f64 2069 7320 6f6e 6c79 2075 7365 thod is only use │ │ │ │ -000201b0: 6420 6279 2074 6865 0a63 6f6d 6d61 6e64 d by the.command │ │ │ │ -000201c0: 7320 2a6e 6f74 6520 4353 4d3a 2043 534d s *note CSM: CSM │ │ │ │ -000201d0: 2c20 616e 6420 2a6e 6f74 6520 4575 6c65 , and *note Eule │ │ │ │ -000201e0: 723a 2045 756c 6572 2c20 616e 6420 6f6e r: Euler, and on │ │ │ │ -000201f0: 6c79 2069 6e20 636f 6d62 696e 6174 696f ly in combinatio │ │ │ │ -00020200: 6e20 7769 7468 0a2a 6e6f 7465 2043 6f6d n with.*note Com │ │ │ │ -00020210: 704d 6574 686f 643a 2043 6f6d 704d 6574 pMethod: CompMet │ │ │ │ -00020220: 686f 642c 3d3e 5072 6f6a 6563 7469 7665 hod,=>Projective │ │ │ │ -00020230: 4465 6772 6565 2e20 5468 6520 4d65 7468 Degree. The Meth │ │ │ │ -00020240: 6f64 2049 6e63 6c75 7369 6f6e 4578 636c od InclusionExcl │ │ │ │ -00020250: 7573 696f 6e0a 7769 6c6c 2061 6c77 6179 usion.will alway │ │ │ │ -00020260: 7320 6265 2075 7365 6420 7769 7468 202a s be used with * │ │ │ │ -00020270: 6e6f 7465 2043 6f6d 704d 6574 686f 643a note CompMethod: │ │ │ │ -00020280: 2043 6f6d 704d 6574 686f 642c 2050 6e52 CompMethod, PnR │ │ │ │ -00020290: 6573 6964 7561 6c20 6f72 2062 6572 7469 esidual or berti │ │ │ │ -000202a0: 6e69 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ni...+---------- │ │ │ │ -000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000202e0: 7c69 3120 3a20 5220 3d20 5a5a 2f33 3237 |i1 : R = ZZ/327 │ │ │ │ -000202f0: 3439 5b78 5f30 2e2e 785f 365d 2020 2020 49[x_0..x_6] │ │ │ │ -00020300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020310: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001fed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fef0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +0001ff00: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +0001ff10: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +0001ff20: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +0001ff30: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +0001ff40: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ +0001ff50: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +0001ff60: 2f0a 4368 6172 6163 7465 7269 7374 6963 /.Characteristic │ │ │ │ +0001ff70: 436c 6173 7365 732e 6d32 3a31 3937 353a Classes.m2:1975: │ │ │ │ +0001ff80: 302e 0a1f 0a46 696c 653a 2043 6861 7261 0....File: Chara │ │ │ │ +0001ff90: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +0001ffa0: 2e69 6e66 6f2c 204e 6f64 653a 204d 6574 .info, Node: Met │ │ │ │ +0001ffb0: 686f 642c 204e 6578 743a 204d 756c 7469 hod, Next: Multi │ │ │ │ +0001ffc0: 5072 6f6a 436f 6f72 6452 696e 672c 2050 ProjCoordRing, P │ │ │ │ +0001ffd0: 7265 763a 2069 734d 756c 7469 486f 6d6f rev: isMultiHomo │ │ │ │ +0001ffe0: 6765 6e65 6f75 732c 2055 703a 2054 6f70 geneous, Up: Top │ │ │ │ +0001fff0: 0a0a 4d65 7468 6f64 0a2a 2a2a 2a2a 2a0a ..Method.******. │ │ │ │ +00020000: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +00020010: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f70 ========..The op │ │ │ │ +00020020: 7469 6f6e 204d 6574 686f 6420 6973 206f tion Method is o │ │ │ │ +00020030: 6e6c 7920 7573 6564 2062 7920 7468 6520 nly used by the │ │ │ │ +00020040: 636f 6d6d 616e 6473 202a 6e6f 7465 2043 commands *note C │ │ │ │ +00020050: 534d 3a20 4353 4d2c 2061 6e64 202a 6e6f SM: CSM, and *no │ │ │ │ +00020060: 7465 2045 756c 6572 3a0a 4575 6c65 722c te Euler:.Euler, │ │ │ │ +00020070: 2061 6e64 206f 6e6c 7920 696e 2063 6f6d and only in com │ │ │ │ +00020080: 6269 6e61 7469 6f6e 2077 6974 6820 2a6e bination with *n │ │ │ │ +00020090: 6f74 6520 436f 6d70 4d65 7468 6f64 3a0a ote CompMethod:. │ │ │ │ +000200a0: 436f 6d70 4d65 7468 6f64 2c3d 3e50 726f CompMethod,=>Pro │ │ │ │ +000200b0: 6a65 6374 6976 6544 6567 7265 652e 2054 jectiveDegree. T │ │ │ │ +000200c0: 6865 204d 6574 686f 6420 496e 636c 7573 he Method Inclus │ │ │ │ +000200d0: 696f 6e45 7863 6c75 7369 6f6e 2077 696c ionExclusion wil │ │ │ │ +000200e0: 6c20 616c 7761 7973 2062 650a 7573 6564 l always be.used │ │ │ │ +000200f0: 2077 6974 6820 2a6e 6f74 6520 436f 6d70 with *note Comp │ │ │ │ +00020100: 4d65 7468 6f64 3a20 436f 6d70 4d65 7468 Method: CompMeth │ │ │ │ +00020110: 6f64 2c20 506e 5265 7369 6475 616c 206f od, PnResidual o │ │ │ │ +00020120: 7220 6265 7274 696e 692e 2057 6865 6e20 r bertini. When │ │ │ │ +00020130: 7468 6520 696e 7075 740a 6964 6561 6c20 the input.ideal │ │ │ │ +00020140: 6973 2061 2063 6f6d 706c 6574 6520 696e is a complete in │ │ │ │ +00020150: 7465 7273 6563 7469 6f6e 206f 6e65 206d tersection one m │ │ │ │ +00020160: 6179 2c20 706f 7465 6e74 6961 6c6c 792c ay, potentially, │ │ │ │ +00020170: 2073 7065 6564 2075 7020 7468 6520 636f speed up the co │ │ │ │ +00020180: 6d70 7574 6174 696f 6e0a 6279 2073 6574 mputation.by set │ │ │ │ +00020190: 7469 6e67 204d 6574 686f 643d 3e20 4469 ting Method=> Di │ │ │ │ +000201a0: 7265 6374 436f 6d70 6c65 7465 496e 742e rectCompleteInt. │ │ │ │ +000201b0: 2054 6865 206f 7074 696f 6e20 4d65 7468 The option Meth │ │ │ │ +000201c0: 6f64 2069 7320 6f6e 6c79 2075 7365 6420 od is only used │ │ │ │ +000201d0: 6279 2074 6865 0a63 6f6d 6d61 6e64 7320 by the.commands │ │ │ │ +000201e0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20 *note CSM: CSM, │ │ │ │ +000201f0: 616e 6420 2a6e 6f74 6520 4575 6c65 723a and *note Euler: │ │ │ │ +00020200: 2045 756c 6572 2c20 616e 6420 6f6e 6c79 Euler, and only │ │ │ │ +00020210: 2069 6e20 636f 6d62 696e 6174 696f 6e20 in combination │ │ │ │ +00020220: 7769 7468 0a2a 6e6f 7465 2043 6f6d 704d with.*note CompM │ │ │ │ +00020230: 6574 686f 643a 2043 6f6d 704d 6574 686f ethod: CompMetho │ │ │ │ +00020240: 642c 3d3e 5072 6f6a 6563 7469 7665 4465 d,=>ProjectiveDe │ │ │ │ +00020250: 6772 6565 2e20 5468 6520 4d65 7468 6f64 gree. The Method │ │ │ │ +00020260: 2049 6e63 6c75 7369 6f6e 4578 636c 7573 InclusionExclus │ │ │ │ +00020270: 696f 6e0a 7769 6c6c 2061 6c77 6179 7320 ion.will always │ │ │ │ +00020280: 6265 2075 7365 6420 7769 7468 202a 6e6f be used with *no │ │ │ │ +00020290: 7465 2043 6f6d 704d 6574 686f 643a 2043 te CompMethod: C │ │ │ │ +000202a0: 6f6d 704d 6574 686f 642c 2050 6e52 6573 ompMethod, PnRes │ │ │ │ +000202b0: 6964 7561 6c20 6f72 2062 6572 7469 6e69 idual or bertini │ │ │ │ +000202c0: 2e0a 0a2b 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 2b0a 7c69 ------------+.|i │ │ │ │ +00020300: 3120 3a20 5220 3d20 5a5a 2f33 3237 3439 1 : R = ZZ/32749 │ │ │ │ +00020310: 5b78 5f30 2e2e 785f 365d 2020 2020 2020 [x_0..x_6] │ │ │ │ 00020320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020330: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00020340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020350: 2020 2020 7c0a 7c6f 3120 3d20 5220 2020 |.|o1 = R │ │ │ │ +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 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020390: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000203a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020370: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000203b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -000203d0: 3a20 506f 6c79 6e6f 6d69 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│ │ -00020c00: 2062 7920 7370 6563 6966 7969 6e67 2077 by specifying w │ │ │ │ -00020c10: 6861 7420 7375 6273 6574 206f 6620 7468 hat subset of th │ │ │ │ -00020c20: 6520 6765 6e65 7261 746f 7273 206f 6620 e generators of │ │ │ │ -00020c30: 7468 6520 696e 7075 7420 6964 6561 6c0a the input ideal. │ │ │ │ -00020c40: 6465 6669 6e65 2061 2073 6d6f 6f74 6820 define a smooth │ │ │ │ -00020c50: 7375 6273 6368 656d 6520 2869 6620 7468 subscheme (if th │ │ │ │ -00020c60: 6973 2069 7320 6b6e 6f77 6e29 2c20 7365 is is known), se │ │ │ │ -00020c70: 6520 2a6e 6f74 6520 496e 6473 4f66 536d e *note IndsOfSm │ │ │ │ -00020c80: 6f6f 7468 3a0a 496e 6473 4f66 536d 6f6f ooth:.IndsOfSmoo │ │ │ │ -00020c90: 7468 2c2e 0a0a 4675 6e63 7469 6f6e 7320 th,...Functions │ │ │ │ -00020ca0: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ -00020cb0: 6775 6d65 6e74 206e 616d 6564 204d 6574 gument named Met │ │ │ │ -00020cc0: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d hod:.=========== │ │ │ │ -00020cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020ce0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020cf0: 3d3d 3d0a 0a20 202a 2022 4353 4d28 2e2e ===.. * "CSM(.. │ │ │ │ -00020d00: 2e2c 4d65 7468 6f64 3d3e 2e2e 2e29 2220 .,Method=>...)" │ │ │ │ -00020d10: 2d2d 2073 6565 202a 6e6f 7465 2043 534d -- see *note CSM │ │ │ │ -00020d20: 3a20 4353 4d2c 202d 2d20 5468 650a 2020 : CSM, -- The. │ │ │ │ -00020d30: 2020 4368 6572 6e2d 5363 6877 6172 747a Chern-Schwartz │ │ │ │ -00020d40: 2d4d 6163 5068 6572 736f 6e20 636c 6173 -MacPherson clas │ │ │ │ -00020d50: 730a 2020 2a20 4575 6c65 7228 2e2e 2e2c s. * Euler(..., │ │ │ │ -00020d60: 4d65 7468 6f64 3d3e 2e2e 2e29 2028 6d69 Method=>...) (mi │ │ │ │ -00020d70: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -00020d80: 696f 6e29 0a0a 466f 7220 7468 6520 7072 ion)..For the pr │ │ │ │ -00020d90: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00020da0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00020db0: 206f 626a 6563 7420 2a6e 6f74 6520 4d65 object *note Me │ │ │ │ -00020dc0: 7468 6f64 3a20 4d65 7468 6f64 2c20 6973 thod: Method, is │ │ │ │ -00020dd0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ -00020de0: 2028 4d61 6361 756c 6179 3244 6f63 2953 (Macaulay2Doc)S │ │ │ │ -00020df0: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ -00020e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020e10: 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 2b0a --------------+. │ │ │ │ +00020bc0: 0a57 6865 6e20 7573 696e 6720 7468 6520 .When using the │ │ │ │ +00020bd0: 4469 7265 6374 436f 6d70 6c65 7465 496e DirectCompleteIn │ │ │ │ +00020be0: 7420 6d65 7468 6f64 206f 6e65 206d 6179 t method one may │ │ │ │ +00020bf0: 2070 6f74 656e 7469 616c 6c79 2066 7572 potentially fur │ │ │ │ +00020c00: 7468 6572 2073 7065 6564 2075 700a 636f ther speed up.co │ │ │ │ +00020c10: 6d70 7574 6174 696f 6e20 7469 6d65 2062 mputation time b │ │ │ │ +00020c20: 7920 7370 6563 6966 7969 6e67 2077 6861 y specifying wha │ │ │ │ +00020c30: 7420 7375 6273 6574 206f 6620 7468 6520 t subset of the │ │ │ │ +00020c40: 6765 6e65 7261 746f 7273 206f 6620 7468 generators of th │ │ │ │ +00020c50: 6520 696e 7075 7420 6964 6561 6c0a 6465 e input ideal.de │ │ │ │ +00020c60: 6669 6e65 2061 2073 6d6f 6f74 6820 7375 fine a smooth su │ │ │ │ +00020c70: 6273 6368 656d 6520 2869 6620 7468 6973 bscheme (if this │ │ │ │ +00020c80: 2069 7320 6b6e 6f77 6e29 2c20 7365 6520 is known), see │ │ │ │ +00020c90: 2a6e 6f74 6520 496e 6473 4f66 536d 6f6f *note IndsOfSmoo │ │ │ │ +00020ca0: 7468 3a0a 496e 6473 4f66 536d 6f6f 7468 th:.IndsOfSmooth │ │ │ │ +00020cb0: 2c2e 0a0a 4675 6e63 7469 6f6e 7320 7769 ,...Functions wi │ │ │ │ +00020cc0: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ +00020cd0: 6d65 6e74 206e 616d 6564 204d 6574 686f ment named Metho │ │ │ │ +00020ce0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d d:.============= │ │ │ │ +00020cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00020d00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00020d10: 3d0a 0a20 202a 2022 4353 4d28 2e2e 2e2c =.. * "CSM(..., │ │ │ │ +00020d20: 4d65 7468 6f64 3d3e 2e2e 2e29 2220 2d2d Method=>...)" -- │ │ │ │ +00020d30: 2073 6565 202a 6e6f 7465 2043 534d 3a20 see *note CSM: │ │ │ │ +00020d40: 4353 4d2c 202d 2d20 5468 650a 2020 2020 CSM, -- The. │ │ │ │ +00020d50: 4368 6572 6e2d 5363 6877 6172 747a 2d4d Chern-Schwartz-M │ │ │ │ +00020d60: 6163 5068 6572 736f 6e20 636c 6173 730a acPherson class. │ │ │ │ +00020d70: 2020 2a20 4575 6c65 7228 2e2e 2e2c 4d65 * Euler(...,Me │ │ │ │ +00020d80: 7468 6f64 3d3e 2e2e 2e29 2028 6d69 7373 thod=>...) (miss │ │ │ │ +00020d90: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +00020da0: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ +00020db0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00020dc0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00020dd0: 626a 6563 7420 2a6e 6f74 6520 4d65 7468 bject *note Meth │ │ │ │ +00020de0: 6f64 3a20 4d65 7468 6f64 2c20 6973 2061 od: Method, is a │ │ │ │ +00020df0: 202a 6e6f 7465 2073 796d 626f 6c3a 2028 *note symbol: ( │ │ │ │ +00020e00: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ +00020e10: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 00020e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020e40: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00020e50: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00020e60: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00020e70: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00020e80: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00020e90: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ -00020ea0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00020eb0: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -00020ec0: 6c61 7373 6573 2e6d 323a 3233 3935 3a30 lasses.m2:2395:0 │ │ │ │ -00020ed0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ -00020ee0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -00020ef0: 696e 666f 2c20 4e6f 6465 3a20 4d75 6c74 info, Node: Mult │ │ │ │ -00020f00: 6950 726f 6a43 6f6f 7264 5269 6e67 2c20 iProjCoordRing, │ │ │ │ -00020f10: 4e65 7874 3a20 4f75 7470 7574 2c20 5072 Next: Output, Pr │ │ │ │ -00020f20: 6576 3a20 4d65 7468 6f64 2c20 5570 3a20 ev: Method, Up: │ │ │ │ -00020f30: 546f 700a 0a4d 756c 7469 5072 6f6a 436f Top..MultiProjCo │ │ │ │ -00020f40: 6f72 6452 696e 6720 2d2d 2041 2071 7569 ordRing -- A qui │ │ │ │ -00020f50: 636b 2077 6179 2074 6f20 6275 696c 6420 ck way to build │ │ │ │ -00020f60: 7468 6520 636f 6f72 6469 6e61 7465 2072 the coordinate r │ │ │ │ -00020f70: 696e 6720 6f66 2061 2070 726f 6475 6374 ing of a product │ │ │ │ -00020f80: 206f 6620 7072 6f6a 6563 7469 7665 2073 of projective s │ │ │ │ -00020f90: 7061 6365 730a 2a2a 2a2a 2a2a 2a2a 2a2a paces.********** │ │ │ │ -00020fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00020fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00020e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020e60: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00020e70: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00020e80: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +00020e90: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +00020ea0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +00020eb0: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ +00020ec0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ +00020ed0: 6861 7261 6374 6572 6973 7469 6343 6c61 haracteristicCla │ │ │ │ +00020ee0: 7373 6573 2e6d 323a 3233 3935 3a30 2e0a sses.m2:2395:0.. │ │ │ │ +00020ef0: 1f0a 4669 6c65 3a20 4368 6172 6163 7465 ..File: Characte │ │ │ │ +00020f00: 7269 7374 6963 436c 6173 7365 732e 696e risticClasses.in │ │ │ │ +00020f10: 666f 2c20 4e6f 6465 3a20 4d75 6c74 6950 fo, Node: MultiP │ │ │ │ +00020f20: 726f 6a43 6f6f 7264 5269 6e67 2c20 4e65 rojCoordRing, Ne │ │ │ │ +00020f30: 7874 3a20 4f75 7470 7574 2c20 5072 6576 xt: Output, Prev │ │ │ │ +00020f40: 3a20 4d65 7468 6f64 2c20 5570 3a20 546f : Method, Up: To │ │ │ │ +00020f50: 700a 0a4d 756c 7469 5072 6f6a 436f 6f72 p..MultiProjCoor │ │ │ │ +00020f60: 6452 696e 6720 2d2d 2041 2071 7569 636b dRing -- A quick │ │ │ │ +00020f70: 2077 6179 2074 6f20 6275 696c 6420 7468 way to build th │ │ │ │ +00020f80: 6520 636f 6f72 6469 6e61 7465 2072 696e e coordinate rin │ │ │ │ +00020f90: 6720 6f66 2061 2070 726f 6475 6374 206f g of a product o │ │ │ │ +00020fa0: 6620 7072 6f6a 6563 7469 7665 2073 7061 f projective spa │ │ │ │ +00020fb0: 6365 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ces.************ │ │ │ │ 00020fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00020fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00020fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00020ff0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00021000: 653a 200a 2020 2020 2020 2020 4d75 6c74 e: . Mult │ │ │ │ -00021010: 6950 726f 6a43 6f6f 7264 5269 6e67 2044 iProjCoordRing D │ │ │ │ -00021020: 696d 730a 2020 2020 2020 2020 4d75 6c74 ims. Mult │ │ │ │ -00021030: 6950 726f 6a43 6f6f 7264 5269 6e67 2028 iProjCoordRing ( │ │ │ │ -00021040: 436f 6566 6652 696e 672c 4469 6d73 290a CoeffRing,Dims). │ │ │ │ -00021050: 2020 2020 2020 2020 4d75 6c74 6950 726f MultiPro │ │ │ │ -00021060: 6a43 6f6f 7264 5269 6e67 2028 7661 722c jCoordRing (var, │ │ │ │ -00021070: 4469 6d73 290a 2020 2020 2020 2020 4d75 Dims). Mu │ │ │ │ -00021080: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -00021090: 2028 436f 6566 6652 696e 672c 7661 722c (CoeffRing,var, │ │ │ │ -000210a0: 4469 6d73 290a 2020 2a20 496e 7075 7473 Dims). * Inputs │ │ │ │ -000210b0: 3a0a 2020 2020 2020 2a20 4469 6d73 2c20 :. * Dims, │ │ │ │ -000210c0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -000210d0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -000210e0: 2c2c 2072 6570 7265 7365 6e74 696e 6720 ,, representing │ │ │ │ -000210f0: 7468 6520 6469 6d65 6e73 696f 6e73 206f the dimensions o │ │ │ │ -00021100: 660a 2020 2020 2020 2020 7468 6520 7072 f. the pr │ │ │ │ -00021110: 6f6a 6563 7469 7665 2073 7061 6365 732c ojective spaces, │ │ │ │ -00021120: 2069 2e65 2e20 7b6e 5f31 2c2e 2e2e 2c6e i.e. {n_1,...,n │ │ │ │ -00021130: 5f6d 7d20 636f 7272 6573 706f 6e64 7320 _m} corresponds │ │ │ │ -00021140: 746f 205c 5050 5e7b 6e5f 317d 0a20 2020 to \PP^{n_1}. │ │ │ │ -00021150: 2020 2020 2078 2e2e 2e2e 2078 205c 5050 x.... x \PP │ │ │ │ -00021160: 5e7b 6e5f 6d7d 0a20 2020 2020 202a 2043 ^{n_m}. * C │ │ │ │ -00021170: 6f65 6666 5269 6e67 2c20 6120 2a6e 6f74 oeffRing, a *not │ │ │ │ -00021180: 6520 7269 6e67 3a20 284d 6163 6175 6c61 e ring: (Macaula │ │ │ │ -00021190: 7932 446f 6329 5269 6e67 2c2c 2074 6865 y2Doc)Ring,, the │ │ │ │ -000211a0: 2063 6f65 6666 6963 6965 6e74 2072 696e coefficient rin │ │ │ │ -000211b0: 6720 6f66 0a20 2020 2020 2020 2074 6865 g of. the │ │ │ │ -000211c0: 2067 7261 6465 6420 706f 6c79 6e6f 6d69 graded polynomi │ │ │ │ -000211d0: 616c 2072 696e 6720 746f 2062 6520 6275 al ring to be bu │ │ │ │ -000211e0: 696c 7420 6279 2074 6865 206d 6574 686f ilt by the metho │ │ │ │ -000211f0: 642c 2062 7920 6465 6661 756c 7420 7468 d, by default th │ │ │ │ -00021200: 6973 0a20 2020 2020 2020 2069 7320 5c5a is. is \Z │ │ │ │ -00021210: 5a2f 3332 3734 390a 2020 2020 2020 2a20 Z/32749. * │ │ │ │ -00021220: 7661 722c 2061 202a 6e6f 7465 2073 796d var, a *note sym │ │ │ │ -00021230: 626f 6c3a 2028 4d61 6361 756c 6179 3244 bol: (Macaulay2D │ │ │ │ -00021240: 6f63 2953 796d 626f 6c2c 2c20 746f 2062 oc)Symbol,, to b │ │ │ │ -00021250: 6520 7573 6564 2066 6f72 2074 6865 0a20 e used for the. │ │ │ │ -00021260: 2020 2020 2020 2069 6e74 6572 6d65 6469 intermedi │ │ │ │ -00021270: 6174 6573 206f 6620 7468 6520 6772 6164 ates of the grad │ │ │ │ -00021280: 6564 2070 6f6c 796e 6f6d 6961 6c20 7269 ed polynomial ri │ │ │ │ -00021290: 6e67 2074 6f20 6265 2062 7569 6c74 2062 ng to be built b │ │ │ │ -000212a0: 7920 7468 6520 6d65 7468 6f64 0a20 202a y the method. * │ │ │ │ -000212b0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -000212c0: 2a20 6120 2a6e 6f74 6520 7269 6e67 3a20 * a *note ring: │ │ │ │ -000212d0: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ -000212e0: 6e67 2c2c 2074 6865 2067 7261 6465 6420 ng,, the graded │ │ │ │ -000212f0: 636f 6f72 6469 6e61 7465 2072 696e 6720 coordinate ring │ │ │ │ -00021300: 6f66 2074 6865 0a20 2020 2020 2020 205c of the. \ │ │ │ │ -00021310: 5050 5e7b 6e5f 317d 2078 2e2e 2e2e 2078 PP^{n_1} x.... x │ │ │ │ -00021320: 205c 5050 5e7b 6e5f 6d7d 2077 6865 7265 \PP^{n_m} where │ │ │ │ -00021330: 207b 6e5f 312c 2e2e 2e2c 6e5f 6d7d 2069 {n_1,...,n_m} i │ │ │ │ -00021340: 7320 7468 6520 696e 7075 7420 6c69 7374 s the input list │ │ │ │ -00021350: 206f 660a 2020 2020 2020 2020 6469 6d65 of. dime │ │ │ │ -00021360: 6e73 696f 6e73 0a0a 4465 7363 7269 7074 nsions..Descript │ │ │ │ -00021370: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00021380: 0a43 6f6d 7075 7465 7320 7468 6520 6772 .Computes the gr │ │ │ │ -00021390: 6164 6564 2063 6f6f 7264 696e 6174 6520 aded coordinate │ │ │ │ -000213a0: 7269 6e67 206f 6620 7468 6520 5c50 505e ring of the \PP^ │ │ │ │ -000213b0: 7b6e 5f31 7d20 782e 2e2e 2e20 7820 5c50 {n_1} x.... x \P │ │ │ │ -000213c0: 505e 7b6e 5f6d 7d20 7768 6572 650a 7b6e P^{n_m} where.{n │ │ │ │ -000213d0: 5f31 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 _1,...,n_m} is t │ │ │ │ -000213e0: 6865 2069 6e70 7574 206c 6973 7420 6f66 he input list of │ │ │ │ -000213f0: 2064 696d 656e 7369 6f6e 732e 2054 6869 dimensions. Thi │ │ │ │ -00021400: 7320 6d65 7468 6f64 2069 7320 7573 6564 s method is used │ │ │ │ -00021410: 2074 6f20 7175 6963 6b6c 790a 6275 696c to quickly.buil │ │ │ │ -00021420: 6420 7468 6520 636f 6f72 6469 6e61 7465 d the coordinate │ │ │ │ -00021430: 2072 696e 6720 6f66 2061 2070 726f 6475 ring of a produ │ │ │ │ -00021440: 6374 206f 6620 7072 6f6a 6563 7469 7665 ct of projective │ │ │ │ -00021450: 2073 7061 6365 7320 666f 7220 7573 6520 spaces for use │ │ │ │ -00021460: 696e 0a63 6f6d 7075 7461 7469 6f6e 732e in.computations. │ │ │ │ -00021470: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021010: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ +00021020: 200a 2020 2020 2020 2020 4d75 6c74 6950 . MultiP │ │ │ │ +00021030: 726f 6a43 6f6f 7264 5269 6e67 2044 696d rojCoordRing Dim │ │ │ │ +00021040: 730a 2020 2020 2020 2020 4d75 6c74 6950 s. MultiP │ │ │ │ +00021050: 726f 6a43 6f6f 7264 5269 6e67 2028 436f rojCoordRing (Co │ │ │ │ +00021060: 6566 6652 696e 672c 4469 6d73 290a 2020 effRing,Dims). │ │ │ │ +00021070: 2020 2020 2020 4d75 6c74 6950 726f 6a43 MultiProjC │ │ │ │ +00021080: 6f6f 7264 5269 6e67 2028 7661 722c 4469 oordRing (var,Di │ │ │ │ +00021090: 6d73 290a 2020 2020 2020 2020 4d75 6c74 ms). Mult │ │ │ │ +000210a0: 6950 726f 6a43 6f6f 7264 5269 6e67 2028 iProjCoordRing ( │ │ │ │ +000210b0: 436f 6566 6652 696e 672c 7661 722c 4469 CoeffRing,var,Di │ │ │ │ +000210c0: 6d73 290a 2020 2a20 496e 7075 7473 3a0a ms). * Inputs:. │ │ │ │ +000210d0: 2020 2020 2020 2a20 4469 6d73 2c20 6120 * Dims, a │ │ │ │ +000210e0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +000210f0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00021100: 2072 6570 7265 7365 6e74 696e 6720 7468 representing th │ │ │ │ +00021110: 6520 6469 6d65 6e73 696f 6e73 206f 660a e dimensions of. │ │ │ │ +00021120: 2020 2020 2020 2020 7468 6520 7072 6f6a the proj │ │ │ │ +00021130: 6563 7469 7665 2073 7061 6365 732c 2069 ective spaces, i │ │ │ │ +00021140: 2e65 2e20 7b6e 5f31 2c2e 2e2e 2c6e 5f6d .e. {n_1,...,n_m │ │ │ │ +00021150: 7d20 636f 7272 6573 706f 6e64 7320 746f } corresponds to │ │ │ │ +00021160: 205c 5050 5e7b 6e5f 317d 0a20 2020 2020 \PP^{n_1}. │ │ │ │ +00021170: 2020 2078 2e2e 2e2e 2078 205c 5050 5e7b x.... x \PP^{ │ │ │ │ +00021180: 6e5f 6d7d 0a20 2020 2020 202a 2043 6f65 n_m}. * Coe │ │ │ │ +00021190: 6666 5269 6e67 2c20 6120 2a6e 6f74 6520 ffRing, a *note │ │ │ │ +000211a0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +000211b0: 446f 6329 5269 6e67 2c2c 2074 6865 2063 Doc)Ring,, the c │ │ │ │ +000211c0: 6f65 6666 6963 6965 6e74 2072 696e 6720 oefficient ring │ │ │ │ +000211d0: 6f66 0a20 2020 2020 2020 2074 6865 2067 of. the g │ │ │ │ +000211e0: 7261 6465 6420 706f 6c79 6e6f 6d69 616c raded polynomial │ │ │ │ +000211f0: 2072 696e 6720 746f 2062 6520 6275 696c ring to be buil │ │ │ │ +00021200: 7420 6279 2074 6865 206d 6574 686f 642c t by the method, │ │ │ │ +00021210: 2062 7920 6465 6661 756c 7420 7468 6973 by default this │ │ │ │ +00021220: 0a20 2020 2020 2020 2069 7320 5c5a 5a2f . is \ZZ/ │ │ │ │ +00021230: 3332 3734 390a 2020 2020 2020 2a20 7661 32749. * va │ │ │ │ +00021240: 722c 2061 202a 6e6f 7465 2073 796d 626f r, a *note symbo │ │ │ │ +00021250: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ +00021260: 2953 796d 626f 6c2c 2c20 746f 2062 6520 )Symbol,, to be │ │ │ │ +00021270: 7573 6564 2066 6f72 2074 6865 0a20 2020 used for the. │ │ │ │ +00021280: 2020 2020 2069 6e74 6572 6d65 6469 6174 intermediat │ │ │ │ +00021290: 6573 206f 6620 7468 6520 6772 6164 6564 es of the graded │ │ │ │ +000212a0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +000212b0: 2074 6f20 6265 2062 7569 6c74 2062 7920 to be built by │ │ │ │ +000212c0: 7468 6520 6d65 7468 6f64 0a20 202a 204f the method. * O │ │ │ │ +000212d0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +000212e0: 6120 2a6e 6f74 6520 7269 6e67 3a20 284d a *note ring: (M │ │ │ │ +000212f0: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ +00021300: 2c2c 2074 6865 2067 7261 6465 6420 636f ,, the graded co │ │ │ │ +00021310: 6f72 6469 6e61 7465 2072 696e 6720 6f66 ordinate ring of │ │ │ │ +00021320: 2074 6865 0a20 2020 2020 2020 205c 5050 the. \PP │ │ │ │ +00021330: 5e7b 6e5f 317d 2078 2e2e 2e2e 2078 205c ^{n_1} x.... x \ │ │ │ │ +00021340: 5050 5e7b 6e5f 6d7d 2077 6865 7265 207b PP^{n_m} where { │ │ │ │ +00021350: 6e5f 312c 2e2e 2e2c 6e5f 6d7d 2069 7320 n_1,...,n_m} is │ │ │ │ +00021360: 7468 6520 696e 7075 7420 6c69 7374 206f the input list o │ │ │ │ +00021370: 660a 2020 2020 2020 2020 6469 6d65 6e73 f. dimens │ │ │ │ +00021380: 696f 6e73 0a0a 4465 7363 7269 7074 696f ions..Descriptio │ │ │ │ +00021390: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 n.===========..C │ │ │ │ +000213a0: 6f6d 7075 7465 7320 7468 6520 6772 6164 omputes the grad │ │ │ │ +000213b0: 6564 2063 6f6f 7264 696e 6174 6520 7269 ed coordinate ri │ │ │ │ +000213c0: 6e67 206f 6620 7468 6520 5c50 505e 7b6e ng of the \PP^{n │ │ │ │ +000213d0: 5f31 7d20 782e 2e2e 2e20 7820 5c50 505e _1} x.... x \PP^ │ │ │ │ +000213e0: 7b6e 5f6d 7d20 7768 6572 650a 7b6e 5f31 {n_m} where.{n_1 │ │ │ │ +000213f0: 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 6865 ,...,n_m} is the │ │ │ │ +00021400: 2069 6e70 7574 206c 6973 7420 6f66 2064 input list of d │ │ │ │ +00021410: 696d 656e 7369 6f6e 732e 2054 6869 7320 imensions. This │ │ │ │ +00021420: 6d65 7468 6f64 2069 7320 7573 6564 2074 method is used t │ │ │ │ +00021430: 6f20 7175 6963 6b6c 790a 6275 696c 6420 o quickly.build │ │ │ │ +00021440: 7468 6520 636f 6f72 6469 6e61 7465 2072 the coordinate r │ │ │ │ +00021450: 696e 6720 6f66 2061 2070 726f 6475 6374 ing of a product │ │ │ │ +00021460: 206f 6620 7072 6f6a 6563 7469 7665 2073 of projective s │ │ │ │ +00021470: 7061 6365 7320 666f 7220 7573 6520 696e paces for use in │ │ │ │ +00021480: 0a63 6f6d 7075 7461 7469 6f6e 732e 0a0a .computations... │ │ │ │ +00021490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000214a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000214b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000214c0: 2b0a 7c69 3120 3a20 533d 4d75 6c74 6950 +.|i1 : S=MultiP │ │ │ │ -000214d0: 726f 6a43 6f6f 7264 5269 6e67 2851 512c rojCoordRing(QQ, │ │ │ │ -000214e0: 7379 6d62 6f6c 207a 2c7b 312c 332c 337d symbol z,{1,3,3} │ │ │ │ -000214f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00021500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021510: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000214d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000214e0: 7c69 3120 3a20 533d 4d75 6c74 6950 726f |i1 : S=MultiPro │ │ │ │ +000214f0: 6a43 6f6f 7264 5269 6e67 2851 512c 7379 jCoordRing(QQ,sy │ │ │ │ +00021500: 6d62 6f6c 207a 2c7b 312c 332c 337d 2920 mbol z,{1,3,3}) │ │ │ │ +00021510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021530: 7c20 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: 7c0a 7c6f 3120 3d20 5320 2020 2020 2020 |.|o1 = S │ │ │ │ -00021570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021580: 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 7c0a |. │ │ │ │ +00021580: 7c6f 3120 3d20 5320 2020 2020 2020 2020 |o1 = S │ │ │ │ 00021590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215d0: 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 7c0a |. │ │ │ │ +000215d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000215e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021600: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ -00021610: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ -00021620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021620: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ +00021630: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00021640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00021660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021660: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021670: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000216a0: 2b0a 7c69 3220 3a20 6465 6772 6565 7320 +.|i2 : degrees │ │ │ │ -000216b0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -000216c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000216a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000216b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000216c0: 7c69 3220 3a20 6465 6772 6565 7320 5320 |i2 : degrees S │ │ │ │ 000216d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021710: 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 7c0a |. │ │ │ │ +00021710: 7c20 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: 7c0a 7c6f 3220 3d20 7b7b 312c 2030 2c20 |.|o2 = {{1, 0, │ │ │ │ -00021750: 307d 2c20 7b31 2c20 302c 2030 7d2c 207b 0}, {1, 0, 0}, { │ │ │ │ -00021760: 302c 2031 2c20 307d 2c20 7b30 2c20 312c 0, 1, 0}, {0, 1, │ │ │ │ -00021770: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ -00021780: 7b30 2c20 312c 2030 7d2c 207b 302c 2020 {0, 1, 0}, {0, │ │ │ │ -00021790: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d |.| -------- │ │ │ │ -000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021760: 7c6f 3220 3d20 7b7b 312c 2030 2c20 307d |o2 = {{1, 0, 0} │ │ │ │ +00021770: 2c20 7b31 2c20 302c 2030 7d2c 207b 302c , {1, 0, 0}, {0, │ │ │ │ +00021780: 2031 2c20 307d 2c20 7b30 2c20 312c 2030 1, 0}, {0, 1, 0 │ │ │ │ +00021790: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30 }, {0, 1, 0}, {0 │ │ │ │ +000217a0: 2c20 312c 2030 7d2c 207b 302c 2020 7c0a , 1, 0}, {0, |. │ │ │ │ +000217b0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ 000217c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000217d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217e0: 7c0a 7c20 2020 2020 302c 2031 7d2c 207b |.| 0, 1}, { │ │ │ │ -000217f0: 302c 2030 2c20 317d 2c20 7b30 2c20 302c 0, 0, 1}, {0, 0, │ │ │ │ -00021800: 2031 7d2c 207b 302c 2030 2c20 317d 7d20 1}, {0, 0, 1}} │ │ │ │ -00021810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021830: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000217e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000217f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00021800: 7c20 2020 2020 302c 2031 7d2c 207b 302c | 0, 1}, {0, │ │ │ │ +00021810: 2030 2c20 317d 2c20 7b30 2c20 302c 2031 0, 1}, {0, 0, 1 │ │ │ │ +00021820: 7d2c 207b 302c 2030 2c20 317d 7d20 2020 }, {0, 0, 1}} │ │ │ │ +00021830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021850: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021880: 7c0a 7c6f 3220 3a20 4c69 7374 2020 2020 |.|o2 : List │ │ │ │ -00021890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218a0: 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 7c0a |. │ │ │ │ +000218a0: 7c6f 3220 3a20 4c69 7374 2020 2020 2020 |o2 : List │ │ │ │ 000218b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -000218e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000218f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000218d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000218f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021920: 2b0a 7c69 3320 3a20 523d 4d75 6c74 6950 +.|i3 : R=MultiP │ │ │ │ -00021930: 726f 6a43 6f6f 7264 5269 6e67 207b 322c rojCoordRing {2, │ │ │ │ -00021940: 337d 2020 2020 2020 2020 2020 2020 2020 3} │ │ │ │ -00021950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00021940: 7c69 3320 3a20 523d 4d75 6c74 6950 726f |i3 : R=MultiPro │ │ │ │ +00021950: 6a43 6f6f 7264 5269 6e67 207b 322c 337d jCoordRing {2,3} │ │ │ │ 00021960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021970: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021990: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000219a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000219b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219c0: 7c0a 7c6f 3320 3d20 5220 2020 2020 2020 |.|o3 = R │ │ │ │ -000219d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219e0: 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 7c0a |. │ │ │ │ +000219e0: 7c6f 3320 3d20 5220 2020 2020 2020 2020 |o3 = R │ │ │ │ 000219f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a30: 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 7c0a |. │ │ │ │ +00021a30: 7c20 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: 7c0a 7c6f 3320 3a20 506f 6c79 6e6f 6d69 |.|o3 : Polynomi │ │ │ │ -00021a70: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ -00021a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021a70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021a80: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c |o3 : Polynomial │ │ │ │ +00021a90: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00021aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021ab0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021ad0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021b00: 2b0a 7c69 3420 3a20 636f 6566 6669 6369 +.|i4 : coeffici │ │ │ │ -00021b10: 656e 7452 696e 6720 5220 2020 2020 2020 entRing R │ │ │ │ -00021b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00021b20: 7c69 3420 3a20 636f 6566 6669 6369 656e |i4 : coefficien │ │ │ │ +00021b30: 7452 696e 6720 5220 2020 2020 2020 2020 tRing R │ │ │ │ 00021b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b70: 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 7c0a |. │ │ │ │ +00021b70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b90: 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│ -00021ce0: 7c0a 7c6f 3420 3a20 5175 6f74 6965 6e74 |.|o4 : Quotient │ │ │ │ -00021cf0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -00021d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d10: 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 7c0a |. │ │ │ │ +00021d00: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269 |o4 : QuotientRi │ │ │ │ +00021d10: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00021d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00021d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021d40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021d50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00021d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021d80: 2b0a 7c69 3520 3a20 6465 7363 7269 6265 +.|i5 : describe │ │ │ │ -00021d90: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00021da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00021da0: 7c69 3520 3a20 6465 7363 7269 6265 2052 |i5 : describe R │ │ │ │ 00021db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021de0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021df0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021e20: 7c0a 7c20 2020 2020 2020 5a5a 2020 2020 |.| ZZ │ │ │ │ -00021e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021e30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021e40: 7c20 2020 2020 2020 5a5a 2020 2020 2020 | ZZ │ │ │ │ 00021e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021e70: 7c0a 7c6f 3520 3d20 2d2d 2d2d 2d5b 7820 |.|o5 = -----[x │ │ │ │ -00021e80: 2e2e 7820 2c20 4465 6772 6565 7320 3d3e ..x , Degrees => │ │ │ │ -00021e90: 207b 333a 7b31 7d2c 2034 3a7b 307d 7d2c {3:{1}, 4:{0}}, │ │ │ │ -00021ea0: 2048 6566 7420 3d3e 207b 323a 317d 5d20 Heft => {2:1}] │ │ │ │ -00021eb0: 2020 2020 2020 2020 2020 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│ │ +00021f80: 7c69 3620 3a20 413d 4368 6f77 5269 6e67 |i6 : A=ChowRing │ │ │ │ +00021f90: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00021fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021fb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021fd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022000: 7c0a 7c6f 3620 3d20 4120 2020 2020 2020 |.|o6 = A │ │ │ │ -00022010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022020: 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: 7c6f 3620 3d20 4120 2020 2020 2020 2020 |o6 = A │ │ │ │ 00022030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022050: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00022060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022060: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022070: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00022080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000220a0: 7c0a 7c6f 3620 3a20 5175 6f74 6965 6e74 |.|o6 : Quotient │ │ │ │ -000220b0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -000220c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000220d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000220a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000220b0: 2020 2020 2020 2020 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+00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00022160: 7c69 3720 3a20 6465 7363 7269 6265 2041 |i7 : describe A │ │ │ │ 00022170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022190: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000221a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000221b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000221c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000221d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221e0: 7c0a 7c20 2020 2020 5a5a 5b68 202e 2e68 |.| ZZ[h ..h │ │ │ │ -000221f0: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00022200: 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 7c0a |. │ │ │ │ +00022200: 7c20 2020 2020 5a5a 5b68 202e 2e68 205d | ZZ[h ..h ] │ │ │ │ 00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022230: 7c0a 7c20 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ -00022240: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00022250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022250: 7c20 2020 2020 2020 2020 3120 2020 3220 | 1 2 │ │ │ │ 00022260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022280: 7c0a 7c6f 3720 3d20 2d2d 2d2d 2d2d 2d2d |.|o7 = -------- │ │ │ │ -00022290: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ -000222a0: 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 7c0a |. │ │ │ │ +000222a0: 7c6f 3720 3d20 2d2d 2d2d 2d2d 2d2d 2d2d |o7 = ---------- │ │ │ │ 000222b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222d0: 7c0a 7c20 2020 2020 2020 2033 2020 2034 |.| 3 4 │ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222f0: 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 7c0a |. │ │ │ │ +000222f0: 7c20 2020 2020 2020 2033 2020 2034 2020 | 3 4 │ │ │ │ 00022300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022320: 7c0a 7c20 2020 2020 2028 6820 2c20 6820 |.| (h , h │ │ │ │ -00022330: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00022340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022330: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022340: 7c20 2020 2020 2028 6820 2c20 6820 2920 | (h , h ) │ │ │ │ 00022350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022370: 7c0a 7c20 2020 2020 2020 2031 2020 2032 |.| 1 2 │ │ │ │ -00022380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022380: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022390: 7c20 2020 2020 2020 2031 2020 2032 2020 | 1 2 │ │ │ │ 000223a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -000223d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000223e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000223d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000223e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000223f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022410: 2b0a 7c69 3820 3a20 5365 6772 6528 412c +.|i8 : Segre(A, │ │ │ │ -00022420: 6964 6561 6c20 7261 6e64 6f6d 287b 312c ideal random({1, │ │ │ │ -00022430: 317d 2c52 2929 2020 2020 2020 2020 2020 1},R)) │ │ │ │ -00022440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022460: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00022470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00022430: 7c69 3820 3a20 5365 6772 6528 412c 6964 |i8 : Segre(A,id │ │ │ │ +00022440: 6561 6c20 7261 6e64 6f6d 287b 312c 317d eal random({1,1} │ │ │ │ +00022450: 2c52 2929 2020 2020 2020 2020 2020 2020 ,R)) │ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022480: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00022490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000224a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224b0: 7c0a 7c20 2020 2020 2020 2032 2033 2020 |.| 2 3 │ │ │ │ -000224c0: 2020 2032 2032 2020 2020 2020 2033 2020 2 2 3 │ │ │ │ -000224d0: 2020 2032 2020 2020 2020 2020 2032 2020 2 2 │ │ │ │ -000224e0: 2020 3320 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ -000224f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00022500: 7c0a 7c6f 3820 3d20 3130 6820 6820 202d |.|o8 = 10h h - │ │ │ │ -00022510: 2036 6820 6820 202d 2034 6820 6820 202b 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│ +000224f0: 2032 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +00022500: 3320 2020 2032 2020 2020 2020 2020 2020 3 2 │ │ │ │ +00022510: 2020 3220 2020 2020 2020 2020 2020 7c0a 2 |. │ │ │ │ +00022520: 7c6f 3820 3d20 3130 6820 6820 202d 2036 |o8 = 10h h - 6 │ │ │ │ +00022530: 6820 6820 202d 2034 6820 6820 202b 2033 h h - 4h h + 3 │ │ │ │ +00022540: 6820 6820 202b 2033 6820 6820 202b 2068 h h + 3h h + h │ │ │ │ +00022550: 2020 2d20 6820 202d 2032 6820 6820 202d - h - 2h h - │ │ │ │ +00022560: 2068 2020 2b20 6820 202b 2068 2020 7c0a h + h + h |. │ │ │ │ +00022570: 7c20 2020 2020 2020 2031 2032 2020 2020 | 1 2 │ │ │ │ +00022580: 2031 2032 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +00022590: 2031 2032 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +000225a0: 3220 2020 2031 2020 2020 2031 2032 2020 2 1 1 2 │ │ │ │ +000225b0: 2020 3220 2020 2031 2020 2020 3220 7c0a 2 1 2 |. │ │ │ │ +000225c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000225d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000225e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000225f0: 7c0a 7c6f 3820 3a20 4120 2020 2020 2020 |.|o8 : A │ │ │ │ -00022600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022600: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022610: 7c6f 3820 3a20 4120 2020 2020 2020 2020 |o8 : A │ │ │ │ 00022620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022640: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00022650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022650: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022660: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022690: 2b0a 0a57 6179 7320 746f 2075 7365 204d +..Ways to use M │ │ │ │ -000226a0: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -000226b0: 673a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d g:.============= │ │ │ │ -000226c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000226d0: 3d3d 0a0a 2020 2a20 224d 756c 7469 5072 ==.. * "MultiPr │ │ │ │ -000226e0: 6f6a 436f 6f72 6452 696e 6728 4c69 7374 ojCoordRing(List │ │ │ │ -000226f0: 2922 0a20 202a 2022 4d75 6c74 6950 726f )". * "MultiPro │ │ │ │ -00022700: 6a43 6f6f 7264 5269 6e67 2852 696e 672c jCoordRing(Ring, │ │ │ │ -00022710: 4c69 7374 2922 0a20 202a 2022 4d75 6c74 List)". * "Mult │ │ │ │ -00022720: 6950 726f 6a43 6f6f 7264 5269 6e67 2852 iProjCoordRing(R │ │ │ │ -00022730: 696e 672c 5379 6d62 6f6c 2c4c 6973 7429 ing,Symbol,List) │ │ │ │ -00022740: 220a 2020 2a20 224d 756c 7469 5072 6f6a ". * "MultiProj │ │ │ │ -00022750: 436f 6f72 6452 696e 6728 5379 6d62 6f6c CoordRing(Symbol │ │ │ │ -00022760: 2c4c 6973 7429 220a 0a46 6f72 2074 6865 ,List)"..For the │ │ │ │ -00022770: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00022780: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00022790: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -000227a0: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ -000227b0: 696e 673a 204d 756c 7469 5072 6f6a 436f ing: MultiProjCo │ │ │ │ -000227c0: 6f72 6452 696e 672c 2069 7320 6120 2a6e ordRing, is a *n │ │ │ │ -000227d0: 6f74 6520 6d65 7468 6f64 0a66 756e 6374 ote method.funct │ │ │ │ -000227e0: 696f 6e3a 2028 4d61 6361 756c 6179 3244 ion: (Macaulay2D │ │ │ │ -000227f0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00022800: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ -00022810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000226b0: 0a57 6179 7320 746f 2075 7365 204d 756c .Ways to use Mul │ │ │ │ +000226c0: 7469 5072 6f6a 436f 6f72 6452 696e 673a tiProjCoordRing: │ │ │ │ +000226d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +000226e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000226f0: 0a0a 2020 2a20 224d 756c 7469 5072 6f6a .. * "MultiProj │ │ │ │ +00022700: 436f 6f72 6452 696e 6728 4c69 7374 2922 CoordRing(List)" │ │ │ │ +00022710: 0a20 202a 2022 4d75 6c74 6950 726f 6a43 . * "MultiProjC │ │ │ │ +00022720: 6f6f 7264 5269 6e67 2852 696e 672c 4c69 oordRing(Ring,Li │ │ │ │ +00022730: 7374 2922 0a20 202a 2022 4d75 6c74 6950 st)". * "MultiP │ │ │ │ +00022740: 726f 6a43 6f6f 7264 5269 6e67 2852 696e rojCoordRing(Rin │ │ │ │ +00022750: 672c 5379 6d62 6f6c 2c4c 6973 7429 220a g,Symbol,List)". │ │ │ │ +00022760: 2020 2a20 224d 756c 7469 5072 6f6a 436f * "MultiProjCo │ │ │ │ +00022770: 6f72 6452 696e 6728 5379 6d62 6f6c 2c4c ordRing(Symbol,L │ │ │ │ +00022780: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +00022790: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +000227a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +000227b0: 6520 6f62 6a65 6374 202a 6e6f 7465 204d e object *note M │ │ │ │ +000227c0: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ +000227d0: 673a 204d 756c 7469 5072 6f6a 436f 6f72 g: MultiProjCoor │ │ │ │ +000227e0: 6452 696e 672c 2069 7320 6120 2a6e 6f74 dRing, is a *not │ │ │ │ +000227f0: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ +00022800: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ +00022810: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +00022820: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 00022830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022850: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00022860: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00022870: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00022880: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00022890: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -000228a0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -000228b0: 7932 2f70 6163 6b61 6765 732f 0a43 6861 y2/packages/.Cha │ │ │ │ -000228c0: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -000228d0: 6573 2e6d 323a 3230 3133 3a30 2e0a 1f0a es.m2:2013:0.... │ │ │ │ -000228e0: 4669 6c65 3a20 4368 6172 6163 7465 7269 File: Characteri │ │ │ │ -000228f0: 7374 6963 436c 6173 7365 732e 696e 666f sticClasses.info │ │ │ │ -00022900: 2c20 4e6f 6465 3a20 4f75 7470 7574 2c20 , Node: Output, │ │ │ │ -00022910: 4e65 7874 3a20 7072 6f62 6162 696c 6973 Next: probabilis │ │ │ │ -00022920: 7469 6320 616c 676f 7269 7468 6d2c 2050 tic algorithm, P │ │ │ │ -00022930: 7265 763a 204d 756c 7469 5072 6f6a 436f rev: MultiProjCo │ │ │ │ -00022940: 6f72 6452 696e 672c 2055 703a 2054 6f70 ordRing, Up: Top │ │ │ │ -00022950: 0a0a 4f75 7470 7574 0a2a 2a2a 2a2a 2a0a ..Output.******. │ │ │ │ -00022960: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -00022970: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f70 ========..The op │ │ │ │ -00022980: 7469 6f6e 204f 7574 7075 7420 6973 206f tion Output is o │ │ │ │ -00022990: 6e6c 7920 7573 6564 2062 7920 7468 6520 nly used by the │ │ │ │ -000229a0: 636f 6d6d 616e 6473 202a 6e6f 7465 2043 commands *note C │ │ │ │ -000229b0: 534d 3a20 4353 4d2c 2c20 2a6e 6f74 6520 SM: CSM,, *note │ │ │ │ -000229c0: 5365 6772 653a 0a53 6567 7265 2c2c 202a Segre:.Segre,, * │ │ │ │ -000229d0: 6e6f 7465 2043 6865 726e 3a20 4368 6572 note Chern: Cher │ │ │ │ -000229e0: 6e2c 2061 6e64 202a 6e6f 7465 2045 756c n, and *note Eul │ │ │ │ -000229f0: 6572 3a20 4575 6c65 722c 2074 6f20 7370 er: Euler, to sp │ │ │ │ -00022a00: 6563 6966 7920 7468 6520 7479 7065 206f ecify the type o │ │ │ │ -00022a10: 660a 6f75 7470 7574 2074 6f20 6265 2072 f.output to be r │ │ │ │ -00022a20: 6574 7572 6e65 6420 746f 2074 6865 2075 eturned to the u │ │ │ │ -00022a30: 7365 642e 2054 6869 7320 6f70 7469 6f6e sed. This option │ │ │ │ -00022a40: 2077 696c 6c20 6265 2069 676e 6f72 6564 will be ignored │ │ │ │ -00022a50: 2077 6865 6e20 7573 6564 2077 6974 680a when used with. │ │ │ │ -00022a60: 2a6e 6f74 6520 436f 6d70 4d65 7468 6f64 *note CompMethod │ │ │ │ -00022a70: 3a20 436f 6d70 4d65 7468 6f64 2c20 506e : CompMethod, Pn │ │ │ │ -00022a80: 5265 7369 6475 616c 206f 7220 6265 7274 Residual or bert │ │ │ │ -00022a90: 696e 692e 2054 6865 206f 7074 696f 6e20 ini. The option │ │ │ │ -00022aa0: 7769 6c6c 2061 6c73 6f20 6265 0a69 676e will also be.ign │ │ │ │ -00022ab0: 6f72 6520 7768 656e 202a 6e6f 7465 204d ore when *note M │ │ │ │ -00022ac0: 6574 686f 643a 204d 6574 686f 642c 3d3e ethod: Method,=> │ │ │ │ -00022ad0: 4469 7265 6374 436f 6d70 6c65 7465 496e DirectCompleteIn │ │ │ │ -00022ae0: 7420 6973 2075 7365 642e 2054 6865 2064 t is used. The d │ │ │ │ -00022af0: 6566 6175 6c74 0a6f 7574 7075 7420 666f efault.output fo │ │ │ │ -00022b00: 7220 616c 6c20 7468 6573 6520 6d65 7468 r all these meth │ │ │ │ -00022b10: 6f64 7320 6973 2043 686f 7752 696e 6745 ods is ChowRingE │ │ │ │ -00022b20: 6c65 6c6d 656e 7420 7768 6963 6820 7769 lelment which wi │ │ │ │ -00022b30: 6c6c 2072 6574 7572 6e20 616e 2065 6c65 ll return an ele │ │ │ │ -00022b40: 6d65 6e74 0a6f 6620 7468 6520 6170 7072 ment.of the appr │ │ │ │ -00022b50: 6f70 7269 6174 6520 4368 6f77 2072 696e opriate Chow rin │ │ │ │ -00022b60: 672e 2041 6c6c 206d 6574 686f 6473 2061 g. All methods a │ │ │ │ -00022b70: 6c73 6f20 6861 7665 2061 6e20 6f70 7469 lso have an opti │ │ │ │ -00022b80: 6f6e 2048 6173 6846 6f72 6d20 7768 6963 on HashForm whic │ │ │ │ -00022b90: 680a 7265 7475 726e 7320 6164 6469 7469 h.returns additi │ │ │ │ -00022ba0: 6f6e 616c 2069 6e66 6f72 6d61 7469 6f6e onal information │ │ │ │ -00022bb0: 2063 6f6d 7075 7465 6420 6279 2074 6865 computed by the │ │ │ │ -00022bc0: 206d 6574 686f 6473 2064 7572 696e 6720 methods during │ │ │ │ -00022bd0: 7468 6569 7220 7374 616e 6461 7264 0a6f their standard.o │ │ │ │ -00022be0: 7065 7261 7469 6f6e 2e0a 0a2b 2d2d 2d2d peration...+---- │ │ │ │ -00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022870: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00022880: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00022890: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +000228a0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +000228b0: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +000228c0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000228d0: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +000228e0: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +000228f0: 2e6d 323a 3230 3133 3a30 2e0a 1f0a 4669 .m2:2013:0....Fi │ │ │ │ +00022900: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ +00022910: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ +00022920: 4e6f 6465 3a20 4f75 7470 7574 2c20 4e65 Node: Output, Ne │ │ │ │ +00022930: 7874 3a20 7072 6f62 6162 696c 6973 7469 xt: probabilisti │ │ │ │ +00022940: 6320 616c 676f 7269 7468 6d2c 2050 7265 c algorithm, Pre │ │ │ │ +00022950: 763a 204d 756c 7469 5072 6f6a 436f 6f72 v: MultiProjCoor │ │ │ │ +00022960: 6452 696e 672c 2055 703a 2054 6f70 0a0a dRing, Up: Top.. │ │ │ │ +00022970: 4f75 7470 7574 0a2a 2a2a 2a2a 2a0a 0a44 Output.******..D │ │ │ │ +00022980: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00022990: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f70 7469 ======..The opti │ │ │ │ +000229a0: 6f6e 204f 7574 7075 7420 6973 206f 6e6c on Output is onl │ │ │ │ +000229b0: 7920 7573 6564 2062 7920 7468 6520 636f y used by the co │ │ │ │ +000229c0: 6d6d 616e 6473 202a 6e6f 7465 2043 534d mmands *note CSM │ │ │ │ +000229d0: 3a20 4353 4d2c 2c20 2a6e 6f74 6520 5365 : CSM,, *note Se │ │ │ │ +000229e0: 6772 653a 0a53 6567 7265 2c2c 202a 6e6f gre:.Segre,, *no │ │ │ │ +000229f0: 7465 2043 6865 726e 3a20 4368 6572 6e2c te Chern: Chern, │ │ │ │ +00022a00: 2061 6e64 202a 6e6f 7465 2045 756c 6572 and *note Euler │ │ │ │ +00022a10: 3a20 4575 6c65 722c 2074 6f20 7370 6563 : Euler, to spec │ │ │ │ +00022a20: 6966 7920 7468 6520 7479 7065 206f 660a ify the type of. │ │ │ │ +00022a30: 6f75 7470 7574 2074 6f20 6265 2072 6574 output to be ret │ │ │ │ +00022a40: 7572 6e65 6420 746f 2074 6865 2075 7365 urned to the use │ │ │ │ +00022a50: 642e 2054 6869 7320 6f70 7469 6f6e 2077 d. This option w │ │ │ │ +00022a60: 696c 6c20 6265 2069 676e 6f72 6564 2077 ill be ignored w │ │ │ │ +00022a70: 6865 6e20 7573 6564 2077 6974 680a 2a6e hen used with.*n │ │ │ │ +00022a80: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20 ote CompMethod: │ │ │ │ +00022a90: 436f 6d70 4d65 7468 6f64 2c20 506e 5265 CompMethod, PnRe │ │ │ │ +00022aa0: 7369 6475 616c 206f 7220 6265 7274 696e sidual or bertin │ │ │ │ +00022ab0: 692e 2054 6865 206f 7074 696f 6e20 7769 i. The option wi │ │ │ │ +00022ac0: 6c6c 2061 6c73 6f20 6265 0a69 676e 6f72 ll also be.ignor │ │ │ │ +00022ad0: 6520 7768 656e 202a 6e6f 7465 204d 6574 e when *note Met │ │ │ │ +00022ae0: 686f 643a 204d 6574 686f 642c 3d3e 4469 hod: Method,=>Di │ │ │ │ +00022af0: 7265 6374 436f 6d70 6c65 7465 496e 7420 rectCompleteInt │ │ │ │ +00022b00: 6973 2075 7365 642e 2054 6865 2064 6566 is used. The def │ │ │ │ +00022b10: 6175 6c74 0a6f 7574 7075 7420 666f 7220 ault.output for │ │ │ │ +00022b20: 616c 6c20 7468 6573 6520 6d65 7468 6f64 all these method │ │ │ │ +00022b30: 7320 6973 2043 686f 7752 696e 6745 6c65 s is ChowRingEle │ │ │ │ +00022b40: 6c6d 656e 7420 7768 6963 6820 7769 6c6c lment which will │ │ │ │ +00022b50: 2072 6574 7572 6e20 616e 2065 6c65 6d65 return an eleme │ │ │ │ +00022b60: 6e74 0a6f 6620 7468 6520 6170 7072 6f70 nt.of the approp │ │ │ │ +00022b70: 7269 6174 6520 4368 6f77 2072 696e 672e riate Chow ring. │ │ │ │ +00022b80: 2041 6c6c 206d 6574 686f 6473 2061 6c73 All methods als │ │ │ │ +00022b90: 6f20 6861 7665 2061 6e20 6f70 7469 6f6e o have an option │ │ │ │ +00022ba0: 2048 6173 6846 6f72 6d20 7768 6963 680a HashForm which. │ │ │ │ +00022bb0: 7265 7475 726e 7320 6164 6469 7469 6f6e returns addition │ │ │ │ +00022bc0: 616c 2069 6e66 6f72 6d61 7469 6f6e 2063 al information c │ │ │ │ +00022bd0: 6f6d 7075 7465 6420 6279 2074 6865 206d omputed by the m │ │ │ │ +00022be0: 6574 686f 6473 2064 7572 696e 6720 7468 ethods during th │ │ │ │ +00022bf0: 6569 7220 7374 616e 6461 7264 0a6f 7065 eir standard.ope │ │ │ │ +00022c00: 7261 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d ration...+------ │ │ │ │ 00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022c30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -00022c40: 2052 203d 205a 5a2f 3332 3734 395b 785f R = ZZ/32749[x_ │ │ │ │ -00022c50: 302e 2e78 5f36 5d20 2020 2020 2020 2020 0..x_6] │ │ │ │ -00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022c50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 -------+.|i1 : R │ │ │ │ +00022c60: 203d 205a 5a2f 3332 3734 395b 785f 302e = ZZ/32749[x_0. │ │ │ │ +00022c70: 2e78 5f36 5d20 2020 2020 2020 2020 2020 .x_6] │ │ │ │ +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 │ │ │ │ +00022ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cd0: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -00022ce0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00022cf0: 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 207c 0a7c 6f31 203d 2052 |.|o1 = R │ │ │ │ 00022d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d20: 2020 2020 2020 2020 207c 0a7c 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 │ │ │ │ +00022d40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00022d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d70: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -00022d80: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022d90: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +00022da0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ 00022db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022dc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022de0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022e10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00022e20: 2041 3d43 686f 7752 696e 6728 5229 2020 A=ChowRing(R) │ │ │ │ -00022e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022e30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2041 -------+.|i2 : A │ │ │ │ +00022e40: 3d43 686f 7752 696e 6728 5229 2020 2020 =ChowRing(R) │ │ │ │ 00022e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e60: 2020 2020 2020 2020 207c 0a7c 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 │ │ │ │ +00022e80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022eb0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ -00022ec0: 2041 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ -00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022ed0: 2020 2020 2020 207c 0a7c 6f32 203d 2041 |.|o2 = A │ │ │ │ 00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022f20: 2020 2020 2020 207c 0a7c 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 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -00022f60: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ -00022f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022f70: 2020 2020 2020 207c 0a7c 6f32 203a 2051 |.|o2 : Q │ │ │ │ +00022f80: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ 00022f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00022fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022fc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00022fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022ff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -00023000: 2049 3d69 6465 616c 2872 616e 646f 6d28 I=ideal(random( │ │ │ │ -00023010: 322c 5229 2c52 5f30 2a52 5f31 2a52 5f36 2,R),R_0*R_1*R_6 │ │ │ │ -00023020: 2d52 5f30 5e33 293b 2020 2020 2020 2020 -R_0^3); │ │ │ │ -00023030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023040: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023010: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ +00023020: 3d69 6465 616c 2872 616e 646f 6d28 322c =ideal(random(2, │ │ │ │ +00023030: 5229 2c52 5f30 2a52 5f31 2a52 5f36 2d52 R),R_0*R_1*R_6-R │ │ │ │ +00023040: 5f30 5e33 293b 2020 2020 2020 2020 2020 _0^3); │ │ │ │ 00023050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023060: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023090: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -000230a0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ -000230b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230b0: 2020 2020 2020 207c 0a7c 6f33 203a 2049 |.|o3 : I │ │ │ │ +000230c0: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ 000230d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000230f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023100: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00023110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023130: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00023140: 2063 736d 3d43 534d 2841 2c49 2c4f 7574 csm=CSM(A,I,Out │ │ │ │ -00023150: 7075 743d 3e48 6173 6846 6f72 6d29 2020 put=>HashForm) │ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023180: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00023130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023150: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2063 -------+.|i4 : c │ │ │ │ +00023160: 736d 3d43 534d 2841 2c49 2c4f 7574 7075 sm=CSM(A,I,Outpu │ │ │ │ +00023170: 743d 3e48 6173 6846 6f72 6d29 2020 2020 t=>HashForm) │ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000231a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000231b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000231c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000231d0: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ -000231e0: 204d 7574 6162 6c65 4861 7368 5461 626c MutableHashTabl │ │ │ │ -000231f0: 657b 2e2e 2e34 2e2e 2e7d 2020 2020 2020 e{...4...} │ │ │ │ -00023200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023220: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000231d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231f0: 2020 2020 2020 207c 0a7c 6f34 203d 204d |.|o4 = M │ │ │ │ +00023200: 7574 6162 6c65 4861 7368 5461 626c 657b utableHashTable{ │ │ │ │ +00023210: 2e2e 2e34 2e2e 2e7d 2020 2020 2020 2020 ...4...} │ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023240: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023270: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ -00023280: 204d 7574 6162 6c65 4861 7368 5461 626c MutableHashTabl │ │ │ │ -00023290: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -000232a0: 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 207c 0a7c 6f34 203a 204d |.|o4 : M │ │ │ │ +000232a0: 7574 6162 6c65 4861 7368 5461 626c 6520 utableHashTable │ │ │ │ 000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000232d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000232e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000232e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 000232f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -00023320: 2070 6565 6b20 6373 6d20 2020 2020 2020 peek csm │ │ │ │ -00023330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023330: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 -------+.|i5 : p │ │ │ │ +00023340: 6565 6b20 6373 6d20 2020 2020 2020 2020 eek csm │ │ │ │ 00023350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023360: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00023360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023380: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000233a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000233b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000233b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000233c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000233d0: 2020 2020 2020 2020 2020 2020 2020 3620 6 │ │ │ │ -000233e0: 2020 2020 2035 2020 2020 2020 3420 2020 5 4 │ │ │ │ -000233f0: 2020 2033 2020 2020 2020 3220 2020 2020 3 2 │ │ │ │ -00023400: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ -00023410: 204d 7574 6162 6c65 4861 7368 5461 626c MutableHashTabl │ │ │ │ -00023420: 657b 7b30 2c20 317d 203d 3e20 3268 2020 e{{0, 1} => 2h │ │ │ │ -00023430: 2b20 3233 6820 202b 2033 3268 2020 2b20 + 23h + 32h + │ │ │ │ -00023440: 3333 6820 202b 2031 3868 2020 2b20 3568 33h + 18h + 5h │ │ │ │ -00023450: 207d 2020 2020 2020 207c 0a7c 2020 2020 } |.| │ │ │ │ -00023460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023470: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -00023480: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ -00023490: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -000234a0: 3120 2020 2020 2020 207c 0a7c 2020 2020 1 |.| │ │ │ │ -000234b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000234c0: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ -000234d0: 2020 2035 2020 2020 2020 3420 2020 2020 5 4 │ │ │ │ -000234e0: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ -000234f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023510: 2020 4353 4d20 3d3e 2031 3068 2020 2b20 CSM => 10h + │ │ │ │ -00023520: 3132 6820 202b 2032 3268 2020 2b20 3136 12h + 22h + 16 │ │ │ │ -00023530: 6820 202b 2036 6820 2020 2020 2020 2020 h + 6h │ │ │ │ -00023540: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023560: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -00023570: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00023580: 2031 2020 2020 2031 2020 2020 2020 2020 1 1 │ │ │ │ -00023590: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000235a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000235b0: 2020 2020 2020 2020 2020 2036 2020 2020 6 │ │ │ │ -000235c0: 2020 3520 2020 2020 2034 2020 2020 2020 5 4 │ │ │ │ -000235d0: 3320 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ -000235e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000235f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023600: 2020 7b30 7d20 3d3e 2036 6820 202b 2031 {0} => 6h + 1 │ │ │ │ -00023610: 3868 2020 2b20 3236 6820 202b 2032 3268 8h + 26h + 22h │ │ │ │ -00023620: 2020 2b20 3130 6820 202b 2032 6820 2020 + 10h + 2h │ │ │ │ -00023630: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| 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|.|1266h │ │ │ │ +000249c0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 000249d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000249f0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00024a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a00: 2020 2020 2020 207c 0a7c 2020 2020 2031 |.| 1 │ │ │ │ 00024a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024a30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00024a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a50: 2020 2020 2020 207c 0a2b 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 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ -00024a90: 3a20 6575 3d45 756c 6572 2820 492c 204f : eu=Euler( I, O │ │ │ │ -00024aa0: 7574 7075 743d 3e48 6173 6846 6f72 6d29 utput=>HashForm) │ │ │ │ -00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ad0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024aa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ +00024ab0: 6575 3d45 756c 6572 2820 492c 204f 7574 eu=Euler( I, Out │ │ │ │ +00024ac0: 7075 743d 3e48 6173 6846 6f72 6d29 2020 put=>HashForm) │ │ │ │ +00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024af0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00024b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b20: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00024b30: 3d20 4d75 7461 626c 6548 6173 6854 6162 = MutableHashTab │ │ │ │ -00024b40: 6c65 7b2e 2e2e 352e 2e2e 7d20 2020 2020 le{...5...} │ │ │ │ -00024b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00024b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024b40: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ +00024b50: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ +00024b60: 7b2e 2e2e 352e 2e2e 7d20 2020 2020 2020 {...5...} │ │ │ │ +00024b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024b90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024bc0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00024bd0: 3a20 4d75 7461 626c 6548 6173 6854 6162 : MutableHashTab │ │ │ │ -00024be0: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024be0: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ +00024bf0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ 00024c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00024c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00024c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024c60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ -00024c70: 3a20 7065 656b 2065 7520 2020 2020 2020 : peek eu │ │ │ │ -00024c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c80: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ +00024c90: 7065 656b 2065 7520 2020 2020 2020 2020 peek eu │ │ │ │ 00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024cb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024cd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00024ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d00: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ -00024d10: 3d20 4d75 7461 626c 6548 6173 6854 6162 = MutableHashTab │ │ │ │ -00024d20: 6c65 7b45 756c 6572 203d 3e20 3130 2020 le{Euler => 10 │ │ │ │ -00024d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d50: 2020 7d20 2020 2020 207c 0a7c 2020 2020 } |.| │ │ │ │ +00024d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d20: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ +00024d30: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ +00024d40: 7b45 756c 6572 203d 3e20 3130 2020 2020 {Euler => 10 │ │ │ │ +00024d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d70: 2020 2020 2020 2020 2020 2020 2020 2036 6 │ │ │ │ -00024d80: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ -00024d90: 2020 2020 3320 2020 2020 2032 2020 2020 3 2 │ │ │ │ -00024da0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024dc0: 2020 207b 302c 2031 7d20 3d3e 2032 6820 {0, 1} => 2h │ │ │ │ -00024dd0: 202b 2032 3368 2020 2b20 3332 6820 202b + 23h + 32h + │ │ │ │ -00024de0: 2033 3368 2020 2b20 3138 6820 202b 2035 33h + 18h + 5 │ │ │ │ -00024df0: 6820 2020 2020 2020 207c 0a7c 2020 2020 h |.| │ │ │ │ -00024e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e10: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00024e20: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00024e30: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024e40: 2031 2020 2020 2020 207c 0a7c 2020 2020 1 |.| │ │ │ │ -00024e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e60: 2020 2020 2020 2020 2020 2020 2036 2020 6 │ │ │ │ -00024e70: 2020 2020 3520 2020 2020 2034 2020 2020 5 4 │ │ │ │ -00024e80: 2020 3320 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ -00024e90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024eb0: 2020 2043 534d 203d 3e20 3130 6820 202b CSM => 10h + │ │ │ │ -00024ec0: 2031 3268 2020 2b20 3232 6820 202b 2031 12h + 22h + 1 │ │ │ │ -00024ed0: 3668 2020 2b20 3668 2020 2020 2020 2020 6h + 6h │ │ │ │ -00024ee0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f00: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -00024f10: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024f20: 2020 3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ -00024f30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f50: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ -00024f60: 2020 2035 2020 2020 2020 3420 2020 2020 5 4 │ │ │ │ -00024f70: 2033 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ -00024f80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024fa0: 2020 207b 307d 203d 3e20 3668 2020 2b20 {0} => 6h + │ │ │ │ -00024fb0: 3138 6820 202b 2032 3668 2020 2b20 3232 18h + 26h + 22 │ │ │ │ -00024fc0: 6820 202b 2031 3068 2020 2b20 3268 2020 h + 10h + 2h │ │ │ │ -00024fd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ff0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -00025000: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00025010: 2031 2020 2020 2020 3120 2020 2020 3120 1 1 1 │ │ │ │ -00025020: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00025030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ -00025050: 2020 2035 2020 2020 2020 3420 2020 2020 5 4 │ │ │ │ -00025060: 2033 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ -00025070: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00025080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025090: 2020 207b 317d 203d 3e20 3668 2020 2b20 {1} => 6h + │ │ │ │ -000250a0: 3137 6820 202b 2032 3868 2020 2b20 3237 17h + 28h + 27 │ │ │ │ -000250b0: 6820 202b 2031 3468 2020 2b20 3368 2020 h + 14h + 3h │ │ │ │ -000250c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000250d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250e0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -000250f0: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00025100: 2031 2020 2020 2020 3120 2020 2020 3120 1 1 1 │ │ │ │ -00025110: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d70: 7d20 2020 2020 207c 0a7c 2020 2020 2020 } |.| │ │ │ │ +00024d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d90: 2020 2020 2020 2020 2020 2020 2036 2020 6 │ │ │ │ +00024da0: 2020 2020 3520 2020 2020 2034 2020 2020 5 4 │ │ │ │ +00024db0: 2020 3320 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +00024dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024de0: 207b 302c 2031 7d20 3d3e 2032 6820 202b {0, 1} => 2h + │ │ │ │ +00024df0: 2032 3368 2020 2b20 3332 6820 202b 2033 23h + 32h + 3 │ │ │ │ +00024e00: 3368 2020 2b20 3138 6820 202b 2035 6820 3h + 18h + 5h │ │ │ │ +00024e10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e30: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +00024e40: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00024e50: 2020 3120 2020 2020 2031 2020 2020 2031 1 1 1 │ │ │ │ +00024e60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2036 2020 2020 6 │ │ │ │ +00024e90: 2020 3520 2020 2020 2034 2020 2020 2020 5 4 │ │ │ │ +00024ea0: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ +00024eb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ed0: 2043 534d 203d 3e20 3130 6820 202b 2031 CSM => 10h + 1 │ │ │ │ +00024ee0: 3268 2020 2b20 3232 6820 202b 2031 3668 2h + 22h + 16h │ │ │ │ +00024ef0: 2020 2b20 3668 2020 2020 2020 2020 2020 + 6h │ │ │ │ +00024f00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024f20: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00024f30: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00024f40: 3120 2020 2020 3120 2020 2020 2020 2020 1 1 │ │ │ │ +00024f50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024f70: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +00024f80: 2035 2020 2020 2020 3420 2020 2020 2033 5 4 3 │ │ │ │ +00024f90: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00024fa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024fc0: 207b 307d 203d 3e20 3668 2020 2b20 3138 {0} => 6h + 18 │ │ │ │ +00024fd0: 6820 202b 2032 3668 2020 2b20 3232 6820 h + 26h + 22h │ │ │ │ +00024fe0: 202b 2031 3068 2020 2b20 3268 2020 2020 + 10h + 2h │ │ │ │ +00024ff0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025010: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00025020: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025030: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ +00025040: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025060: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +00025070: 2035 2020 2020 2020 3420 2020 2020 2033 5 4 3 │ │ │ │ +00025080: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00025090: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000250a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000250b0: 207b 317d 203d 3e20 3668 2020 2b20 3137 {1} => 6h + 17 │ │ │ │ +000250c0: 6820 202b 2032 3868 2020 2b20 3237 6820 h + 28h + 27h │ │ │ │ +000250d0: 202b 2031 3468 2020 2b20 3368 2020 2020 + 14h + 3h │ │ │ │ +000250e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000250f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025100: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00025110: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025120: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ +00025130: 2020 2020 2020 207c 0a2b 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 2d2b 0a0a 5468 6520 ---------+..The │ │ │ │ -00025170: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -00025180: 2072 6574 7572 6e65 6420 7769 7468 2074 returned with t │ │ │ │ -00025190: 6865 206f 7074 696f 6e20 4f75 7470 7574 he option Output │ │ │ │ -000251a0: 3d3e 4861 7368 466f 726d 2063 6f6e 7461 =>HashForm conta │ │ │ │ -000251b0: 696e 730a 6469 6666 6572 656e 7420 696e ins.different in │ │ │ │ -000251c0: 666f 726d 6174 696f 6e20 6465 7065 6e64 formation depend │ │ │ │ -000251d0: 696e 6720 6f6e 2074 6865 206d 6574 686f ing on the metho │ │ │ │ -000251e0: 6420 7769 7468 2077 6869 6368 2069 7420 d with which it │ │ │ │ -000251f0: 6973 2075 7365 642e 0a41 6464 6974 696f is used..Additio │ │ │ │ -00025200: 6e61 6c6c 7920 6966 2074 6865 206f 7074 nally if the opt │ │ │ │ -00025210: 696f 6e20 2a6e 6f74 6520 496e 7075 7449 ion *note InputI │ │ │ │ -00025220: 7353 6d6f 6f74 683a 2049 6e70 7574 4973 sSmooth: InputIs │ │ │ │ -00025230: 536d 6f6f 7468 2c20 6973 2075 7365 6420 Smooth, is used │ │ │ │ -00025240: 7468 656e 2074 6865 0a68 6173 6820 7461 then the.hash ta │ │ │ │ -00025250: 626c 6520 7265 7475 726e 6564 2062 7920 ble returned by │ │ │ │ -00025260: 7468 6520 6d65 7468 6f64 7320 4575 6c65 the methods Eule │ │ │ │ -00025270: 7220 616e 6420 4353 4d20 7769 6c6c 2062 r and CSM will b │ │ │ │ -00025280: 6520 7468 6520 7361 6d65 2061 7320 7468 e the same as th │ │ │ │ -00025290: 6174 0a72 6574 7572 6e65 6420 6279 2043 at.returned by C │ │ │ │ -000252a0: 6865 726e 2e20 5768 656e 2075 7369 6e67 hern. When using │ │ │ │ -000252b0: 2074 6865 202a 6e6f 7465 2043 534d 3a20 the *note CSM: │ │ │ │ -000252c0: 4353 4d2c 2020 636f 6d6d 616e 6420 696e CSM, command in │ │ │ │ -000252d0: 2074 6865 2064 6566 6175 6c74 0a63 6f6e the default.con │ │ │ │ -000252e0: 6669 6775 7261 7469 6f6e 7320 2874 6861 figurations (tha │ │ │ │ -000252f0: 7420 6973 202a 6e6f 7465 204d 6574 686f t is *note Metho │ │ │ │ -00025300: 643a 204d 6574 686f 642c 3d3e 496e 636c d: Method,=>Incl │ │ │ │ -00025310: 7573 696f 6e45 7863 6c75 7369 6f6e 2c20 usionExclusion, │ │ │ │ -00025320: 2a6e 6f74 650a 436f 6d70 4d65 7468 6f64 *note.CompMethod │ │ │ │ -00025330: 3a20 436f 6d70 4d65 7468 6f64 2c3d 3e50 : CompMethod,=>P │ │ │ │ -00025340: 726f 6a65 6374 6976 6544 6567 7265 6529 rojectiveDegree) │ │ │ │ -00025350: 2074 6865 7265 2069 7320 7468 6520 6164 there is the ad │ │ │ │ -00025360: 6469 7469 6f6e 616c 206f 7074 696f 6e20 ditional option │ │ │ │ -00025370: 746f 0a73 6574 204f 7574 7075 743d 3e48 to.set Output=>H │ │ │ │ -00025380: 6173 6846 6f72 6d58 4c2e 2054 6869 7320 ashFormXL. This │ │ │ │ -00025390: 7265 7475 726e 7320 616c 6c20 7468 6520 returns all the │ │ │ │ -000253a0: 7573 7561 6c20 696e 666f 726d 6174 696f usual informatio │ │ │ │ -000253b0: 6e20 7468 6174 0a4f 7574 7075 743d 3e48 n that.Output=>H │ │ │ │ -000253c0: 6173 6846 6f72 6d20 776f 756c 6420 666f ashForm would fo │ │ │ │ -000253d0: 7220 7468 6973 2063 6f6e 6669 6775 7261 r this configura │ │ │ │ -000253e0: 7469 6f6e 2077 6974 6820 7468 6520 6164 tion with the ad │ │ │ │ -000253f0: 6469 7469 6f6e 206f 6620 7468 650a 7072 dition of the.pr │ │ │ │ -00025400: 6f6a 6563 7469 7665 2064 6567 7265 6573 ojective degrees │ │ │ │ -00025410: 2061 6e64 2053 6567 7265 2063 6c61 7373 and Segre class │ │ │ │ -00025420: 6573 206f 6620 7369 6e67 756c 6172 6974 es of singularit │ │ │ │ -00025430: 7920 7375 6273 6368 656d 6573 2067 656e y subschemes gen │ │ │ │ -00025440: 6572 6174 6564 2062 7920 7468 650a 6879 erated by the.hy │ │ │ │ -00025450: 7065 7273 7572 6661 6365 7320 636f 6e73 persurfaces cons │ │ │ │ -00025460: 6964 6572 6564 2069 6e20 7468 6520 696e idered in the in │ │ │ │ -00025470: 636c 7573 696f 6e2f 6578 636c 7573 696f clusion/exclusio │ │ │ │ -00025480: 6e20 7072 6f63 6564 7572 652c 2074 6861 n procedure, tha │ │ │ │ -00025490: 7420 6973 2069 6e0a 6669 6e64 696e 6720 t is in.finding │ │ │ │ -000254a0: 7468 6520 4353 4d20 636c 6173 7320 6f66 the CSM class of │ │ │ │ -000254b0: 2061 6c6c 2068 7970 6572 7375 7266 6163 all hypersurfac │ │ │ │ -000254c0: 6573 2067 656e 6572 6174 6564 2062 7920 es generated by │ │ │ │ -000254d0: 7461 6b69 6e67 2061 2070 726f 6475 6374 taking a product │ │ │ │ -000254e0: 206f 660a 736f 6d65 2073 7562 7365 7473 of.some subsets │ │ │ │ -000254f0: 206f 6620 6765 6e65 7261 746f 7273 206f of generators o │ │ │ │ -00025500: 6620 7468 6520 696e 7075 7420 6964 6561 f the input idea │ │ │ │ -00025510: 6c2e 204e 6f74 6520 7468 6174 2c20 7369 l. Note that, si │ │ │ │ -00025520: 6e63 6520 7468 6520 4353 4d20 636c 6173 nce the CSM clas │ │ │ │ -00025530: 730a 6f66 2061 2073 7562 7363 6865 6d65 s.of a subscheme │ │ │ │ -00025540: 2065 7175 616c 7320 7468 6520 4353 4d20 equals the CSM │ │ │ │ -00025550: 636c 6173 7320 6f66 2069 7473 2072 6564 class of its red │ │ │ │ -00025560: 7563 6564 2073 6368 656d 652c 206f 7220 uced scheme, or │ │ │ │ -00025570: 6571 7569 7661 6c65 6e74 6c79 2066 6f72 equivalently for │ │ │ │ -00025580: 0a75 7320 7468 6520 4353 4d20 636c 6173 .us the CSM clas │ │ │ │ -00025590: 7320 636f 7272 6573 706f 6e64 696e 6720 s corresponding │ │ │ │ -000255a0: 746f 2061 6e20 6964 6561 6c20 4920 6571 to an ideal I eq │ │ │ │ -000255b0: 7561 6c73 2074 6865 2043 534d 2063 6c61 uals the CSM cla │ │ │ │ -000255c0: 7373 206f 6620 7468 650a 7261 6469 6361 ss of the.radica │ │ │ │ -000255d0: 6c20 6f66 2049 2c20 7468 656e 2069 6e74 l of I, then int │ │ │ │ -000255e0: 6572 6e61 6c6c 7920 7765 2061 6c77 6179 ernally we alway │ │ │ │ -000255f0: 7320 776f 726b 2077 6974 6820 7261 6469 s work with radi │ │ │ │ -00025600: 6361 6c20 6964 6561 6c73 2028 666f 720a cal ideals (for. │ │ │ │ -00025610: 6566 6669 6369 656e 6379 2072 6561 736f efficiency reaso │ │ │ │ -00025620: 6e73 292e 2048 656e 6365 2074 6865 2070 ns). Hence the p │ │ │ │ -00025630: 726f 6a65 6374 6976 6520 6465 6772 6565 rojective degree │ │ │ │ -00025640: 7320 616e 6420 5365 6772 6520 636c 6173 s and Segre clas │ │ │ │ -00025650: 7365 7320 636f 6d70 7574 6564 0a69 6e74 ses computed.int │ │ │ │ -00025660: 6572 6e61 6c6c 7920 7769 6c6c 2062 6520 ernally will be │ │ │ │ -00025670: 7468 6f73 6520 6f66 2074 6865 2072 6164 those of the rad │ │ │ │ -00025680: 6963 616c 206f 6620 616e 2069 6465 616c ical of an ideal │ │ │ │ -00025690: 2064 6566 696e 6564 2062 7920 6120 706f defined by a po │ │ │ │ -000256a0: 6c79 6e6f 6d69 616c 0a77 6869 6368 2069 lynomial.which i │ │ │ │ -000256b0: 7320 6120 7072 6f64 7563 7420 6f66 2073 s a product of s │ │ │ │ -000256c0: 6f6d 6520 7375 6273 6574 206f 6620 7468 ome subset of th │ │ │ │ -000256d0: 6520 6765 6e65 7261 746f 7273 2e20 5765 e generators. We │ │ │ │ -000256e0: 2069 6c6c 7573 7472 6174 6520 7468 6973 illustrate this │ │ │ │ -000256f0: 2077 6974 6820 616e 0a65 7861 6d70 6c65 with an.example │ │ │ │ -00025700: 2062 656c 6f77 2e0a 0a2b 2d2d 2d2d 2d2d below...+------ │ │ │ │ -00025710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025180: 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6520 4d75 -------+..The Mu │ │ │ │ +00025190: 7461 626c 6548 6173 6854 6162 6c65 2072 tableHashTable r │ │ │ │ +000251a0: 6574 7572 6e65 6420 7769 7468 2074 6865 eturned with the │ │ │ │ +000251b0: 206f 7074 696f 6e20 4f75 7470 7574 3d3e option Output=> │ │ │ │ +000251c0: 4861 7368 466f 726d 2063 6f6e 7461 696e HashForm contain │ │ │ │ +000251d0: 730a 6469 6666 6572 656e 7420 696e 666f s.different info │ │ │ │ +000251e0: 726d 6174 696f 6e20 6465 7065 6e64 696e rmation dependin │ │ │ │ +000251f0: 6720 6f6e 2074 6865 206d 6574 686f 6420 g on the method │ │ │ │ +00025200: 7769 7468 2077 6869 6368 2069 7420 6973 with which it is │ │ │ │ +00025210: 2075 7365 642e 0a41 6464 6974 696f 6e61 used..Additiona │ │ │ │ +00025220: 6c6c 7920 6966 2074 6865 206f 7074 696f lly if the optio │ │ │ │ +00025230: 6e20 2a6e 6f74 6520 496e 7075 7449 7353 n *note InputIsS │ │ │ │ +00025240: 6d6f 6f74 683a 2049 6e70 7574 4973 536d mooth: InputIsSm │ │ │ │ +00025250: 6f6f 7468 2c20 6973 2075 7365 6420 7468 ooth, is used th │ │ │ │ +00025260: 656e 2074 6865 0a68 6173 6820 7461 626c en the.hash tabl │ │ │ │ +00025270: 6520 7265 7475 726e 6564 2062 7920 7468 e returned by th │ │ │ │ +00025280: 6520 6d65 7468 6f64 7320 4575 6c65 7220 e methods Euler │ │ │ │ +00025290: 616e 6420 4353 4d20 7769 6c6c 2062 6520 and CSM will be │ │ │ │ +000252a0: 7468 6520 7361 6d65 2061 7320 7468 6174 the same as that │ │ │ │ +000252b0: 0a72 6574 7572 6e65 6420 6279 2043 6865 .returned by Che │ │ │ │ +000252c0: 726e 2e20 5768 656e 2075 7369 6e67 2074 rn. When using t │ │ │ │ +000252d0: 6865 202a 6e6f 7465 2043 534d 3a20 4353 he *note CSM: CS │ │ │ │ +000252e0: 4d2c 2020 636f 6d6d 616e 6420 696e 2074 M, command in t │ │ │ │ +000252f0: 6865 2064 6566 6175 6c74 0a63 6f6e 6669 he default.confi │ │ │ │ +00025300: 6775 7261 7469 6f6e 7320 2874 6861 7420 gurations (that │ │ │ │ +00025310: 6973 202a 6e6f 7465 204d 6574 686f 643a is *note Method: │ │ │ │ +00025320: 204d 6574 686f 642c 3d3e 496e 636c 7573 Method,=>Inclus │ │ │ │ +00025330: 696f 6e45 7863 6c75 7369 6f6e 2c20 2a6e ionExclusion, *n │ │ │ │ +00025340: 6f74 650a 436f 6d70 4d65 7468 6f64 3a20 ote.CompMethod: │ │ │ │ +00025350: 436f 6d70 4d65 7468 6f64 2c3d 3e50 726f CompMethod,=>Pro │ │ │ │ +00025360: 6a65 6374 6976 6544 6567 7265 6529 2074 jectiveDegree) t │ │ │ │ +00025370: 6865 7265 2069 7320 7468 6520 6164 6469 here is the addi │ │ │ │ +00025380: 7469 6f6e 616c 206f 7074 696f 6e20 746f tional option to │ │ │ │ +00025390: 0a73 6574 204f 7574 7075 743d 3e48 6173 .set Output=>Has │ │ │ │ +000253a0: 6846 6f72 6d58 4c2e 2054 6869 7320 7265 hFormXL. This re │ │ │ │ +000253b0: 7475 726e 7320 616c 6c20 7468 6520 7573 turns all the us │ │ │ │ +000253c0: 7561 6c20 696e 666f 726d 6174 696f 6e20 ual information │ │ │ │ +000253d0: 7468 6174 0a4f 7574 7075 743d 3e48 6173 that.Output=>Has │ │ │ │ +000253e0: 6846 6f72 6d20 776f 756c 6420 666f 7220 hForm would for │ │ │ │ +000253f0: 7468 6973 2063 6f6e 6669 6775 7261 7469 this configurati │ │ │ │ +00025400: 6f6e 2077 6974 6820 7468 6520 6164 6469 on with the addi │ │ │ │ +00025410: 7469 6f6e 206f 6620 7468 650a 7072 6f6a tion of the.proj │ │ │ │ +00025420: 6563 7469 7665 2064 6567 7265 6573 2061 ective degrees a │ │ │ │ +00025430: 6e64 2053 6567 7265 2063 6c61 7373 6573 nd Segre classes │ │ │ │ +00025440: 206f 6620 7369 6e67 756c 6172 6974 7920 of singularity │ │ │ │ +00025450: 7375 6273 6368 656d 6573 2067 656e 6572 subschemes gener │ │ │ │ +00025460: 6174 6564 2062 7920 7468 650a 6879 7065 ated by the.hype │ │ │ │ +00025470: 7273 7572 6661 6365 7320 636f 6e73 6964 rsurfaces consid │ │ │ │ +00025480: 6572 6564 2069 6e20 7468 6520 696e 636c ered in the incl │ │ │ │ +00025490: 7573 696f 6e2f 6578 636c 7573 696f 6e20 usion/exclusion │ │ │ │ +000254a0: 7072 6f63 6564 7572 652c 2074 6861 7420 procedure, that │ │ │ │ +000254b0: 6973 2069 6e0a 6669 6e64 696e 6720 7468 is in.finding th │ │ │ │ +000254c0: 6520 4353 4d20 636c 6173 7320 6f66 2061 e CSM class of a │ │ │ │ +000254d0: 6c6c 2068 7970 6572 7375 7266 6163 6573 ll hypersurfaces │ │ │ │ +000254e0: 2067 656e 6572 6174 6564 2062 7920 7461 generated by ta │ │ │ │ +000254f0: 6b69 6e67 2061 2070 726f 6475 6374 206f king a product o │ │ │ │ +00025500: 660a 736f 6d65 2073 7562 7365 7473 206f f.some subsets o │ │ │ │ +00025510: 6620 6765 6e65 7261 746f 7273 206f 6620 f generators of │ │ │ │ +00025520: 7468 6520 696e 7075 7420 6964 6561 6c2e the input ideal. │ │ │ │ +00025530: 204e 6f74 6520 7468 6174 2c20 7369 6e63 Note that, sinc │ │ │ │ +00025540: 6520 7468 6520 4353 4d20 636c 6173 730a e the CSM class. │ │ │ │ +00025550: 6f66 2061 2073 7562 7363 6865 6d65 2065 of a subscheme e │ │ │ │ +00025560: 7175 616c 7320 7468 6520 4353 4d20 636c quals the CSM cl │ │ │ │ +00025570: 6173 7320 6f66 2069 7473 2072 6564 7563 ass of its reduc │ │ │ │ +00025580: 6564 2073 6368 656d 652c 206f 7220 6571 ed scheme, or eq │ │ │ │ +00025590: 7569 7661 6c65 6e74 6c79 2066 6f72 0a75 uivalently for.u │ │ │ │ +000255a0: 7320 7468 6520 4353 4d20 636c 6173 7320 s the CSM class │ │ │ │ +000255b0: 636f 7272 6573 706f 6e64 696e 6720 746f corresponding to │ │ │ │ +000255c0: 2061 6e20 6964 6561 6c20 4920 6571 7561 an ideal I equa │ │ │ │ +000255d0: 6c73 2074 6865 2043 534d 2063 6c61 7373 ls the CSM class │ │ │ │ +000255e0: 206f 6620 7468 650a 7261 6469 6361 6c20 of the.radical │ │ │ │ +000255f0: 6f66 2049 2c20 7468 656e 2069 6e74 6572 of I, then inter │ │ │ │ +00025600: 6e61 6c6c 7920 7765 2061 6c77 6179 7320 nally we always │ │ │ │ +00025610: 776f 726b 2077 6974 6820 7261 6469 6361 work with radica │ │ │ │ +00025620: 6c20 6964 6561 6c73 2028 666f 720a 6566 l ideals (for.ef │ │ │ │ +00025630: 6669 6369 656e 6379 2072 6561 736f 6e73 ficiency reasons │ │ │ │ +00025640: 292e 2048 656e 6365 2074 6865 2070 726f ). Hence the pro │ │ │ │ +00025650: 6a65 6374 6976 6520 6465 6772 6565 7320 jective degrees │ │ │ │ +00025660: 616e 6420 5365 6772 6520 636c 6173 7365 and Segre classe │ │ │ │ +00025670: 7320 636f 6d70 7574 6564 0a69 6e74 6572 s computed.inter │ │ │ │ +00025680: 6e61 6c6c 7920 7769 6c6c 2062 6520 7468 nally will be th │ │ │ │ +00025690: 6f73 6520 6f66 2074 6865 2072 6164 6963 ose of the radic │ │ │ │ +000256a0: 616c 206f 6620 616e 2069 6465 616c 2064 al of an ideal d │ │ │ │ +000256b0: 6566 696e 6564 2062 7920 6120 706f 6c79 efined by a poly │ │ │ │ +000256c0: 6e6f 6d69 616c 0a77 6869 6368 2069 7320 nomial.which is │ │ │ │ +000256d0: 6120 7072 6f64 7563 7420 6f66 2073 6f6d a product of som │ │ │ │ +000256e0: 6520 7375 6273 6574 206f 6620 7468 6520 e subset of the │ │ │ │ +000256f0: 6765 6e65 7261 746f 7273 2e20 5765 2069 generators. We i │ │ │ │ +00025700: 6c6c 7573 7472 6174 6520 7468 6973 2077 llustrate this w │ │ │ │ +00025710: 6974 6820 616e 0a65 7861 6d70 6c65 2062 ith an.example b │ │ │ │ +00025720: 656c 6f77 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d elow...+-------- │ │ │ │ 00025730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025750: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ -00025760: 6373 6d58 4c68 6173 683d 4353 4d28 412c csmXLhash=CSM(A, │ │ │ │ -00025770: 492c 4f75 7470 7574 3d3e 4861 7368 466f I,Output=>HashFo │ │ │ │ -00025780: 726d 584c 2920 2020 2020 2020 2020 2020 rmXL) │ │ │ │ -00025790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025770: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 6373 -----+.|i14 : cs │ │ │ │ +00025780: 6d58 4c68 6173 683d 4353 4d28 412c 492c mXLhash=CSM(A,I, │ │ │ │ +00025790: 4f75 7470 7574 3d3e 4861 7368 466f 726d Output=>HashForm │ │ │ │ +000257a0: 584c 2920 2020 2020 2020 2020 2020 2020 XL) │ │ │ │ 000257b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000257c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000257d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000257e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257f0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ -00025800: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -00025810: 7b2e 2e2e 3130 2e2e 2e7d 2020 2020 2020 {...10...} │ │ │ │ -00025820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025840: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000257f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025810: 2020 2020 207c 0a7c 6f31 3420 3d20 4d75 |.|o14 = Mu │ │ │ │ +00025820: 7461 626c 6548 6173 6854 6162 6c65 7b2e tableHashTable{. │ │ │ │ +00025830: 2e2e 3130 2e2e 2e7d 2020 2020 2020 2020 ..10...} │ │ │ │ +00025840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025860: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00025870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025890: 2020 2020 2020 207c 0a7c 6f31 3420 3a20 |.|o14 : │ │ │ │ -000258a0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -000258b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000258a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000258b0: 2020 2020 207c 0a7c 6f31 3420 3a20 4d75 |.|o14 : Mu │ │ │ │ +000258c0: 7461 626c 6548 6173 6854 6162 6c65 2020 tableHashTable │ │ │ │ 000258d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000258e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000258f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025900: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025930: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ -00025940: 7065 656b 2063 736d 584c 6861 7368 2020 peek csmXLhash │ │ │ │ -00025950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025950: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7065 -----+.|i15 : pe │ │ │ │ +00025960: 656b 2063 736d 584c 6861 7368 2020 2020 ek csmXLhash │ │ │ │ 00025970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025980: 2020 2020 2020 207c 0a7c 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 │ │ │ │ +000259a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000259b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000259c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000259d0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20 |.|o15 = │ │ │ │ -000259e0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -000259f0: 7b47 284a 6163 6f62 6961 6e29 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2020 2020 2020 2020 2020 2020 │ │ │ │ -000265b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -000265c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000265b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026600: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ -00026610: 4b3d 6964 6561 6c20 495f 302a 495f 313b K=ideal I_0*I_1; │ │ │ │ -00026620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026620: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4b3d -----+.|i16 : K= │ │ │ │ +00026630: 6964 6561 6c20 495f 302a 495f 313b 2020 ideal I_0*I_1; │ │ │ │ 00026640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026650: 2020 2020 2020 207c 0a7c 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 │ │ │ │ +00026670: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266a0: 2020 2020 2020 207c 0a7c 6f31 3620 3a20 |.|o16 : │ │ │ │ -000266b0: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ -000266c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000266a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000266b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000266c0: 2020 2020 207c 0a7c 6f31 3620 3a20 4964 |.|o16 : Id │ │ │ │ +000266d0: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ 000266e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000266f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026710: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00026720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026740: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 -------+.|i17 : │ │ │ │ -00026750: 4353 4d28 412c 7261 6469 6361 6c20 4b29 CSM(A,radical K) │ │ │ │ -00026760: 3d3d 4353 4d28 412c 4b29 2020 2020 2020 ==CSM(A,K) │ │ │ │ -00026770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026790: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00026740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026760: 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 4353 -----+.|i17 : CS │ │ │ │ +00026770: 4d28 412c 7261 6469 6361 6c20 4b29 3d3d M(A,radical K)== │ │ │ │ +00026780: 4353 4d28 412c 4b29 2020 2020 2020 2020 CSM(A,K) │ │ │ │ +00026790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000267b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000267b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000267c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000267e0: 2020 2020 2020 207c 0a7c 6f31 3720 3d20 |.|o17 = │ │ │ │ -000267f0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ -00026800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026810: 2020 2020 2020 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 207c 0a7c 6f31 3720 3d20 7472 |.|o17 = tr │ │ │ │ +00026810: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 00026820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026830: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026850: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00026860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026880: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 -------+.|i18 : │ │ │ │ -00026890: 4a3d 6964 6561 6c20 6a61 636f 6269 616e J=ideal jacobian │ │ │ │ -000268a0: 2072 6164 6963 616c 204b 3b20 2020 2020 radical K; │ │ │ │ -000268b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00026880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000268a0: 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 4a3d -----+.|i18 : J= │ │ │ │ +000268b0: 6964 6561 6c20 6a61 636f 6269 616e 2072 ideal jacobian r │ │ │ │ +000268c0: 6164 6963 616c 204b 3b20 2020 2020 2020 adical K; │ │ │ │ +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 │ │ │ │ +000268f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026920: 2020 2020 2020 207c 0a7c 6f31 3820 3a20 |.|o18 : │ │ │ │ -00026930: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ -00026940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026940: 2020 2020 207c 0a7c 6f31 3820 3a20 4964 |.|o18 : Id │ │ │ │ +00026950: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ 00026960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026970: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026990: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 000269a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000269b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000269c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 -------+.|i19 : │ │ │ │ -000269d0: 7365 674a 3d53 6567 7265 2841 2c4a 2c4f segJ=Segre(A,J,O │ │ │ │ -000269e0: 7574 7075 743d 3e48 6173 6846 6f72 6d29 utput=>HashForm) │ │ │ │ -000269f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000269c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000269d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000269e0: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 7365 -----+.|i19 : se │ │ │ │ +000269f0: 674a 3d53 6567 7265 2841 2c4a 2c4f 7574 gJ=Segre(A,J,Out │ │ │ │ +00026a00: 7075 743d 3e48 6173 6846 6f72 6d29 2020 put=>HashForm) │ │ │ │ +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 2020 │ │ │ │ +00026a30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a60: 2020 2020 2020 207c 0a7c 6f31 3920 3d20 |.|o19 = │ │ │ │ -00026a70: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -00026a80: 7b2e 2e2e 342e 2e2e 7d20 2020 2020 2020 {...4...} │ │ │ │ -00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00026a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026a80: 2020 2020 207c 0a7c 6f31 3920 3d20 4d75 |.|o19 = Mu │ │ │ │ +00026a90: 7461 626c 6548 6173 6854 6162 6c65 7b2e tableHashTable{. │ │ │ │ +00026aa0: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020 ..4...} │ │ │ │ +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 2020 │ │ │ │ +00026ad0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b00: 2020 2020 2020 207c 0a7c 6f31 3920 3a20 |.|o19 : │ │ │ │ -00026b10: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -00026b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b20: 2020 2020 207c 0a7c 6f31 3920 3a20 4d75 |.|o19 : Mu │ │ │ │ +00026b30: 7461 626c 6548 6173 6854 6162 6c65 2020 tableHashTable │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00026b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026ba0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 -------+.|i20 : │ │ │ │ -00026bb0: 6373 6d58 4c68 6173 6823 2822 4728 4a61 csmXLhash#("G(Ja │ │ │ │ -00026bc0: 636f 6269 616e 2922 7c74 6f53 7472 696e cobian)"|toStrin │ │ │ │ -00026bd0: 6728 7b30 2c31 7d29 293d 3d73 6567 4a23 g({0,1}))==segJ# │ │ │ │ -00026be0: 2247 2220 2020 2020 2020 2020 2020 2020 "G" │ │ │ │ -00026bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00026c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026bc0: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 6373 -----+.|i20 : cs │ │ │ │ +00026bd0: 6d58 4c68 6173 6823 2822 4728 4a61 636f mXLhash#("G(Jaco │ │ │ │ +00026be0: 6269 616e 2922 7c74 6f53 7472 696e 6728 bian)"|toString( │ │ │ │ +00026bf0: 7b30 2c31 7d29 293d 3d73 6567 4a23 2247 {0,1}))==segJ#"G │ │ │ │ +00026c00: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +00026c10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c40: 2020 2020 2020 207c 0a7c 6f32 3020 3d20 |.|o20 = │ │ │ │ -00026c50: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ -00026c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c60: 2020 2020 207c 0a7c 6f32 3020 3d20 7472 |.|o20 = tr │ │ │ │ +00026c70: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 00026c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026cb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00026cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026ce0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 -------+.|i21 : │ │ │ │ -00026cf0: 6373 6d58 4c68 6173 6823 2822 5365 6772 csmXLhash#("Segr │ │ │ │ -00026d00: 6528 4a61 636f 6269 616e 2922 7c74 6f53 e(Jacobian)"|toS │ │ │ │ -00026d10: 7472 696e 6728 7b30 2c31 7d29 293d 3d73 tring({0,1}))==s │ │ │ │ -00026d20: 6567 4a23 2253 6567 7265 2220 2020 2020 egJ#"Segre" │ │ │ │ -00026d30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00026d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026d00: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 6373 -----+.|i21 : cs │ │ │ │ +00026d10: 6d58 4c68 6173 6823 2822 5365 6772 6528 mXLhash#("Segre( │ │ │ │ +00026d20: 4a61 636f 6269 616e 2922 7c74 6f53 7472 Jacobian)"|toStr │ │ │ │ +00026d30: 696e 6728 7b30 2c31 7d29 293d 3d73 6567 ing({0,1}))==seg │ │ │ │ +00026d40: 4a23 2253 6567 7265 2220 2020 2020 2020 J#"Segre" │ │ │ │ +00026d50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d80: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ -00026d90: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ -00026da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026da0: 2020 2020 207c 0a7c 6f32 3120 3d20 7472 |.|o21 = tr │ │ │ │ +00026db0: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00026de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026df0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00026e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e20: 2d2d 2d2d 2d2d 2d2b 0a0a 4675 6e63 7469 -------+..Functi │ │ │ │ -00026e30: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00026e40: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ -00026e50: 204f 7574 7075 743a 0a3d 3d3d 3d3d 3d3d Output:.======= │ │ │ │ -00026e60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00026e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00026e80: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4368 =======.. * "Ch │ │ │ │ -00026e90: 6572 6e28 2e2e 2e2c 4f75 7470 7574 3d3e ern(...,Output=> │ │ │ │ -00026ea0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ -00026eb0: 7465 2043 6865 726e 3a20 4368 6572 6e2c te Chern: Chern, │ │ │ │ -00026ec0: 202d 2d20 5468 6520 4368 6572 6e20 636c -- The Chern cl │ │ │ │ -00026ed0: 6173 730a 2020 2a20 2243 534d 282e 2e2e ass. * "CSM(... │ │ │ │ -00026ee0: 2c4f 7574 7075 743d 3e2e 2e2e 2922 202d ,Output=>...)" - │ │ │ │ -00026ef0: 2d20 7365 6520 2a6e 6f74 6520 4353 4d3a - see *note CSM: │ │ │ │ -00026f00: 2043 534d 2c20 2d2d 2054 6865 0a20 2020 CSM, -- The. │ │ │ │ -00026f10: 2043 6865 726e 2d53 6368 7761 7274 7a2d Chern-Schwartz- │ │ │ │ -00026f20: 4d61 6350 6865 7273 6f6e 2063 6c61 7373 MacPherson class │ │ │ │ -00026f30: 0a20 202a 2022 4575 6c65 7228 2e2e 2e2c . * "Euler(..., │ │ │ │ -00026f40: 4f75 7470 7574 3d3e 2e2e 2e29 2220 2d2d Output=>...)" -- │ │ │ │ -00026f50: 2073 6565 202a 6e6f 7465 2045 756c 6572 see *note Euler │ │ │ │ -00026f60: 3a20 4575 6c65 722c 202d 2d20 5468 6520 : Euler, -- The │ │ │ │ -00026f70: 4575 6c65 720a 2020 2020 4368 6172 6163 Euler. Charac │ │ │ │ -00026f80: 7465 7269 7374 6963 0a20 202a 2022 5365 teristic. * "Se │ │ │ │ -00026f90: 6772 6528 2e2e 2e2c 4f75 7470 7574 3d3e gre(...,Output=> │ │ │ │ -00026fa0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ -00026fb0: 7465 2053 6567 7265 3a20 5365 6772 652c te Segre: Segre, │ │ │ │ -00026fc0: 202d 2d20 5468 6520 5365 6772 6520 636c -- The Segre cl │ │ │ │ -00026fd0: 6173 7320 6f66 2061 0a20 2020 2073 7562 ass of a. sub │ │ │ │ -00026fe0: 7363 6865 6d65 0a0a 466f 7220 7468 6520 scheme..For the │ │ │ │ -00026ff0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00027000: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00027010: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00027020: 4f75 7470 7574 3a20 4f75 7470 7574 2c20 Output: Output, │ │ │ │ -00027030: 6973 2061 202a 6e6f 7465 2073 796d 626f is a *note symbo │ │ │ │ -00027040: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -00027050: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d )Symbol,...----- │ │ │ │ -00027060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026e40: 2d2d 2d2d 2d2b 0a0a 4675 6e63 7469 6f6e -----+..Function │ │ │ │ +00026e50: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +00026e60: 6172 6775 6d65 6e74 206e 616d 6564 204f argument named O │ │ │ │ +00026e70: 7574 7075 743a 0a3d 3d3d 3d3d 3d3d 3d3d utput:.========= │ │ │ │ +00026e80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00026e90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00026ea0: 3d3d 3d3d 3d0a 0a20 202a 2022 4368 6572 =====.. * "Cher │ │ │ │ +00026eb0: 6e28 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e n(...,Output=>.. │ │ │ │ +00026ec0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ +00026ed0: 2043 6865 726e 3a20 4368 6572 6e2c 202d Chern: Chern, - │ │ │ │ +00026ee0: 2d20 5468 6520 4368 6572 6e20 636c 6173 - The Chern clas │ │ │ │ +00026ef0: 730a 2020 2a20 2243 534d 282e 2e2e 2c4f s. * "CSM(...,O │ │ │ │ +00026f00: 7574 7075 743d 3e2e 2e2e 2922 202d 2d20 utput=>...)" -- │ │ │ │ +00026f10: 7365 6520 2a6e 6f74 6520 4353 4d3a 2043 see *note CSM: C │ │ │ │ +00026f20: 534d 2c20 2d2d 2054 6865 0a20 2020 2043 SM, -- The. C │ │ │ │ +00026f30: 6865 726e 2d53 6368 7761 7274 7a2d 4d61 hern-Schwartz-Ma │ │ │ │ +00026f40: 6350 6865 7273 6f6e 2063 6c61 7373 0a20 cPherson class. │ │ │ │ +00026f50: 202a 2022 4575 6c65 7228 2e2e 2e2c 4f75 * "Euler(...,Ou │ │ │ │ +00026f60: 7470 7574 3d3e 2e2e 2e29 2220 2d2d 2073 tput=>...)" -- s │ │ │ │ +00026f70: 6565 202a 6e6f 7465 2045 756c 6572 3a20 ee *note Euler: │ │ │ │ +00026f80: 4575 6c65 722c 202d 2d20 5468 6520 4575 Euler, -- The Eu │ │ │ │ +00026f90: 6c65 720a 2020 2020 4368 6172 6163 7465 ler. Characte │ │ │ │ +00026fa0: 7269 7374 6963 0a20 202a 2022 5365 6772 ristic. * "Segr │ │ │ │ +00026fb0: 6528 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e e(...,Output=>.. │ │ │ │ +00026fc0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ +00026fd0: 2053 6567 7265 3a20 5365 6772 652c 202d Segre: Segre, - │ │ │ │ +00026fe0: 2d20 5468 6520 5365 6772 6520 636c 6173 - The Segre clas │ │ │ │ +00026ff0: 7320 6f66 2061 0a20 2020 2073 7562 7363 s of a. subsc │ │ │ │ +00027000: 6865 6d65 0a0a 466f 7220 7468 6520 7072 heme..For the pr │ │ │ │ +00027010: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00027020: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00027030: 206f 626a 6563 7420 2a6e 6f74 6520 4f75 object *note Ou │ │ │ │ +00027040: 7470 7574 3a20 4f75 7470 7574 2c20 6973 tput: Output, is │ │ │ │ +00027050: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ +00027060: 2028 4d61 6361 756c 6179 3244 6f63 2953 (Macaulay2Doc)S │ │ │ │ +00027070: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ 00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000270b0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -000270c0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -000270d0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -000270e0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -000270f0: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ -00027100: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00027110: 732f 0a43 6861 7261 6374 6572 6973 7469 s/.Characteristi │ │ │ │ -00027120: 6343 6c61 7373 6573 2e6d 323a 3234 3332 cClasses.m2:2432 │ │ │ │ -00027130: 3a30 2e0a 1f0a 4669 6c65 3a20 4368 6172 :0....File: Char │ │ │ │ -00027140: 6163 7465 7269 7374 6963 436c 6173 7365 acteristicClasse │ │ │ │ -00027150: 732e 696e 666f 2c20 4e6f 6465 3a20 7072 s.info, Node: pr │ │ │ │ -00027160: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ -00027170: 7269 7468 6d2c 204e 6578 743a 2053 6567 rithm, Next: Seg │ │ │ │ -00027180: 7265 2c20 5072 6576 3a20 4f75 7470 7574 re, Prev: Output │ │ │ │ -00027190: 2c20 5570 3a20 546f 700a 0a70 726f 6261 , Up: Top..proba │ │ │ │ -000271a0: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -000271b0: 686d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a hm.************* │ │ │ │ -000271c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5468 6520 **********..The │ │ │ │ -000271d0: 616c 676f 7269 7468 6d73 2075 7365 6420 algorithms used │ │ │ │ -000271e0: 666f 7220 7468 6520 636f 6d70 7574 6174 for the computat │ │ │ │ -000271f0: 696f 6e20 6f66 2063 6861 7261 6374 6572 ion of character │ │ │ │ -00027200: 6973 7469 6320 636c 6173 7365 7320 6172 istic classes ar │ │ │ │ -00027210: 650a 7072 6f62 6162 696c 6973 7469 632e e.probabilistic. │ │ │ │ -00027220: 2054 6865 6f72 6574 6963 616c 6c79 2c20 Theoretically, │ │ │ │ -00027230: 7468 6579 2063 616c 6375 6c61 7465 2074 they calculate t │ │ │ │ -00027240: 6865 2063 6c61 7373 6573 2063 6f72 7265 he classes corre │ │ │ │ -00027250: 6374 6c79 2066 6f72 2061 0a67 656e 6572 ctly for a.gener │ │ │ │ -00027260: 616c 2063 686f 6963 6520 6f66 2063 6572 al choice of cer │ │ │ │ -00027270: 7461 696e 2070 6f6c 796e 6f6d 6961 6c73 tain polynomials │ │ │ │ -00027280: 2e20 5468 6174 2069 732c 2074 6865 7265 . That is, there │ │ │ │ -00027290: 2069 7320 616e 206f 7065 6e20 6465 6e73 is an open dens │ │ │ │ -000272a0: 6520 5a61 7269 736b 690a 7365 7420 666f e Zariski.set fo │ │ │ │ -000272b0: 7220 7768 6963 6820 7468 6520 616c 676f r which the algo │ │ │ │ -000272c0: 7269 7468 6d20 7969 656c 6473 2074 6865 rithm yields the │ │ │ │ -000272d0: 2063 6f72 7265 6374 2063 6c61 7373 2c20 correct class, │ │ │ │ -000272e0: 692e 652e 2c20 7468 6520 636f 7272 6563 i.e., the correc │ │ │ │ -000272f0: 7420 636c 6173 730a 6973 2063 616c 6375 t class.is calcu │ │ │ │ -00027300: 6c61 7465 6420 7769 7468 2070 726f 6261 lated with proba │ │ │ │ -00027310: 6269 6c69 7479 2031 2e20 486f 7765 7665 bility 1. Howeve │ │ │ │ -00027320: 722c 2073 696e 6365 2074 6865 2069 6d70 r, since the imp │ │ │ │ -00027330: 6c65 6d65 6e74 6174 696f 6e20 776f 726b lementation work │ │ │ │ -00027340: 7320 6f76 6572 0a61 2064 6973 6372 6574 s over.a discret │ │ │ │ -00027350: 6520 7072 6f62 6162 696c 6974 7920 7370 e probability sp │ │ │ │ -00027360: 6163 6520 7468 6572 6520 6973 2061 2076 ace there is a v │ │ │ │ -00027370: 6572 7920 736d 616c 6c2c 2062 7574 206e ery small, but n │ │ │ │ -00027380: 6f6e 2d7a 6572 6f2c 2070 726f 6261 6269 on-zero, probabi │ │ │ │ -00027390: 6c69 7479 0a6f 6620 6e6f 7420 636f 6d70 lity.of not comp │ │ │ │ -000273a0: 7574 696e 6720 7468 6520 636f 7272 6563 uting the correc │ │ │ │ -000273b0: 7420 636c 6173 732e 2053 6b65 7074 6963 t class. Skeptic │ │ │ │ -000273c0: 616c 2075 7365 7273 2073 686f 756c 6420 al users should │ │ │ │ -000273d0: 7265 7065 6174 2063 616c 6375 6c61 7469 repeat calculati │ │ │ │ -000273e0: 6f6e 730a 7365 7665 7261 6c20 7469 6d65 ons.several time │ │ │ │ -000273f0: 7320 746f 2069 6e63 7265 6173 6520 7468 s to increase th │ │ │ │ -00027400: 6520 7072 6f62 6162 696c 6974 7920 6f66 e probability of │ │ │ │ -00027410: 2063 6f6d 7075 7469 6e67 2074 6865 2063 computing the c │ │ │ │ -00027420: 6f72 7265 6374 2063 6c61 7373 2e0a 0a49 orrect class...I │ │ │ │ -00027430: 6e20 7468 6520 6361 7365 206f 6620 7468 n the case of th │ │ │ │ -00027440: 6520 7379 6d62 6f6c 6963 2069 6d70 6c65 e symbolic imple │ │ │ │ -00027450: 6d65 6e74 6174 696f 6e20 6f66 2074 6865 mentation of the │ │ │ │ -00027460: 2050 726f 6a65 6374 6976 6544 6567 7265 ProjectiveDegre │ │ │ │ -00027470: 6520 6d65 7468 6f64 0a70 7261 6374 6963 e method.practic │ │ │ │ -00027480: 616c 2065 7870 6572 6965 6e63 6520 616e al experience an │ │ │ │ -00027490: 6420 616c 676f 7269 7468 6d20 7465 7374 d algorithm test │ │ │ │ -000274a0: 696e 6720 696e 6469 6361 7465 2074 6861 ing indicate tha │ │ │ │ -000274b0: 7420 6120 6669 6e69 7465 2066 6965 6c64 t a finite field │ │ │ │ -000274c0: 2077 6974 680a 6f76 6572 2032 3530 3030 with.over 25000 │ │ │ │ -000274d0: 2065 6c65 6d65 6e74 7320 6973 206d 6f72 elements is mor │ │ │ │ -000274e0: 6520 7468 616e 2073 7566 6669 6369 656e e than sufficien │ │ │ │ -000274f0: 7420 746f 2065 7870 6563 7420 6120 636f t to expect a co │ │ │ │ -00027500: 7272 6563 7420 7265 7375 6c74 2077 6974 rrect result wit │ │ │ │ -00027510: 680a 6869 6768 2070 726f 6261 6269 6c69 h.high probabili │ │ │ │ -00027520: 7479 2c20 692e 652e 2075 7369 6e67 2074 ty, i.e. using t │ │ │ │ -00027530: 6865 2066 696e 6974 6520 6669 656c 6420 he finite field │ │ │ │ -00027540: 6b6b 3d5a 5a2f 3235 3037 3320 7468 6520 kk=ZZ/25073 the │ │ │ │ -00027550: 6578 7065 7269 6d65 6e74 616c 0a63 6861 experimental.cha │ │ │ │ -00027560: 6e63 6520 6f66 2066 6169 6c75 7265 2077 nce of failure w │ │ │ │ -00027570: 6974 6820 7468 6520 5072 6f6a 6563 7469 ith the Projecti │ │ │ │ -00027580: 7665 4465 6772 6565 2061 6c67 6f72 6974 veDegree algorit │ │ │ │ -00027590: 686d 206f 6e20 6120 7661 7269 6574 7920 hm on a variety │ │ │ │ -000275a0: 6f66 2065 7861 6d70 6c65 730a 7761 7320 of examples.was │ │ │ │ -000275b0: 6c65 7373 2074 6861 6e20 312f 3230 3030 less than 1/2000 │ │ │ │ -000275c0: 2e20 5573 696e 6720 7468 6520 6669 6e69 . Using the fini │ │ │ │ -000275d0: 7465 2066 6965 6c64 206b 6b3d 5a5a 2f33 te field kk=ZZ/3 │ │ │ │ -000275e0: 3237 3439 2072 6573 756c 7465 6420 696e 2749 resulted in │ │ │ │ -000275f0: 206e 6f0a 6661 696c 7572 6573 2069 6e20 no.failures in │ │ │ │ -00027600: 6f76 6572 2031 3030 3030 2061 7474 656d over 10000 attem │ │ │ │ -00027610: 7074 7320 6f66 2073 6576 6572 616c 2064 pts of several d │ │ │ │ -00027620: 6966 6665 7265 6e74 2065 7861 6d70 6c65 ifferent example │ │ │ │ -00027630: 732e 0a0a 5765 2069 6c6c 7573 7472 6174 s...We illustrat │ │ │ │ -00027640: 6520 7468 6520 7072 6f62 6162 696c 6973 e the probabilis │ │ │ │ -00027650: 7469 6320 6265 6861 7669 6f75 7220 7769 tic behaviour wi │ │ │ │ -00027660: 7468 2061 6e20 6578 616d 706c 6520 7768 th an example wh │ │ │ │ -00027670: 6572 6520 7468 6520 6368 6f73 656e 0a72 ere the chosen.r │ │ │ │ -00027680: 616e 646f 6d20 7365 6564 206c 6561 6473 andom seed leads │ │ │ │ -00027690: 2074 6f20 6120 7772 6f6e 6720 7265 7375 to a wrong resu │ │ │ │ -000276a0: 6c74 2069 6e20 7468 6520 6669 7273 7420 lt in the first │ │ │ │ -000276b0: 6361 6c63 756c 6174 696f 6e2e 0a0a 2b2d calculation...+- │ │ │ │ -000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000276f0: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d .|i1 : setRandom │ │ │ │ -00027700: 5365 6564 2031 3231 3b20 2020 2020 2020 Seed 121; │ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027720: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -00027730: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -00027740: 3132 3120 2020 2020 2020 2020 2020 2020 121 │ │ │ │ -00027750: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00027760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027780: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00027790: 5220 3d20 5151 5b78 2c79 2c7a 2c77 5d20 R = QQ[x,y,z,w] │ │ │ │ -000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000270b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000270c0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000270d0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000270e0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000270f0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00027100: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00027110: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ +00027120: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00027130: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +00027140: 6c61 7373 6573 2e6d 323a 3234 3332 3a30 lasses.m2:2432:0 │ │ │ │ +00027150: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +00027160: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00027170: 696e 666f 2c20 4e6f 6465 3a20 7072 6f62 info, Node: prob │ │ │ │ +00027180: 6162 696c 6973 7469 6320 616c 676f 7269 abilistic algori │ │ │ │ +00027190: 7468 6d2c 204e 6578 743a 2053 6567 7265 thm, Next: Segre │ │ │ │ +000271a0: 2c20 5072 6576 3a20 4f75 7470 7574 2c20 , Prev: Output, │ │ │ │ +000271b0: 5570 3a20 546f 700a 0a70 726f 6261 6269 Up: Top..probabi │ │ │ │ +000271c0: 6c69 7374 6963 2061 6c67 6f72 6974 686d listic algorithm │ │ │ │ +000271d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +000271e0: 2a2a 2a2a 2a2a 2a2a 0a0a 5468 6520 616c ********..The al │ │ │ │ +000271f0: 676f 7269 7468 6d73 2075 7365 6420 666f gorithms used fo │ │ │ │ +00027200: 7220 7468 6520 636f 6d70 7574 6174 696f r the computatio │ │ │ │ +00027210: 6e20 6f66 2063 6861 7261 6374 6572 6973 n of characteris │ │ │ │ +00027220: 7469 6320 636c 6173 7365 7320 6172 650a tic classes are. │ │ │ │ +00027230: 7072 6f62 6162 696c 6973 7469 632e 2054 probabilistic. T │ │ │ │ +00027240: 6865 6f72 6574 6963 616c 6c79 2c20 7468 heoretically, th │ │ │ │ +00027250: 6579 2063 616c 6375 6c61 7465 2074 6865 ey calculate the │ │ │ │ +00027260: 2063 6c61 7373 6573 2063 6f72 7265 6374 classes correct │ │ │ │ +00027270: 6c79 2066 6f72 2061 0a67 656e 6572 616c ly for a.general │ │ │ │ +00027280: 2063 686f 6963 6520 6f66 2063 6572 7461 choice of certa │ │ │ │ +00027290: 696e 2070 6f6c 796e 6f6d 6961 6c73 2e20 in polynomials. │ │ │ │ +000272a0: 5468 6174 2069 732c 2074 6865 7265 2069 That is, there i │ │ │ │ +000272b0: 7320 616e 206f 7065 6e20 6465 6e73 6520 s an open dense │ │ │ │ +000272c0: 5a61 7269 736b 690a 7365 7420 666f 7220 Zariski.set for │ │ │ │ +000272d0: 7768 6963 6820 7468 6520 616c 676f 7269 which the algori │ │ │ │ +000272e0: 7468 6d20 7969 656c 6473 2074 6865 2063 thm yields the c │ │ │ │ +000272f0: 6f72 7265 6374 2063 6c61 7373 2c20 692e orrect class, i. │ │ │ │ +00027300: 652e 2c20 7468 6520 636f 7272 6563 7420 e., the correct │ │ │ │ +00027310: 636c 6173 730a 6973 2063 616c 6375 6c61 class.is calcula │ │ │ │ +00027320: 7465 6420 7769 7468 2070 726f 6261 6269 ted with probabi │ │ │ │ +00027330: 6c69 7479 2031 2e20 486f 7765 7665 722c lity 1. However, │ │ │ │ +00027340: 2073 696e 6365 2074 6865 2069 6d70 6c65 since the imple │ │ │ │ +00027350: 6d65 6e74 6174 696f 6e20 776f 726b 7320 mentation works │ │ │ │ +00027360: 6f76 6572 0a61 2064 6973 6372 6574 6520 over.a discrete │ │ │ │ +00027370: 7072 6f62 6162 696c 6974 7920 7370 6163 probability spac │ │ │ │ +00027380: 6520 7468 6572 6520 6973 2061 2076 6572 e there is a ver │ │ │ │ +00027390: 7920 736d 616c 6c2c 2062 7574 206e 6f6e y small, but non │ │ │ │ +000273a0: 2d7a 6572 6f2c 2070 726f 6261 6269 6c69 -zero, probabili │ │ │ │ +000273b0: 7479 0a6f 6620 6e6f 7420 636f 6d70 7574 ty.of not comput │ │ │ │ +000273c0: 696e 6720 7468 6520 636f 7272 6563 7420 ing the correct │ │ │ │ +000273d0: 636c 6173 732e 2053 6b65 7074 6963 616c class. Skeptical │ │ │ │ +000273e0: 2075 7365 7273 2073 686f 756c 6420 7265 users should re │ │ │ │ +000273f0: 7065 6174 2063 616c 6375 6c61 7469 6f6e peat calculation │ │ │ │ +00027400: 730a 7365 7665 7261 6c20 7469 6d65 7320 s.several times │ │ │ │ +00027410: 746f 2069 6e63 7265 6173 6520 7468 6520 to increase the │ │ │ │ +00027420: 7072 6f62 6162 696c 6974 7920 6f66 2063 probability of c │ │ │ │ +00027430: 6f6d 7075 7469 6e67 2074 6865 2063 6f72 omputing the cor │ │ │ │ +00027440: 7265 6374 2063 6c61 7373 2e0a 0a49 6e20 rect class...In │ │ │ │ +00027450: 7468 6520 6361 7365 206f 6620 7468 6520 the case of the │ │ │ │ +00027460: 7379 6d62 6f6c 6963 2069 6d70 6c65 6d65 symbolic impleme │ │ │ │ +00027470: 6e74 6174 696f 6e20 6f66 2074 6865 2050 ntation of the P │ │ │ │ +00027480: 726f 6a65 6374 6976 6544 6567 7265 6520 rojectiveDegree │ │ │ │ +00027490: 6d65 7468 6f64 0a70 7261 6374 6963 616c method.practical │ │ │ │ +000274a0: 2065 7870 6572 6965 6e63 6520 616e 6420 experience and │ │ │ │ +000274b0: 616c 676f 7269 7468 6d20 7465 7374 696e algorithm testin │ │ │ │ +000274c0: 6720 696e 6469 6361 7465 2074 6861 7420 g indicate that │ │ │ │ +000274d0: 6120 6669 6e69 7465 2066 6965 6c64 2077 a finite field w │ │ │ │ +000274e0: 6974 680a 6f76 6572 2032 3530 3030 2065 ith.over 25000 e │ │ │ │ +000274f0: 6c65 6d65 6e74 7320 6973 206d 6f72 6520 lements is more │ │ │ │ +00027500: 7468 616e 2073 7566 6669 6369 656e 7420 than sufficient │ │ │ │ +00027510: 746f 2065 7870 6563 7420 6120 636f 7272 to expect a corr │ │ │ │ +00027520: 6563 7420 7265 7375 6c74 2077 6974 680a ect result with. │ │ │ │ +00027530: 6869 6768 2070 726f 6261 6269 6c69 7479 high probability │ │ │ │ +00027540: 2c20 692e 652e 2075 7369 6e67 2074 6865 , i.e. using the │ │ │ │ +00027550: 2066 696e 6974 6520 6669 656c 6420 6b6b finite field kk │ │ │ │ +00027560: 3d5a 5a2f 3235 3037 3320 7468 6520 6578 =ZZ/25073 the ex │ │ │ │ +00027570: 7065 7269 6d65 6e74 616c 0a63 6861 6e63 perimental.chanc │ │ │ │ +00027580: 6520 6f66 2066 6169 6c75 7265 2077 6974 e of failure wit │ │ │ │ +00027590: 6820 7468 6520 5072 6f6a 6563 7469 7665 h the Projective │ │ │ │ +000275a0: 4465 6772 6565 2061 6c67 6f72 6974 686d Degree algorithm │ │ │ │ +000275b0: 206f 6e20 6120 7661 7269 6574 7920 6f66 on a variety of │ │ │ │ +000275c0: 2065 7861 6d70 6c65 730a 7761 7320 6c65 examples.was le │ │ │ │ +000275d0: 7373 2074 6861 6e20 312f 3230 3030 2e20 ss than 1/2000. │ │ │ │ +000275e0: 5573 696e 6720 7468 6520 6669 6e69 7465 Using the finite │ │ │ │ +000275f0: 2066 6965 6c64 206b 6b3d 5a5a 2f33 3237 field kk=ZZ/327 │ │ │ │ +00027600: 3439 2072 6573 756c 7465 6420 696e 206e 49 resulted in n │ │ │ │ +00027610: 6f0a 6661 696c 7572 6573 2069 6e20 6f76 o.failures in ov │ │ │ │ +00027620: 6572 2031 3030 3030 2061 7474 656d 7074 er 10000 attempt │ │ │ │ +00027630: 7320 6f66 2073 6576 6572 616c 2064 6966 s of several dif │ │ │ │ +00027640: 6665 7265 6e74 2065 7861 6d70 6c65 732e ferent examples. │ │ │ │ +00027650: 0a0a 5765 2069 6c6c 7573 7472 6174 6520 ..We illustrate │ │ │ │ +00027660: 7468 6520 7072 6f62 6162 696c 6973 7469 the probabilisti │ │ │ │ +00027670: 6320 6265 6861 7669 6f75 7220 7769 7468 c behaviour with │ │ │ │ +00027680: 2061 6e20 6578 616d 706c 6520 7768 6572 an example wher │ │ │ │ +00027690: 6520 7468 6520 6368 6f73 656e 0a72 616e e the chosen.ran │ │ │ │ +000276a0: 646f 6d20 7365 6564 206c 6561 6473 2074 dom seed leads t │ │ │ │ +000276b0: 6f20 6120 7772 6f6e 6720 7265 7375 6c74 o a wrong result │ │ │ │ +000276c0: 2069 6e20 7468 6520 6669 7273 7420 6361 in the first ca │ │ │ │ +000276d0: 6c63 756c 6174 696f 6e2e 0a0a 2b2d 2d2d lculation...+--- │ │ │ │ +000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027710: 6931 203a 2073 6574 5261 6e64 6f6d 5365 i1 : setRandomSe │ │ │ │ +00027720: 6564 2031 3231 3b20 2020 2020 2020 2020 ed 121; │ │ │ │ +00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027740: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072 |.| -- setting r │ │ │ │ +00027750: 616e 646f 6d20 7365 6564 2074 6f20 3132 andom seed to 12 │ │ │ │ +00027760: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00027770: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000277a0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5220 ------+.|i2 : R │ │ │ │ +000277b0: 3d20 5151 5b78 2c79 2c7a 2c77 5d20 2020 = QQ[x,y,z,w] │ │ │ │ 000277c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000277f0: 7c6f 3220 3d20 5220 2020 2020 2020 2020 |o2 = R │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027820: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00027830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027850: 2020 2020 7c0a 7c6f 3220 3a20 506f 6c79 |.|o2 : Poly │ │ │ │ -00027860: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ -00027870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000278a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -000278c0: 3a20 4920 3d20 6d69 6e6f 7273 2832 2c6d : I = minors(2,m │ │ │ │ -000278d0: 6174 7269 787b 7b78 2c79 2c7a 7d2c 7b79 atrix{{x,y,z},{y │ │ │ │ -000278e0: 2c7a 2c77 7d7d 2920 2020 2020 207c 0a7c ,z,w}}) |.| │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +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 7c0a 7c6f |.|o │ │ │ │ +00027810: 3220 3d20 5220 2020 2020 2020 2020 2020 2 = R │ │ │ │ +00027820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027830: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027840: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00027850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027870: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ +00027880: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00027890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000278a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000278d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +000278e0: 4920 3d20 6d69 6e6f 7273 2832 2c6d 6174 I = minors(2,mat │ │ │ │ +000278f0: 7269 787b 7b78 2c79 2c7a 7d2c 7b79 2c7a rix{{x,y,z},{y,z │ │ │ │ +00027900: 2c77 7d7d 2920 2020 2020 207c 0a7c 2020 ,w}}) |.| │ │ │ │ 00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027920: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00027930: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027940: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00027950: 2020 207c 0a7c 6f33 203d 2069 6465 616c |.|o3 = ideal │ │ │ │ -00027960: 2028 2d20 7920 202b 2078 2a7a 2c20 2d20 (- y + x*z, - │ │ │ │ -00027970: 792a 7a20 2b20 782a 772c 202d 207a 2020 y*z + x*w, - z │ │ │ │ -00027980: 2b20 792a 7729 7c0a 7c20 2020 2020 2020 + y*w)|.| │ │ │ │ -00027990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279b0: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -000279c0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ -000279d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00027a20: 0a7c 6934 203a 2043 6865 726e 2028 492c .|i4 : Chern (I, │ │ │ │ -00027a30: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265 CompMethod=>PnRe │ │ │ │ -00027a40: 7369 6475 616c 2920 2020 2020 2020 2020 sidual) │ │ │ │ -00027a50: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 2020 2033 |.| 3 │ │ │ │ -00027a90: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00027aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ab0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -00027ac0: 3248 2020 2b20 3348 2020 2020 2020 2020 2H + 3H │ │ │ │ -00027ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027930: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027940: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00027950: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00027960: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00027970: 207c 0a7c 6f33 203d 2069 6465 616c 2028 |.|o3 = ideal ( │ │ │ │ +00027980: 2d20 7920 202b 2078 2a7a 2c20 2d20 792a - y + x*z, - y* │ │ │ │ +00027990: 7a20 2b20 782a 772c 202d 207a 2020 2b20 z + x*w, - z + │ │ │ │ +000279a0: 792a 7729 7c0a 7c20 2020 2020 2020 2020 y*w)|.| │ │ │ │ +000279b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000279c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000279d0: 2020 2020 2020 207c 0a7c 6f33 203a 2049 |.|o3 : I │ │ │ │ +000279e0: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ +000279f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00027a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027a40: 6934 203a 2043 6865 726e 2028 492c 436f i4 : Chern (I,Co │ │ │ │ +00027a50: 6d70 4d65 7468 6f64 3d3e 506e 5265 7369 mpMethod=>PnResi │ │ │ │ +00027a60: 6475 616c 2920 2020 2020 2020 2020 2020 dual) │ │ │ │ +00027a70: 7c0a 7c20 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: 2020 207c 0a7c 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ +00027ab0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00027ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ad0: 2020 2020 2020 7c0a 7c6f 3420 3d20 3248 |.|o4 = 2H │ │ │ │ +00027ae0: 2020 2b20 3348 2020 2020 2020 2020 2020 + 3H │ │ │ │ 00027af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027b20: 7c20 2020 2020 5a5a 5b48 5d20 2020 2020 | ZZ[H] │ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b50: 207c 0a7c 6f34 203a 202d 2d2d 2d2d 2020 |.|o4 : ----- │ │ │ │ -00027b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b80: 2020 2020 7c0a 7c20 2020 2020 2020 2034 |.| 4 │ │ │ │ +00027b00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027b40: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ +00027b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027b70: 0a7c 6f34 203a 202d 2d2d 2d2d 2020 2020 .|o4 : ----- │ │ │ │ +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 2020 2020 │ │ │ │ -00027bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00027bc0: 2048 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ -00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027be0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00027bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00027c20: 6935 203a 2043 6865 726e 2028 492c 436f i5 : Chern (I,Co │ │ │ │ -00027c30: 6d70 4d65 7468 6f64 3d3e 506e 5265 7369 mpMethod=>PnResi │ │ │ │ -00027c40: 6475 616c 2920 2020 2020 2020 2020 2020 dual) │ │ │ │ -00027c50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00027c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c80: 2020 207c 0a7c 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ -00027c90: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027cb0: 2020 2020 2020 7c0a 7c6f 3520 3d20 3248 |.|o5 = 2H │ │ │ │ -00027cc0: 2020 2b20 3348 2020 2020 2020 2020 2020 + 3H │ │ │ │ -00027cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ce0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027ba0: 2020 7c0a 7c20 2020 2020 2020 2034 2020 |.| 4 │ │ │ │ +00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027bd0: 2020 2020 207c 0a7c 2020 2020 2020 2048 |.| H │ │ │ │ +00027be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027c00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 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 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +00027c40: 203a 2043 6865 726e 2028 492c 436f 6d70 : Chern (I,Comp │ │ │ │ +00027c50: 4d65 7468 6f64 3d3e 506e 5265 7369 6475 Method=>PnResidu │ │ │ │ +00027c60: 616c 2920 2020 2020 2020 2020 2020 7c0a al) |. │ │ │ │ +00027c70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00027c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ca0: 207c 0a7c 2020 2020 2020 2033 2020 2020 |.| 3 │ │ │ │ +00027cb0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00027cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027cd0: 2020 2020 7c0a 7c6f 3520 3d20 3248 2020 |.|o5 = 2H │ │ │ │ +00027ce0: 2b20 3348 2020 2020 2020 2020 2020 2020 + 3H │ │ │ │ 00027cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00027d20: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ -00027d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027d50: 0a7c 6f35 203a 202d 2d2d 2d2d 2020 2020 .|o5 : ----- │ │ │ │ -00027d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d80: 2020 7c0a 7c20 2020 2020 2020 2034 2020 |.| 4 │ │ │ │ +00027d00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00027d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027d40: 2020 5a5a 5b48 5d20 2020 2020 2020 2020 ZZ[H] │ │ │ │ +00027d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00027d70: 6f35 203a 202d 2d2d 2d2d 2020 2020 2020 o5 : ----- │ │ │ │ +00027d80: 2020 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 207c 0a7c 2020 2020 2020 2048 |.| H │ │ │ │ +00027da0: 7c0a 7c20 2020 2020 2020 2034 2020 2020 |.| 4 │ │ │ │ +00027db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027de0: 2020 2020 2020 2020 7c0a 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 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -00027e20: 203a 2043 6865 726e 2028 492c 436f 6d70 : Chern (I,Comp │ │ │ │ -00027e30: 4d65 7468 6f64 3d3e 506e 5265 7369 6475 Method=>PnResidu │ │ │ │ -00027e40: 616c 2920 2020 2020 2020 2020 2020 7c0a al) |. │ │ │ │ -00027e50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00027e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027dd0: 2020 207c 0a7c 2020 2020 2020 2048 2020 |.| H │ │ │ │ +00027de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027e00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027e30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +00027e40: 2043 6865 726e 2028 492c 436f 6d70 4d65 Chern (I,CompMe │ │ │ │ +00027e50: 7468 6f64 3d3e 506e 5265 7369 6475 616c thod=>PnResidual │ │ │ │ +00027e60: 2920 2020 2020 2020 2020 2020 7c0a 7c20 ) |.| │ │ │ │ 00027e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027e80: 207c 0a7c 2020 2020 2020 2033 2020 2020 |.| 3 │ │ │ │ -00027e90: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027eb0: 2020 2020 7c0a 7c6f 3620 3d20 3248 2020 |.|o6 = 2H │ │ │ │ -00027ec0: 2b20 3348 2020 2020 2020 2020 2020 2020 + 3H │ │ │ │ -00027ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ee0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00027e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027e90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027ea0: 0a7c 2020 2020 2020 2033 2020 2020 2032 .| 3 2 │ │ │ │ +00027eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ed0: 2020 7c0a 7c6f 3620 3d20 3248 2020 2b20 |.|o6 = 2H + │ │ │ │ +00027ee0: 3348 2020 2020 2020 2020 2020 2020 2020 3H │ │ │ │ 00027ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00027f20: 2020 5a5a 5b48 5d20 2020 2020 2020 2020 ZZ[H] │ │ │ │ -00027f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00027f50: 6f36 203a 202d 2d2d 2d2d 2020 2020 2020 o6 : ----- │ │ │ │ -00027f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f80: 7c0a 7c20 2020 2020 2020 2034 2020 2020 |.| 4 │ │ │ │ -00027f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fb0: 2020 207c 0a7c 2020 2020 2020 2048 2020 |.| H │ │ │ │ +00027f00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00027f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027f30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00027f40: 5a5a 5b48 5d20 2020 2020 2020 2020 2020 ZZ[H] │ │ │ │ +00027f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027f60: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +00027f70: 203a 202d 2d2d 2d2d 2020 2020 2020 2020 : ----- │ │ │ │ +00027f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027f90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027fa0: 7c20 2020 2020 2020 2034 2020 2020 2020 | 4 │ │ │ │ +00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fe0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00027ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028010: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ -00028020: 2043 6865 726e 2849 2c43 6f6d 704d 6574 Chern(I,CompMet │ │ │ │ -00028030: 686f 643d 3e50 726f 6a65 6374 6976 6544 hod=>ProjectiveD │ │ │ │ -00028040: 6567 7265 6529 2020 2020 2020 7c0a 7c20 egree) |.| │ │ │ │ -00028050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028080: 0a7c 2020 2020 2020 2033 2020 2020 2032 .| 3 2 │ │ │ │ -00028090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000280a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000280b0: 2020 7c0a 7c6f 3720 3d20 3268 2020 2b20 |.|o7 = 2h + │ │ │ │ -000280c0: 3368 2020 2020 2020 2020 2020 2020 2020 3h │ │ │ │ -000280d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000280e0: 2020 2020 207c 0a7c 2020 2020 2020 2031 |.| 1 │ │ │ │ -000280f0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00028100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028110: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00027fd0: 207c 0a7c 2020 2020 2020 2048 2020 2020 |.| H │ │ │ │ +00027fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028000: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028030: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2043 -------+.|i7 : C │ │ │ │ +00028040: 6865 726e 2849 2c43 6f6d 704d 6574 686f hern(I,CompMetho │ │ │ │ +00028050: 643d 3e50 726f 6a65 6374 6976 6544 6567 d=>ProjectiveDeg │ │ │ │ +00028060: 7265 6529 2020 2020 2020 7c0a 7c20 2020 ree) |.| │ │ │ │ +00028070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028090: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000280a0: 2020 2020 2020 2033 2020 2020 2032 2020 3 2 │ │ │ │ +000280b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000280c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000280d0: 7c0a 7c6f 3720 3d20 3268 2020 2b20 3368 |.|o7 = 2h + 3h │ │ │ │ +000280e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000280f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028100: 2020 207c 0a7c 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +00028110: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00028120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028140: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028150: 2020 205a 5a5b 6820 5d20 2020 2020 2020 ZZ[h ] │ │ │ │ -00028160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028180: 7c20 2020 2020 2020 2020 3120 2020 2020 | 1 │ │ │ │ -00028190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281b0: 207c 0a7c 6f37 203a 202d 2d2d 2d2d 2d20 |.|o7 : ------ │ │ │ │ -000281c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281e0: 2020 2020 7c0a 7c20 2020 2020 2020 2034 |.| 4 │ │ │ │ +00028130: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00028140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028160: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00028170: 205a 5a5b 6820 5d20 2020 2020 2020 2020 ZZ[h ] │ │ │ │ +00028180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028190: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000281a0: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +000281b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000281c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000281d0: 0a7c 6f37 203a 202d 2d2d 2d2d 2d20 2020 .|o7 : ------ │ │ │ │ +000281e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000281f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028220: 2068 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ -00028230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028240: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00028250: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00028260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028270: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00028280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028200: 2020 7c0a 7c20 2020 2020 2020 2034 2020 |.| 4 │ │ │ │ +00028210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028230: 2020 2020 207c 0a7c 2020 2020 2020 2068 |.| h │ │ │ │ +00028240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028260: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028270: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00028280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028290: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000282a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000282b0: 2b0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +.-------------- │ │ │ │ -000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000282d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028300: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00028310: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00028320: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00028330: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00028340: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ -00028350: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00028360: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ -00028370: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -00028380: 6d32 3a32 3334 313a 302e 0a1f 0a46 696c m2:2341:0....Fil │ │ │ │ -00028390: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ -000283a0: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ -000283b0: 6f64 653a 2053 6567 7265 2c20 4e65 7874 ode: Segre, Next │ │ │ │ -000283c0: 3a20 546f 7269 6343 686f 7752 696e 672c : ToricChowRing, │ │ │ │ -000283d0: 2050 7265 763a 2070 726f 6261 6269 6c69 Prev: probabili │ │ │ │ -000283e0: 7374 6963 2061 6c67 6f72 6974 686d 2c20 stic algorithm, │ │ │ │ -000283f0: 5570 3a20 546f 700a 0a53 6567 7265 202d Up: Top..Segre - │ │ │ │ -00028400: 2d20 5468 6520 5365 6772 6520 636c 6173 - The Segre clas │ │ │ │ -00028410: 7320 6f66 2061 2073 7562 7363 6865 6d65 s of a subscheme │ │ │ │ -00028420: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00028430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00028440: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -00028450: 6167 653a 200a 2020 2020 2020 2020 5365 age: . Se │ │ │ │ -00028460: 6772 6520 490a 2020 2020 2020 2020 5365 gre I. Se │ │ │ │ -00028470: 6772 6528 412c 4929 0a20 2020 2020 2020 gre(A,I). │ │ │ │ -00028480: 2053 6567 7265 2858 2c4a 290a 2020 2020 Segre(X,J). │ │ │ │ -00028490: 2020 2020 5365 6772 6528 4368 2c58 2c4a Segre(Ch,X,J │ │ │ │ -000284a0: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ -000284b0: 2020 2020 2a20 492c 2061 6e20 2a6e 6f74 * I, an *not │ │ │ │ -000284c0: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ -000284d0: 6179 3244 6f63 2949 6465 616c 2c2c 2061 ay2Doc)Ideal,, a │ │ │ │ -000284e0: 206d 756c 7469 2d68 6f6d 6f67 656e 656f multi-homogeneo │ │ │ │ -000284f0: 7573 2069 6465 616c 2069 6e20 610a 2020 us ideal in a. │ │ │ │ -00028500: 2020 2020 2020 6772 6164 6564 2070 6f6c graded pol │ │ │ │ -00028510: 796e 6f6d 6961 6c20 7269 6e67 206f 7665 ynomial ring ove │ │ │ │ -00028520: 7220 6120 6669 656c 6420 6465 6669 6e69 r a field defini │ │ │ │ -00028530: 6e67 2061 2063 6c6f 7365 6420 7375 6273 ng a closed subs │ │ │ │ -00028540: 6368 656d 6520 5620 6f66 0a20 2020 2020 cheme V of. │ │ │ │ -00028550: 2020 205c 5050 5e7b 6e5f 317d 782e 2e2e \PP^{n_1}x... │ │ │ │ -00028560: 785c 5050 5e7b 6e5f 6d7d 0a20 2020 2020 x\PP^{n_m}. │ │ │ │ -00028570: 202a 2041 2c20 6120 2a6e 6f74 6520 7175 * A, a *note qu │ │ │ │ -00028580: 6f74 6965 6e74 2072 696e 673a 2028 4d61 otient ring: (Ma │ │ │ │ -00028590: 6361 756c 6179 3244 6f63 2951 756f 7469 caulay2Doc)Quoti │ │ │ │ -000285a0: 656e 7452 696e 672c 2c0a 2020 2020 2020 entRing,,. │ │ │ │ -000285b0: 2020 413d 5c5a 5a5b 685f 312c 2e2e 2e2c A=\ZZ[h_1,..., │ │ │ │ -000285c0: 685f 6d5d 2f28 685f 315e 7b6e 5f31 2b31 h_m]/(h_1^{n_1+1 │ │ │ │ -000285d0: 7d2c 2e2e 2e2c 685f 6d5e 7b6e 5f6d 2b31 },...,h_m^{n_m+1 │ │ │ │ -000285e0: 7d29 2071 756f 7469 656e 7420 7269 6e67 }) quotient ring │ │ │ │ -000285f0: 0a20 2020 2020 2020 2072 6570 7265 7365 . represe │ │ │ │ -00028600: 6e74 696e 6720 7468 6520 4368 6f77 2072 nting the Chow r │ │ │ │ -00028610: 696e 6720 6f66 205c 5050 5e7b 6e5f 317d ing of \PP^{n_1} │ │ │ │ -00028620: 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 2c20 x...x\PP^{n_m}, │ │ │ │ -00028630: 7468 6973 2072 696e 6720 7368 6f75 6c64 this ring should │ │ │ │ -00028640: 0a20 2020 2020 2020 2062 6520 6275 696c . be buil │ │ │ │ -00028650: 7420 7573 696e 6720 7468 6520 2a6e 6f74 t using the *not │ │ │ │ -00028660: 6520 4368 6f77 5269 6e67 3a20 4368 6f77 e ChowRing: Chow │ │ │ │ -00028670: 5269 6e67 2c20 636f 6d6d 616e 640a 2020 Ring, command. │ │ │ │ -00028680: 2020 2020 2a20 4a2c 2061 6e20 2a6e 6f74 * J, an *not │ │ │ │ -00028690: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ -000286a0: 6179 3244 6f63 2949 6465 616c 2c2c 2069 ay2Doc)Ideal,, i │ │ │ │ -000286b0: 6e20 7468 6520 6772 6164 6564 2070 6f6c n the graded pol │ │ │ │ -000286c0: 796e 6f6d 6961 6c20 7269 6e67 0a20 2020 ynomial ring. │ │ │ │ -000286d0: 2020 2020 2077 6869 6368 2069 7320 636f which is co │ │ │ │ -000286e0: 6f72 6469 6e61 7465 2072 696e 6720 6f66 ordinate ring of │ │ │ │ -000286f0: 2074 6865 204e 6f72 6d61 6c20 546f 7269 the Normal Tori │ │ │ │ -00028700: 6320 5661 7269 6574 7920 580a 2020 2020 c Variety X. │ │ │ │ -00028710: 2020 2a20 582c 2061 202a 6e6f 7465 206e * X, a *note n │ │ │ │ -00028720: 6f72 6d61 6c20 746f 7269 6320 7661 7269 ormal toric vari │ │ │ │ -00028730: 6574 793a 0a20 2020 2020 2020 2028 4e6f ety:. (No │ │ │ │ -00028740: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ -00028750: 6573 294e 6f72 6d61 6c54 6f72 6963 5661 es)NormalToricVa │ │ │ │ -00028760: 7269 6574 792c 2c20 7768 6963 6820 6973 riety,, which is │ │ │ │ -00028770: 2074 6865 2061 6d62 6965 6e74 2073 7061 the ambient spa │ │ │ │ -00028780: 6365 0a20 2020 2020 2020 2077 6869 6368 ce. which │ │ │ │ -00028790: 2063 6f6e 7461 696e 7320 5628 4a29 0a20 contains V(J). │ │ │ │ -000287a0: 2020 2020 202a 2043 682c 2061 202a 6e6f * Ch, a *no │ │ │ │ -000287b0: 7465 2071 756f 7469 656e 7420 7269 6e67 te quotient ring │ │ │ │ -000287c0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000287d0: 5175 6f74 6965 6e74 5269 6e67 2c2c 2074 QuotientRing,, t │ │ │ │ -000287e0: 6865 2043 686f 7720 7269 6e67 0a20 2020 he Chow ring. │ │ │ │ -000287f0: 2020 2020 206f 6620 7468 6520 746f 7269 of the tori │ │ │ │ -00028800: 6320 7661 7269 6574 7920 582c 2043 683d c variety X, Ch= │ │ │ │ -00028810: 2872 696e 6720 4a29 2f28 5352 2b4c 5229 (ring J)/(SR+LR) │ │ │ │ -00028820: 2077 6865 7265 2053 5220 6973 2074 6865 where SR is the │ │ │ │ -00028830: 0a20 2020 2020 2020 2053 7461 6e6c 6579 . Stanley │ │ │ │ -00028840: 2d52 6569 736e 6572 2069 6465 616c 206f -Reisner ideal o │ │ │ │ -00028850: 6620 7468 6520 6661 6e20 6465 6669 6e69 f the fan defini │ │ │ │ -00028860: 6e67 2058 2061 6e64 204c 5220 6973 2074 ng X and LR is t │ │ │ │ -00028870: 6865 206c 696e 6561 720a 2020 2020 2020 he linear. │ │ │ │ -00028880: 2020 7265 6c61 7469 6f6e 7320 6964 6561 relations idea │ │ │ │ -00028890: 6c2c 2074 6869 7320 7269 6e67 2073 686f l, this ring sho │ │ │ │ -000288a0: 756c 6420 6265 2062 7569 6c74 2075 7369 uld be built usi │ │ │ │ -000288b0: 6e67 2074 6865 202a 6e6f 7465 0a20 2020 ng the *note. │ │ │ │ -000288c0: 2020 2020 2054 6f72 6963 4368 6f77 5269 ToricChowRi │ │ │ │ -000288d0: 6e67 3a20 546f 7269 6343 686f 7752 696e ng: ToricChowRin │ │ │ │ -000288e0: 672c 2063 6f6d 6d61 6e64 0a20 202a 202a g, command. * * │ │ │ │ -000288f0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -00028900: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00028910: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00028920: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00028930: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00028940: 202a 2043 6f6d 704d 6574 686f 6420 286d * CompMethod (m │ │ │ │ -00028950: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ -00028960: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ -00028970: 6661 756c 7420 7661 6c75 650a 2020 2020 fault value. │ │ │ │ -00028980: 2020 2020 5072 6f6a 6563 7469 7665 4465 ProjectiveDe │ │ │ │ -00028990: 6772 6565 2c20 5072 6f6a 6563 7469 7665 gree, Projective │ │ │ │ -000289a0: 4465 6772 6565 2c20 7468 6973 2061 6c67 Degree, this alg │ │ │ │ -000289b0: 6f72 6974 686d 206d 6179 2062 6520 7573 orithm may be us │ │ │ │ -000289c0: 6564 2066 6f72 0a20 2020 2020 2020 2073 ed for. s │ │ │ │ -000289d0: 7562 7363 6865 6d65 7320 6f66 2061 6e79 ubschemes of any │ │ │ │ -000289e0: 2061 7070 6c69 6361 626c 6520 746f 7269 applicable tori │ │ │ │ -000289f0: 6320 7661 7269 6574 7920 2874 6869 7320 c variety (this │ │ │ │ -00028a00: 6d61 7920 6265 2063 6865 636b 6564 2075 may be checked u │ │ │ │ -00028a10: 7369 6e67 0a20 2020 2020 2020 2074 6865 sing. the │ │ │ │ -00028a20: 202a 6e6f 7465 2043 6865 636b 546f 7269 *note CheckTori │ │ │ │ -00028a30: 6356 6172 6965 7479 5661 6c69 643a 2043 cVarietyValid: C │ │ │ │ -00028a40: 6865 636b 546f 7269 6356 6172 6965 7479 heckToricVariety │ │ │ │ -00028a50: 5661 6c69 642c 2063 6f6d 6d61 6e64 290a Valid, command). │ │ │ │ -00028a60: 2020 2020 2020 2a20 436f 6d70 4d65 7468 * CompMeth │ │ │ │ -00028a70: 6f64 2028 6d69 7373 696e 6720 646f 6375 od (missing docu │ │ │ │ -00028a80: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -00028a90: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00028aa0: 0a20 2020 2020 2020 2050 726f 6a65 6374 . Project │ │ │ │ -00028ab0: 6976 6544 6567 7265 652c 2050 6e52 6573 iveDegree, PnRes │ │ │ │ -00028ac0: 6964 7561 6c2c 2074 6869 7320 616c 676f idual, this algo │ │ │ │ -00028ad0: 7269 7468 6d20 6d61 7920 6265 2075 7365 rithm may be use │ │ │ │ -00028ae0: 6420 666f 7220 7375 6273 6368 656d 6573 d for subschemes │ │ │ │ -00028af0: 0a20 2020 2020 2020 206f 6620 5c50 505e . of \PP^ │ │ │ │ -00028b00: 6e20 6f6e 6c79 0a20 2020 2020 202a 204f n only. * O │ │ │ │ -00028b10: 7574 7075 7420 3d3e 202e 2e2e 2c20 6465 utput => ..., de │ │ │ │ -00028b20: 6661 756c 7420 7661 6c75 6520 4368 6f77 fault value Chow │ │ │ │ -00028b30: 5269 6e67 456c 656d 656e 742c 2043 686f RingElement, Cho │ │ │ │ -00028b40: 7752 696e 6745 6c65 6d65 6e74 2c20 7265 wRingElement, re │ │ │ │ -00028b50: 7475 726e 730a 2020 2020 2020 2020 6120 turns. a │ │ │ │ -00028b60: 5269 6e67 456c 656d 656e 7420 696e 2074 RingElement in t │ │ │ │ -00028b70: 6865 2043 686f 7720 7269 6e67 206f 6620 he Chow ring of │ │ │ │ -00028b80: 7468 6520 6170 7072 6f70 7269 6174 6520 the appropriate │ │ │ │ -00028b90: 616d 6269 656e 7420 7370 6163 650a 2020 ambient space. │ │ │ │ -00028ba0: 2020 2020 2a20 4f75 7470 7574 203d 3e20 * Output => │ │ │ │ -00028bb0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00028bc0: 7565 2043 686f 7752 696e 6745 6c65 6d65 ue ChowRingEleme │ │ │ │ -00028bd0: 6e74 2c20 4861 7368 466f 726d 2c20 4861 nt, HashForm, Ha │ │ │ │ -00028be0: 7368 466f 726d 0a20 2020 2020 2020 2072 shForm. r │ │ │ │ -00028bf0: 6574 7572 6e73 2061 204d 7574 6162 6c65 eturns a Mutable │ │ │ │ -00028c00: 4861 7368 5461 626c 6520 636f 6e74 6169 HashTable contai │ │ │ │ -00028c10: 6e69 6e67 2074 6865 2066 6f6c 6c6f 7769 ning the followi │ │ │ │ -00028c20: 6e67 206b 6579 733a 2022 4722 2028 7468 ng keys: "G" (th │ │ │ │ -00028c30: 650a 2020 2020 2020 2020 706f 6c79 6e6f e. polyno │ │ │ │ -00028c40: 6d69 616c 2077 6974 6820 636f 6566 6669 mial with coeffi │ │ │ │ -00028c50: 6369 656e 7473 206f 6620 7468 6520 6879 cients of the hy │ │ │ │ -00028c60: 7065 7270 6c61 6e65 2063 6c61 7373 6573 perplane classes │ │ │ │ -00028c70: 2072 6570 7265 7365 6e74 696e 6720 7468 representing th │ │ │ │ -00028c80: 650a 2020 2020 2020 2020 7072 6f6a 6563 e. projec │ │ │ │ -00028c90: 7469 7665 2064 6567 7265 6573 292c 2022 tive degrees), " │ │ │ │ -00028ca0: 476c 6973 7422 2028 7468 6520 6c69 7374 Glist" (the list │ │ │ │ -00028cb0: 2066 6f72 6d20 6f66 2022 4722 2920 2c20 form of "G") , │ │ │ │ -00028cc0: 2253 6567 7265 2220 2874 6865 0a20 2020 "Segre" (the. │ │ │ │ -00028cd0: 2020 2020 2074 6f74 616c 2053 6567 7265 total Segre │ │ │ │ -00028ce0: 2063 6c61 7373 206f 6620 7468 6520 696e class of the in │ │ │ │ -00028cf0: 7075 7429 2c22 5365 6772 654c 6973 7422 put),"SegreList" │ │ │ │ -00028d00: 2028 7468 6520 6c69 7374 2066 6f72 6d20 (the list form │ │ │ │ -00028d10: 6f66 2022 5365 6772 6522 290a 2020 2a20 of "Segre"). * │ │ │ │ -00028d20: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00028d30: 2061 202a 6e6f 7465 2072 696e 6720 656c a *note ring el │ │ │ │ -00028d40: 656d 656e 743a 2028 4d61 6361 756c 6179 ement: (Macaulay │ │ │ │ -00028d50: 3244 6f63 2952 696e 6745 6c65 6d65 6e74 2Doc)RingElement │ │ │ │ -00028d60: 2c2c 2074 6865 2070 7573 6866 6f72 7761 ,, the pushforwa │ │ │ │ -00028d70: 7264 206f 660a 2020 2020 2020 2020 7468 rd of. th │ │ │ │ -00028d80: 6520 746f 7461 6c20 5365 6772 6520 636c e total Segre cl │ │ │ │ -00028d90: 6173 7320 6f66 2074 6865 2073 6368 656d ass of the schem │ │ │ │ -00028da0: 6520 5620 6465 6669 6e65 6420 6279 2074 e V defined by t │ │ │ │ -00028db0: 6865 2069 6e70 7574 2069 6465 616c 2074 he input ideal t │ │ │ │ -00028dc0: 6f20 7468 650a 2020 2020 2020 2020 6170 o the. ap │ │ │ │ -00028dd0: 7072 6f70 7269 6174 6520 4368 6f77 2072 propriate Chow r │ │ │ │ -00028de0: 696e 670a 0a44 6573 6372 6970 7469 6f6e ing..Description │ │ │ │ -00028df0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f .===========..Fo │ │ │ │ -00028e00: 7220 6120 7375 6273 6368 656d 6520 5620 r a subscheme V │ │ │ │ -00028e10: 6f66 2061 6e20 6170 706c 6963 6162 6c65 of an applicable │ │ │ │ -00028e20: 2074 6f72 6963 2076 6172 6965 7479 2058 toric variety X │ │ │ │ -00028e30: 2074 6869 7320 636f 6d6d 616e 6420 636f this command co │ │ │ │ -00028e40: 6d70 7574 6573 2074 6865 0a70 7573 682d mputes the.push- │ │ │ │ -00028e50: 666f 7277 6172 6420 6f66 2074 6865 2074 forward of the t │ │ │ │ -00028e60: 6f74 616c 2053 6567 7265 2063 6c61 7373 otal Segre class │ │ │ │ -00028e70: 2073 2856 2c58 2920 6f66 2056 2069 6e20 s(V,X) of V in │ │ │ │ -00028e80: 5820 746f 2074 6865 2043 686f 7720 7269 X to the Chow ri │ │ │ │ -00028e90: 6e67 206f 6620 582e 0a0a 2b2d 2d2d 2d2d ng of X...+----- │ │ │ │ -00028ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ec0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00028ed0: 7365 7452 616e 646f 6d53 6565 6420 3732 setRandomSeed 72 │ │ │ │ -00028ee0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -00028ef0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073 |.| -- s │ │ │ │ -00028f00: 6574 7469 6e67 2072 616e 646f 6d20 7365 etting random se │ │ │ │ -00028f10: 6564 2074 6f20 3732 2020 2020 2020 2020 ed to 72 │ │ │ │ -00028f20: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00028f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00028f60: 5220 3d20 5a5a 2f33 3237 3439 5b77 2c79 R = ZZ/32749[w,y │ │ │ │ -00028f70: 2c7a 5d20 2020 2020 2020 2020 2020 2020 ,z] │ │ │ │ -00028f80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fb0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -00028fc0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00028fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00028320: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00028330: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00028340: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00028350: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00028360: 756c 6179 322d 312e 3235 2e30 362b 6473 ulay2-1.25.06+ds │ │ │ │ +00028370: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00028380: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ +00028390: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ +000283a0: 3a32 3334 313a 302e 0a1f 0a46 696c 653a :2341:0....File: │ │ │ │ +000283b0: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ +000283c0: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ +000283d0: 653a 2053 6567 7265 2c20 4e65 7874 3a20 e: Segre, Next: │ │ │ │ +000283e0: 546f 7269 6343 686f 7752 696e 672c 2050 ToricChowRing, P │ │ │ │ +000283f0: 7265 763a 2070 726f 6261 6269 6c69 7374 rev: probabilist │ │ │ │ +00028400: 6963 2061 6c67 6f72 6974 686d 2c20 5570 ic algorithm, Up │ │ │ │ +00028410: 3a20 546f 700a 0a53 6567 7265 202d 2d20 : Top..Segre -- │ │ │ │ +00028420: 5468 6520 5365 6772 6520 636c 6173 7320 The Segre class │ │ │ │ +00028430: 6f66 2061 2073 7562 7363 6865 6d65 0a2a of a subscheme.* │ │ │ │ +00028440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00028450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00028460: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ +00028470: 653a 200a 2020 2020 2020 2020 5365 6772 e: . Segr │ │ │ │ +00028480: 6520 490a 2020 2020 2020 2020 5365 6772 e I. Segr │ │ │ │ +00028490: 6528 412c 4929 0a20 2020 2020 2020 2053 e(A,I). S │ │ │ │ +000284a0: 6567 7265 2858 2c4a 290a 2020 2020 2020 egre(X,J). │ │ │ │ +000284b0: 2020 5365 6772 6528 4368 2c58 2c4a 290a Segre(Ch,X,J). │ │ │ │ +000284c0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +000284d0: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ +000284e0: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ +000284f0: 3244 6f63 2949 6465 616c 2c2c 2061 206d 2Doc)Ideal,, a m │ │ │ │ +00028500: 756c 7469 2d68 6f6d 6f67 656e 656f 7573 ulti-homogeneous │ │ │ │ +00028510: 2069 6465 616c 2069 6e20 610a 2020 2020 ideal in a. │ │ │ │ +00028520: 2020 2020 6772 6164 6564 2070 6f6c 796e graded polyn │ │ │ │ +00028530: 6f6d 6961 6c20 7269 6e67 206f 7665 7220 omial ring over │ │ │ │ +00028540: 6120 6669 656c 6420 6465 6669 6e69 6e67 a field defining │ │ │ │ +00028550: 2061 2063 6c6f 7365 6420 7375 6273 6368 a closed subsch │ │ │ │ +00028560: 656d 6520 5620 6f66 0a20 2020 2020 2020 eme V of. │ │ │ │ +00028570: 205c 5050 5e7b 6e5f 317d 782e 2e2e 785c \PP^{n_1}x...x\ │ │ │ │ +00028580: 5050 5e7b 6e5f 6d7d 0a20 2020 2020 202a PP^{n_m}. * │ │ │ │ +00028590: 2041 2c20 6120 2a6e 6f74 6520 7175 6f74 A, a *note quot │ │ │ │ +000285a0: 6965 6e74 2072 696e 673a 2028 4d61 6361 ient ring: (Maca │ │ │ │ +000285b0: 756c 6179 3244 6f63 2951 756f 7469 656e ulay2Doc)Quotien │ │ │ │ +000285c0: 7452 696e 672c 2c0a 2020 2020 2020 2020 tRing,,. │ │ │ │ +000285d0: 413d 5c5a 5a5b 685f 312c 2e2e 2e2c 685f A=\ZZ[h_1,...,h_ │ │ │ │ +000285e0: 6d5d 2f28 685f 315e 7b6e 5f31 2b31 7d2c m]/(h_1^{n_1+1}, │ │ │ │ +000285f0: 2e2e 2e2c 685f 6d5e 7b6e 5f6d 2b31 7d29 ...,h_m^{n_m+1}) │ │ │ │ +00028600: 2071 756f 7469 656e 7420 7269 6e67 0a20 quotient ring. │ │ │ │ +00028610: 2020 2020 2020 2072 6570 7265 7365 6e74 represent │ │ │ │ +00028620: 696e 6720 7468 6520 4368 6f77 2072 696e ing the Chow rin │ │ │ │ +00028630: 6720 6f66 205c 5050 5e7b 6e5f 317d 782e g of \PP^{n_1}x. │ │ │ │ +00028640: 2e2e 785c 5050 5e7b 6e5f 6d7d 2c20 7468 ..x\PP^{n_m}, th │ │ │ │ +00028650: 6973 2072 696e 6720 7368 6f75 6c64 0a20 is ring should. │ │ │ │ +00028660: 2020 2020 2020 2062 6520 6275 696c 7420 be built │ │ │ │ +00028670: 7573 696e 6720 7468 6520 2a6e 6f74 6520 using the *note │ │ │ │ +00028680: 4368 6f77 5269 6e67 3a20 4368 6f77 5269 ChowRing: ChowRi │ │ │ │ +00028690: 6e67 2c20 636f 6d6d 616e 640a 2020 2020 ng, command. │ │ │ │ +000286a0: 2020 2a20 4a2c 2061 6e20 2a6e 6f74 6520 * J, an *note │ │ │ │ +000286b0: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ +000286c0: 3244 6f63 2949 6465 616c 2c2c 2069 6e20 2Doc)Ideal,, in │ │ │ │ +000286d0: 7468 6520 6772 6164 6564 2070 6f6c 796e the graded polyn │ │ │ │ +000286e0: 6f6d 6961 6c20 7269 6e67 0a20 2020 2020 omial ring. │ │ │ │ +000286f0: 2020 2077 6869 6368 2069 7320 636f 6f72 which is coor │ │ │ │ +00028700: 6469 6e61 7465 2072 696e 6720 6f66 2074 dinate ring of t │ │ │ │ +00028710: 6865 204e 6f72 6d61 6c20 546f 7269 6320 he Normal Toric │ │ │ │ +00028720: 5661 7269 6574 7920 580a 2020 2020 2020 Variety X. │ │ │ │ +00028730: 2a20 582c 2061 202a 6e6f 7465 206e 6f72 * X, a *note nor │ │ │ │ +00028740: 6d61 6c20 746f 7269 6320 7661 7269 6574 mal toric variet │ │ │ │ +00028750: 793a 0a20 2020 2020 2020 2028 4e6f 726d y:. (Norm │ │ │ │ +00028760: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ +00028770: 294e 6f72 6d61 6c54 6f72 6963 5661 7269 )NormalToricVari │ │ │ │ +00028780: 6574 792c 2c20 7768 6963 6820 6973 2074 ety,, which is t │ │ │ │ +00028790: 6865 2061 6d62 6965 6e74 2073 7061 6365 he ambient space │ │ │ │ +000287a0: 0a20 2020 2020 2020 2077 6869 6368 2063 . which c │ │ │ │ +000287b0: 6f6e 7461 696e 7320 5628 4a29 0a20 2020 ontains V(J). │ │ │ │ +000287c0: 2020 202a 2043 682c 2061 202a 6e6f 7465 * Ch, a *note │ │ │ │ +000287d0: 2071 756f 7469 656e 7420 7269 6e67 3a20 quotient ring: │ │ │ │ +000287e0: 284d 6163 6175 6c61 7932 446f 6329 5175 (Macaulay2Doc)Qu │ │ │ │ +000287f0: 6f74 6965 6e74 5269 6e67 2c2c 2074 6865 otientRing,, the │ │ │ │ +00028800: 2043 686f 7720 7269 6e67 0a20 2020 2020 Chow ring. │ │ │ │ +00028810: 2020 206f 6620 7468 6520 746f 7269 6320 of the toric │ │ │ │ +00028820: 7661 7269 6574 7920 582c 2043 683d 2872 variety X, Ch=(r │ │ │ │ +00028830: 696e 6720 4a29 2f28 5352 2b4c 5229 2077 ing J)/(SR+LR) w │ │ │ │ +00028840: 6865 7265 2053 5220 6973 2074 6865 0a20 here SR is the. │ │ │ │ +00028850: 2020 2020 2020 2053 7461 6e6c 6579 2d52 Stanley-R │ │ │ │ +00028860: 6569 736e 6572 2069 6465 616c 206f 6620 eisner ideal of │ │ │ │ +00028870: 7468 6520 6661 6e20 6465 6669 6e69 6e67 the fan defining │ │ │ │ +00028880: 2058 2061 6e64 204c 5220 6973 2074 6865 X and LR is the │ │ │ │ +00028890: 206c 696e 6561 720a 2020 2020 2020 2020 linear. │ │ │ │ +000288a0: 7265 6c61 7469 6f6e 7320 6964 6561 6c2c relations ideal, │ │ │ │ +000288b0: 2074 6869 7320 7269 6e67 2073 686f 756c this ring shoul │ │ │ │ +000288c0: 6420 6265 2062 7569 6c74 2075 7369 6e67 d be built using │ │ │ │ +000288d0: 2074 6865 202a 6e6f 7465 0a20 2020 2020 the *note. │ │ │ │ +000288e0: 2020 2054 6f72 6963 4368 6f77 5269 6e67 ToricChowRing │ │ │ │ +000288f0: 3a20 546f 7269 6343 686f 7752 696e 672c : ToricChowRing, │ │ │ │ +00028900: 2063 6f6d 6d61 6e64 0a20 202a 202a 6e6f command. * *no │ │ │ │ +00028910: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +00028920: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +00028930: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +00028940: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +00028950: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +00028960: 2043 6f6d 704d 6574 686f 6420 286d 6973 CompMethod (mis │ │ │ │ +00028970: 7369 6e67 2064 6f63 756d 656e 7461 7469 sing documentati │ │ │ │ +00028980: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661 on) => ..., defa │ │ │ │ +00028990: 756c 7420 7661 6c75 650a 2020 2020 2020 ult value. │ │ │ │ +000289a0: 2020 5072 6f6a 6563 7469 7665 4465 6772 ProjectiveDegr │ │ │ │ +000289b0: 6565 2c20 5072 6f6a 6563 7469 7665 4465 ee, ProjectiveDe │ │ │ │ +000289c0: 6772 6565 2c20 7468 6973 2061 6c67 6f72 gree, this algor │ │ │ │ +000289d0: 6974 686d 206d 6179 2062 6520 7573 6564 ithm may be used │ │ │ │ +000289e0: 2066 6f72 0a20 2020 2020 2020 2073 7562 for. sub │ │ │ │ +000289f0: 7363 6865 6d65 7320 6f66 2061 6e79 2061 schemes of any a │ │ │ │ +00028a00: 7070 6c69 6361 626c 6520 746f 7269 6320 pplicable toric │ │ │ │ +00028a10: 7661 7269 6574 7920 2874 6869 7320 6d61 variety (this ma │ │ │ │ +00028a20: 7920 6265 2063 6865 636b 6564 2075 7369 y be checked usi │ │ │ │ +00028a30: 6e67 0a20 2020 2020 2020 2074 6865 202a ng. the * │ │ │ │ +00028a40: 6e6f 7465 2043 6865 636b 546f 7269 6356 note CheckToricV │ │ │ │ +00028a50: 6172 6965 7479 5661 6c69 643a 2043 6865 arietyValid: Che │ │ │ │ +00028a60: 636b 546f 7269 6356 6172 6965 7479 5661 ckToricVarietyVa │ │ │ │ +00028a70: 6c69 642c 2063 6f6d 6d61 6e64 290a 2020 lid, command). │ │ │ │ +00028a80: 2020 2020 2a20 436f 6d70 4d65 7468 6f64 * CompMethod │ │ │ │ +00028a90: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +00028aa0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +00028ab0: 2064 6566 6175 6c74 2076 616c 7565 0a20 default value. │ │ │ │ +00028ac0: 2020 2020 2020 2050 726f 6a65 6374 6976 Projectiv │ │ │ │ +00028ad0: 6544 6567 7265 652c 2050 6e52 6573 6964 eDegree, PnResid │ │ │ │ +00028ae0: 7561 6c2c 2074 6869 7320 616c 676f 7269 ual, this algori │ │ │ │ +00028af0: 7468 6d20 6d61 7920 6265 2075 7365 6420 thm may be used │ │ │ │ +00028b00: 666f 7220 7375 6273 6368 656d 6573 0a20 for subschemes. │ │ │ │ +00028b10: 2020 2020 2020 206f 6620 5c50 505e 6e20 of \PP^n │ │ │ │ +00028b20: 6f6e 6c79 0a20 2020 2020 202a 204f 7574 only. * Out │ │ │ │ +00028b30: 7075 7420 3d3e 202e 2e2e 2c20 6465 6661 put => ..., defa │ │ │ │ +00028b40: 756c 7420 7661 6c75 6520 4368 6f77 5269 ult value ChowRi │ │ │ │ +00028b50: 6e67 456c 656d 656e 742c 2043 686f 7752 ngElement, ChowR │ │ │ │ +00028b60: 696e 6745 6c65 6d65 6e74 2c20 7265 7475 ingElement, retu │ │ │ │ +00028b70: 726e 730a 2020 2020 2020 2020 6120 5269 rns. a Ri │ │ │ │ +00028b80: 6e67 456c 656d 656e 7420 696e 2074 6865 ngElement in the │ │ │ │ +00028b90: 2043 686f 7720 7269 6e67 206f 6620 7468 Chow ring of th │ │ │ │ +00028ba0: 6520 6170 7072 6f70 7269 6174 6520 616d e appropriate am │ │ │ │ +00028bb0: 6269 656e 7420 7370 6163 650a 2020 2020 bient space. │ │ │ │ +00028bc0: 2020 2a20 4f75 7470 7574 203d 3e20 2e2e * Output => .. │ │ │ │ +00028bd0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +00028be0: 2043 686f 7752 696e 6745 6c65 6d65 6e74 ChowRingElement │ │ │ │ +00028bf0: 2c20 4861 7368 466f 726d 2c20 4861 7368 , HashForm, Hash │ │ │ │ +00028c00: 466f 726d 0a20 2020 2020 2020 2072 6574 Form. ret │ │ │ │ +00028c10: 7572 6e73 2061 204d 7574 6162 6c65 4861 urns a MutableHa │ │ │ │ +00028c20: 7368 5461 626c 6520 636f 6e74 6169 6e69 shTable containi │ │ │ │ +00028c30: 6e67 2074 6865 2066 6f6c 6c6f 7769 6e67 ng the following │ │ │ │ +00028c40: 206b 6579 733a 2022 4722 2028 7468 650a keys: "G" (the. │ │ │ │ +00028c50: 2020 2020 2020 2020 706f 6c79 6e6f 6d69 polynomi │ │ │ │ +00028c60: 616c 2077 6974 6820 636f 6566 6669 6369 al with coeffici │ │ │ │ +00028c70: 656e 7473 206f 6620 7468 6520 6879 7065 ents of the hype │ │ │ │ +00028c80: 7270 6c61 6e65 2063 6c61 7373 6573 2072 rplane classes r │ │ │ │ +00028c90: 6570 7265 7365 6e74 696e 6720 7468 650a epresenting the. │ │ │ │ +00028ca0: 2020 2020 2020 2020 7072 6f6a 6563 7469 projecti │ │ │ │ +00028cb0: 7665 2064 6567 7265 6573 292c 2022 476c ve degrees), "Gl │ │ │ │ +00028cc0: 6973 7422 2028 7468 6520 6c69 7374 2066 ist" (the list f │ │ │ │ +00028cd0: 6f72 6d20 6f66 2022 4722 2920 2c20 2253 orm of "G") , "S │ │ │ │ +00028ce0: 6567 7265 2220 2874 6865 0a20 2020 2020 egre" (the. │ │ │ │ +00028cf0: 2020 2074 6f74 616c 2053 6567 7265 2063 total Segre c │ │ │ │ +00028d00: 6c61 7373 206f 6620 7468 6520 696e 7075 lass of the inpu │ │ │ │ +00028d10: 7429 2c22 5365 6772 654c 6973 7422 2028 t),"SegreList" ( │ │ │ │ +00028d20: 7468 6520 6c69 7374 2066 6f72 6d20 6f66 the list form of │ │ │ │ +00028d30: 2022 5365 6772 6522 290a 2020 2a20 4f75 "Segre"). * Ou │ │ │ │ +00028d40: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00028d50: 202a 6e6f 7465 2072 696e 6720 656c 656d *note ring elem │ │ │ │ +00028d60: 656e 743a 2028 4d61 6361 756c 6179 3244 ent: (Macaulay2D │ │ │ │ +00028d70: 6f63 2952 696e 6745 6c65 6d65 6e74 2c2c oc)RingElement,, │ │ │ │ +00028d80: 2074 6865 2070 7573 6866 6f72 7761 7264 the pushforward │ │ │ │ +00028d90: 206f 660a 2020 2020 2020 2020 7468 6520 of. the │ │ │ │ +00028da0: 746f 7461 6c20 5365 6772 6520 636c 6173 total Segre clas │ │ │ │ +00028db0: 7320 6f66 2074 6865 2073 6368 656d 6520 s of the scheme │ │ │ │ +00028dc0: 5620 6465 6669 6e65 6420 6279 2074 6865 V defined by the │ │ │ │ +00028dd0: 2069 6e70 7574 2069 6465 616c 2074 6f20 input ideal to │ │ │ │ +00028de0: 7468 650a 2020 2020 2020 2020 6170 7072 the. appr │ │ │ │ +00028df0: 6f70 7269 6174 6520 4368 6f77 2072 696e opriate Chow rin │ │ │ │ +00028e00: 670a 0a44 6573 6372 6970 7469 6f6e 0a3d g..Description.= │ │ │ │ +00028e10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f 7220 ==========..For │ │ │ │ +00028e20: 6120 7375 6273 6368 656d 6520 5620 6f66 a subscheme V of │ │ │ │ +00028e30: 2061 6e20 6170 706c 6963 6162 6c65 2074 an applicable t │ │ │ │ +00028e40: 6f72 6963 2076 6172 6965 7479 2058 2074 oric variety X t │ │ │ │ +00028e50: 6869 7320 636f 6d6d 616e 6420 636f 6d70 his command comp │ │ │ │ +00028e60: 7574 6573 2074 6865 0a70 7573 682d 666f utes the.push-fo │ │ │ │ +00028e70: 7277 6172 6420 6f66 2074 6865 2074 6f74 rward of the tot │ │ │ │ +00028e80: 616c 2053 6567 7265 2063 6c61 7373 2073 al Segre class s │ │ │ │ +00028e90: 2856 2c58 2920 6f66 2056 2069 6e20 5820 (V,X) of V in X │ │ │ │ +00028ea0: 746f 2074 6865 2043 686f 7720 7269 6e67 to the Chow ring │ │ │ │ +00028eb0: 206f 6620 582e 0a0a 2b2d 2d2d 2d2d 2d2d of X...+------- │ │ │ │ +00028ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ee0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 ------+.|i1 : se │ │ │ │ +00028ef0: 7452 616e 646f 6d53 6565 6420 3732 3b20 tRandomSeed 72; │ │ │ │ +00028f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028f10: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574 |.| -- set │ │ │ │ +00028f20: 7469 6e67 2072 616e 646f 6d20 7365 6564 ting random seed │ │ │ │ +00028f30: 2074 6f20 3732 2020 2020 2020 2020 2020 to 72 │ │ │ │ +00028f40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00028f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f70: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5220 ------+.|i2 : R │ │ │ │ +00028f80: 3d20 5a5a 2f33 3237 3439 5b77 2c79 2c7a = ZZ/32749[w,y,z │ │ │ │ +00028f90: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00028fa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00028fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028fd0: 2020 2020 2020 7c0a 7c6f 3220 3d20 5220 |.|o2 = R │ │ │ │ +00028fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029010: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00029020: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -00029030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029040: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00029050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029070: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00029080: 5365 6772 6528 6964 6561 6c28 772a 7929 Segre(ideal(w*y) │ │ │ │ -00029090: 2c43 6f6d 704d 6574 686f 643d 3e50 6e52 ,CompMethod=>PnR │ │ │ │ -000290a0: 6573 6964 7561 6c29 7c0a 7c20 2020 2020 esidual)|.| │ │ │ │ -000290b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000290c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000290d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000290e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -000290f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029100: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00029110: 2d20 3448 2020 2b20 3248 2020 2020 2020 - 4H + 2H │ │ │ │ -00029120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029130: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00029000: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029030: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f |.|o2 : Po │ │ │ │ +00029040: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020 lynomialRing │ │ │ │ +00029050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029060: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00029070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029090: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5365 ------+.|i3 : Se │ │ │ │ +000290a0: 6772 6528 6964 6561 6c28 772a 7929 2c43 gre(ideal(w*y),C │ │ │ │ +000290b0: 6f6d 704d 6574 686f 643d 3e50 6e52 6573 ompMethod=>PnRes │ │ │ │ +000290c0: 6964 7561 6c29 7c0a 7c20 2020 2020 2020 idual)|.| │ │ │ │ +000290d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000290e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000290f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029100: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029120: 2020 2020 2020 7c0a 7c6f 3320 3d20 2d20 |.|o3 = - │ │ │ │ +00029130: 3448 2020 2b20 3248 2020 2020 2020 2020 4H + 2H │ │ │ │ 00029140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029160: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00029170: 5a5a 5b48 5d20 2020 2020 2020 2020 2020 ZZ[H] │ │ │ │ -00029180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029190: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -000291a0: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020 ----- │ │ │ │ -000291b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000291c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000291d0: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -000291e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000291f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00029200: 2020 4820 2020 2020 2020 2020 2020 2020 H │ │ │ │ -00029210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029220: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00029230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029250: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00029260: 413d 4368 6f77 5269 6e67 2852 2920 2020 A=ChowRing(R) │ │ │ │ -00029270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029280: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00029150: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029180: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ +00029190: 5b48 5d20 2020 2020 2020 2020 2020 2020 [H] │ │ │ │ +000291a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000291b0: 2020 2020 2020 7c0a 7c6f 3320 3a20 2d2d |.|o3 : -- │ │ │ │ +000291c0: 2d2d 2d20 2020 2020 2020 2020 2020 2020 --- │ │ │ │ +000291d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000291e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000291f0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00029200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029210: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029220: 4820 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ +00029230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029240: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00029250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029270: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 413d ------+.|i4 : A= │ │ │ │ +00029280: 4368 6f77 5269 6e67 2852 2920 2020 2020 ChowRing(R) │ │ │ │ 00029290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292b0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -000292c0: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ -000292d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000292a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000292b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000292c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000292d0: 2020 2020 2020 7c0a 7c6f 3420 3d20 4120 |.|o4 = A │ │ │ │ +000292e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000292f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029310: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00029320: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ -00029330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029340: 2020 2020 2020 2020 7c0a 2b2d 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 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -00029380: 5365 6772 6528 412c 6964 6561 6c28 775e Segre(A,ideal(w^ │ │ │ │ -00029390: 322a 792c 772a 795e 3229 2920 2020 2020 2*y,w*y^2)) │ │ │ │ -000293a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000293b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000293e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -000293f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029400: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -00029410: 2d20 3368 2020 2b20 3268 2020 2020 2020 - 3h + 2h │ │ │ │ -00029420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029430: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00029440: 2020 2020 3120 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00029450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00029300: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029330: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175 |.|o4 : Qu │ │ │ │ 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│ │ +00029430: 3368 2020 2b20 3268 2020 2020 2020 2020 3h + 2h │ │ │ │ +00029440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029450: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029460: 2020 3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ 00029470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029490: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ -000294a0: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ -000294b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000294c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -000294d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000294e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000294f0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e 6f77 2063 --------+..Now c │ │ │ │ -00029500: 6f6e 7369 6465 7220 616e 2065 7861 6d70 onsider an examp │ │ │ │ -00029510: 6c65 2069 6e20 5c50 505e 3220 5c74 696d le in \PP^2 \tim │ │ │ │ -00029520: 6573 205c 5050 5e32 2c20 6966 2077 6520 es \PP^2, if we │ │ │ │ -00029530: 696e 7075 7420 7468 6520 4368 6f77 2072 input the Chow r │ │ │ │ -00029540: 696e 6720 4120 7468 650a 6f75 7470 7574 ing A the.output │ │ │ │ -00029550: 2077 696c 6c20 6265 2072 6574 7572 6e65 will be returne │ │ │ │ -00029560: 6420 696e 2074 6865 2073 616d 6520 7269 d in the same ri │ │ │ │ -00029570: 6e67 2e20 546f 2065 6e73 7572 6520 7072 ng. To ensure pr │ │ │ │ -00029580: 6f70 6572 2066 756e 6374 696f 6e20 6f66 oper function of │ │ │ │ -00029590: 2074 6865 0a6d 6574 686f 6473 2077 6520 the.methods we │ │ │ │ -000295a0: 6275 696c 6420 7468 6520 4368 6f77 2072 build the Chow r │ │ │ │ -000295b0: 696e 6720 7573 696e 6720 7468 6520 2a6e ing using the *n │ │ │ │ -000295c0: 6f74 6520 4368 6f77 5269 6e67 3a20 4368 ote ChowRing: Ch │ │ │ │ -000295d0: 6f77 5269 6e67 2c20 636f 6d6d 616e 642e owRing, command. │ │ │ │ -000295e0: 2057 650a 6d61 7920 616c 736f 2072 6574 We.may also ret │ │ │ │ -000295f0: 7572 6e20 6120 4d75 7461 626c 6548 6173 urn a MutableHas │ │ │ │ -00029600: 6854 6162 6c65 2e0a 0a2b 2d2d 2d2d 2d2d hTable...+------ │ │ │ │ -00029610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029480: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00029490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000294a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000294b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4120 |.|o5 : A │ │ │ │ +000294c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000294d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000294e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000294f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029510: 2d2d 2d2d 2d2d 2b0a 0a4e 6f77 2063 6f6e ------+..Now con │ │ │ │ +00029520: 7369 6465 7220 616e 2065 7861 6d70 6c65 sider an example │ │ │ │ +00029530: 2069 6e20 5c50 505e 3220 5c74 696d 6573 in \PP^2 \times │ │ │ │ +00029540: 205c 5050 5e32 2c20 6966 2077 6520 696e \PP^2, if we in │ │ │ │ +00029550: 7075 7420 7468 6520 4368 6f77 2072 696e put the Chow rin │ │ │ │ +00029560: 6720 4120 7468 650a 6f75 7470 7574 2077 g A the.output w │ │ │ │ +00029570: 696c 6c20 6265 2072 6574 7572 6e65 6420 ill be returned │ │ │ │ +00029580: 696e 2074 6865 2073 616d 6520 7269 6e67 in the same ring │ │ │ │ +00029590: 2e20 546f 2065 6e73 7572 6520 7072 6f70 . 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2020 2020 2020 2020 │ │ │ │ 0002a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a340: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a370: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002a380: 2020 2032 2032 2020 2020 2020 3220 2020 2 2 2 │ │ │ │ -0002a390: 2020 2020 2020 2032 2020 2020 2032 2020 2 2 │ │ │ │ -0002a3a0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3c0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ -0002a3d0: 3732 6820 6820 202d 2032 3468 2068 2020 72h h - 24h h │ │ │ │ -0002a3e0: 2d20 3132 6820 6820 202b 2034 6820 202b - 12h h + 4h + │ │ │ │ -0002a3f0: 2034 6820 6820 202b 2068 2020 2020 2020 4h h + h │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002a420: 2020 2031 2032 2020 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -0002a430: 2020 2020 2031 2032 2020 2020 2031 2020 1 2 1 │ │ │ │ -0002a440: 2020 2031 2032 2020 2020 3220 2020 2020 1 2 2 │ │ │ │ -0002a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a390: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a3a0: 2032 2032 2020 2020 2020 3220 2020 2020 2 2 2 │ │ │ │ +0002a3b0: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0002a3c0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a3e0: 2020 2020 207c 0a7c 6f31 3120 3d20 3732 |.|o11 = 72 │ │ │ │ +0002a3f0: 6820 6820 202d 2032 3468 2068 2020 2d20 h h - 24h h - │ │ │ │ +0002a400: 3132 6820 6820 202b 2034 6820 202b 2034 12h h + 4h + 4 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+0002a4d0: 2020 2020 207c 0a7c 6f31 3120 3a20 4120 |.|o11 : A │ │ │ │ 0002a4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a500: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a520: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0002a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a550: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -0002a560: 5365 6748 6173 683d 5365 6772 6528 412c SegHash=Segre(A, │ │ │ │ -0002a570: 492c 4f75 7470 7574 3d3e 4861 7368 466f I,Output=>HashFo │ │ │ │ -0002a580: 726d 2920 2020 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2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ -0002a6a0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6b0: 2020 2020 207c 0a7c 6f31 3220 3a20 4d75 |.|o12 : Mu │ │ │ │ +0002a6c0: 7461 626c 6548 6173 6854 6162 6c65 2020 tableHashTable │ │ │ │ 0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a700: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0002a710: 2d2d 2d2d 2d2d 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2020 2020 │ │ │ │ -0002a840: 2020 2020 2020 2020 3120 2020 2032 2020 1 2 │ │ │ │ +0002a840: 2020 2020 207c 0a7c 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 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a860: 2020 2020 2020 3120 2020 2032 2020 2020 1 2 │ │ │ │ +0002a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a890: 2047 6c69 7374 203d 3e20 7b31 2c20 3268 Glist => {1, 2h │ │ │ │ -0002a8a0: 2020 2b20 6820 2c20 302c 2030 2c20 307d + h , 0, 0, 0} │ │ │ │ -0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a890: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2047 G │ │ │ │ +0002a8b0: 6c69 7374 203d 3e20 7b31 2c20 3268 2020 list => {1, 2h │ │ │ │ +0002a8c0: 2b20 6820 2c20 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+0002aca0: 2020 2020 207c 0a7c 2020 2032 2020 2020 |.| 2 │ │ │ │ +0002acb0: 2020 2020 2020 3220 2020 2020 3220 3220 2 2 2 │ │ │ │ 0002acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002acd0: 2020 2020 2020 207c 0a7c 3234 6820 6820 |.|24h h │ │ │ │ -0002ace0: 202d 2031 3268 2068 202c 2037 3268 2068 - 12h h , 72h h │ │ │ │ -0002acf0: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -0002ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002acf0: 2020 2020 207c 0a7c 3234 6820 6820 202d |.|24h h - │ │ │ │ +0002ad00: 2031 3268 2068 202c 2037 3268 2068 207d 12h h , 72h h } │ │ │ │ 0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad20: 2020 2020 2020 207c 0a7c 2020 2031 2032 |.| 1 2 │ │ │ │ -0002ad30: 2020 2020 2020 3120 3220 2020 2020 3120 1 2 1 │ │ │ │ -0002ad40: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad40: 2020 2020 207c 0a7c 2020 2031 2032 2020 |.| 1 2 │ │ │ │ +0002ad50: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ 0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002ad80: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002ada0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002adc0: 2020 2020 2020 207c 0a7c 2b20 3468 2068 |.|+ 4h h │ │ │ │ -0002add0: 2020 2b20 6820 2020 2020 2020 2020 2020 + h │ │ │ │ -0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ade0: 2020 2020 207c 0a7c 2b20 3468 2068 2020 |.|+ 4h h │ │ │ │ +0002adf0: 2b20 6820 2020 2020 2020 2020 2020 2020 + h │ │ │ │ 0002ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae10: 2020 2020 2020 207c 0a7c 2020 2020 3120 |.| 1 │ │ │ │ -0002ae20: 3220 2020 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ -0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae30: 2020 2020 207c 0a7c 2020 2020 3120 3220 |.| 1 2 │ │ │ │ +0002ae40: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0002ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aeb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ -0002aec0: 7331 3d3d 5365 6748 6173 6823 2253 6567 s1==SegHash#"Seg │ │ │ │ -0002aed0: 7265 2220 2020 2020 2020 2020 2020 2020 re" │ │ │ │ -0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002aed0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 7331 -----+.|i14 : s1 │ │ │ │ +0002aee0: 3d3d 5365 6748 6173 6823 2253 6567 7265 ==SegHash#"Segre │ │ │ │ +0002aef0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +0002af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af50: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ -0002af60: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ -0002af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af70: 2020 2020 207c 0a7c 6f31 3420 3d20 7472 |.|o14 = tr │ │ │ │ +0002af80: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 0002af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afa0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002afc0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0002afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aff0: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 -------+..In the │ │ │ │ -0002b000: 2063 6173 6520 7768 6572 6520 7468 6520 case where the │ │ │ │ -0002b010: 616d 6269 656e 7420 7370 6163 6520 6973 ambient space is │ │ │ │ -0002b020: 2061 2074 6f72 6963 2076 6172 6965 7479 a toric variety │ │ │ │ -0002b030: 2077 6869 6368 2069 7320 6e6f 7420 6120 which is not a │ │ │ │ -0002b040: 7072 6f64 7563 740a 6f66 2070 726f 6a65 product.of proje │ │ │ │ -0002b050: 6374 6976 6520 7370 6163 6573 2077 6520 ctive spaces we │ │ │ │ -0002b060: 6d75 7374 206c 6f61 6420 7468 6520 4e6f must load the No │ │ │ │ -0002b070: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ -0002b080: 6573 2070 6163 6b61 6765 2061 6e64 206d es package and m │ │ │ │ -0002b090: 7573 740a 616c 736f 2069 6e70 7574 2074 ust.also input t │ │ │ │ -0002b0a0: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ -0002b0b0: 2e20 4966 2074 6865 2074 6f72 6963 2076 . If the toric v │ │ │ │ -0002b0c0: 6172 6965 7479 2069 7320 6120 7072 6f64 ariety is a prod │ │ │ │ -0002b0d0: 7563 7420 6f66 2070 726f 6a65 6374 6976 uct of projectiv │ │ │ │ -0002b0e0: 650a 7370 6163 6520 6974 2069 7320 7265 e.space it is re │ │ │ │ -0002b0f0: 636f 6d6d 656e 6465 6420 746f 2075 7365 commended to use │ │ │ │ -0002b100: 2074 6865 2066 6f72 6d20 6162 6f76 6520 the form above │ │ │ │ -0002b110: 7261 7468 6572 2074 6861 6e20 696e 7075 rather than inpu │ │ │ │ -0002b120: 7474 696e 6720 7468 6520 746f 7269 630a tting the toric. │ │ │ │ -0002b130: 7661 7269 6574 7920 666f 7220 6566 6669 variety for effi │ │ │ │ -0002b140: 6369 656e 6379 2072 6561 736f 6e73 2e0a ciency reasons.. │ │ │ │ -0002b150: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b010: 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 2063 -----+..In the c │ │ │ │ +0002b020: 6173 6520 7768 6572 6520 7468 6520 616d ase where the am │ │ │ │ +0002b030: 6269 656e 7420 7370 6163 6520 6973 2061 bient space is a │ │ │ │ +0002b040: 2074 6f72 6963 2076 6172 6965 7479 2077 toric variety w │ │ │ │ +0002b050: 6869 6368 2069 7320 6e6f 7420 6120 7072 hich is not a pr │ │ │ │ +0002b060: 6f64 7563 740a 6f66 2070 726f 6a65 6374 oduct.of project │ │ │ │ +0002b070: 6976 6520 7370 6163 6573 2077 6520 6d75 ive spaces we mu │ │ │ │ +0002b080: 7374 206c 6f61 6420 7468 6520 4e6f 726d st load the Norm │ │ │ │ +0002b090: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ +0002b0a0: 2070 6163 6b61 6765 2061 6e64 206d 7573 package and mus │ │ │ │ +0002b0b0: 740a 616c 736f 2069 6e70 7574 2074 6865 t.also input the │ │ │ │ +0002b0c0: 2074 6f72 6963 2076 6172 6965 7479 2e20 toric variety. │ │ │ │ +0002b0d0: 4966 2074 6865 2074 6f72 6963 2076 6172 If the toric var │ │ │ │ +0002b0e0: 6965 7479 2069 7320 6120 7072 6f64 7563 iety is a produc │ │ │ │ +0002b0f0: 7420 6f66 2070 726f 6a65 6374 6976 650a t of projective. │ │ │ │ +0002b100: 7370 6163 6520 6974 2069 7320 7265 636f space it is reco │ │ │ │ +0002b110: 6d6d 656e 6465 6420 746f 2075 7365 2074 mmended to use t │ │ │ │ +0002b120: 6865 2066 6f72 6d20 6162 6f76 6520 7261 he form above ra │ │ │ │ +0002b130: 7468 6572 2074 6861 6e20 696e 7075 7474 ther than inputt │ │ │ │ +0002b140: 696e 6720 7468 6520 746f 7269 630a 7661 ing the toric.va │ │ │ │ +0002b150: 7269 6574 7920 666f 7220 6566 6669 6369 riety for effici │ │ │ │ +0002b160: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b ency reasons...+ │ │ │ │ 0002b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 ----------+.|i15 │ │ │ │ -0002b1a0: 203a 206e 6565 6473 5061 636b 6167 6520 : needsPackage │ │ │ │ -0002b1b0: 224e 6f72 6d61 6c54 6f72 6963 5661 7269 "NormalToricVari │ │ │ │ -0002b1c0: 6574 6965 7322 2020 2020 2020 2020 2020 eties" │ │ │ │ -0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ +0002b1c0: 206e 6565 6473 5061 636b 6167 6520 224e needsPackage "N │ │ │ │ +0002b1d0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ +0002b1e0: 6965 7322 2020 2020 2020 2020 2020 2020 ies" │ │ │ │ 0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b200: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b230: 7c0a 7c6f 3135 203d 204e 6f72 6d61 6c54 |.|o15 = NormalT │ │ │ │ -0002b240: 6f72 6963 5661 7269 6574 6965 7320 2020 oricVarieties │ │ │ │ -0002b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b270: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b250: 7c6f 3135 203d 204e 6f72 6d61 6c54 6f72 |o15 = NormalTor │ │ │ │ +0002b260: 6963 5661 7269 6574 6965 7320 2020 2020 icVarieties │ │ │ │ +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 │ │ │ │ +0002b290: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2c0: 2020 2020 2020 7c0a 7c6f 3135 203a 2050 |.|o15 : P │ │ │ │ -0002b2d0: 6163 6b61 6765 2020 2020 2020 2020 2020 ackage │ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2f0: 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 7c0a 7c6f 3135 203a 2050 6163 |.|o15 : Pac │ │ │ │ +0002b2f0: 6b61 6765 2020 2020 2020 2020 2020 2020 kage │ │ │ │ 0002b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b310: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b320: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002b360: 3136 203a 2052 686f 203d 207b 7b31 2c30 16 : Rho = {{1,0 │ │ │ │ -0002b370: 2c30 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 ,0},{0,1,0},{0,0 │ │ │ │ -0002b380: 2c31 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 ,1},{-1,-1,0},{0 │ │ │ │ -0002b390: 2c30 2c2d 317d 7d20 2020 2020 2020 2020 ,0,-1}} │ │ │ │ -0002b3a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ +0002b380: 203a 2052 686f 203d 207b 7b31 2c30 2c30 : Rho = {{1,0,0 │ │ │ │ +0002b390: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31 },{0,1,0},{0,0,1 │ │ │ │ +0002b3a0: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30 },{-1,-1,0},{0,0 │ │ │ │ +0002b3b0: 2c2d 317d 7d20 2020 2020 2020 2020 2020 ,-1}} │ │ │ │ +0002b3c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3f0: 2020 7c0a 7c6f 3136 203d 207b 7b31 2c20 |.|o16 = {{1, │ │ │ │ -0002b400: 302c 2030 7d2c 207b 302c 2031 2c20 307d 0, 0}, {0, 1, 0} │ │ │ │ -0002b410: 2c20 7b30 2c20 302c 2031 7d2c 207b 2d31 , {0, 0, 1}, {-1 │ │ │ │ -0002b420: 2c20 2d31 2c20 307d 2c20 7b30 2c20 302c , -1, 0}, {0, 0, │ │ │ │ -0002b430: 202d 317d 7d20 2020 2020 2020 207c 0a7c -1}} |.| │ │ │ │ -0002b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b410: 7c0a 7c6f 3136 203d 207b 7b31 2c20 302c |.|o16 = {{1, 0, │ │ │ │ +0002b420: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ +0002b430: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ +0002b440: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ +0002b450: 317d 7d20 2020 2020 2020 207c 0a7c 2020 1}} |.| │ │ │ │ 0002b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b480: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ -0002b490: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4b0: 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: 2020 2020 2020 7c0a 7c6f 3136 203a 204c |.|o16 : L │ │ │ │ +0002b4b0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0002b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002b520: 7c69 3137 203a 2053 6967 6d61 203d 207b |i17 : Sigma = { │ │ │ │ -0002b530: 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c {0,1,2},{1,2,3}, │ │ │ │ -0002b540: 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c {0,2,3},{0,1,4}, │ │ │ │ -0002b550: 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d {1,3,4},{0,3,4}} │ │ │ │ -0002b560: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002b540: 3137 203a 2053 6967 6d61 203d 207b 7b30 17 : Sigma = {{0 │ │ │ │ +0002b550: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30 ,1,2},{1,2,3},{0 │ │ │ │ +0002b560: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31 ,2,3},{0,1,4},{1 │ │ │ │ +0002b570: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020 ,3,4},{0,3,4}} │ │ │ │ +0002b580: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0002b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5b0: 2020 2020 7c0a 7c6f 3137 203d 207b 7b30 |.|o17 = {{0 │ │ │ │ -0002b5c0: 2c20 312c 2032 7d2c 207b 312c 2032 2c20 , 1, 2}, {1, 2, │ │ │ │ -0002b5d0: 337d 2c20 7b30 2c20 322c 2033 7d2c 207b 3}, {0, 2, 3}, { │ │ │ │ -0002b5e0: 302c 2031 2c20 347d 2c20 7b31 2c20 332c 0, 1, 4}, {1, 3, │ │ │ │ -0002b5f0: 2034 7d2c 207b 302c 2033 2c20 347d 7d7c 4}, {0, 3, 4}}| │ │ │ │ -0002b600: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b610: 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 7c0a 7c6f 3137 203d 207b 7b30 2c20 |.|o17 = {{0, │ │ │ │ +0002b5e0: 312c 2032 7d2c 207b 312c 2032 2c20 337d 1, 2}, {1, 2, 3} │ │ │ │ +0002b5f0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c , {0, 2, 3}, {0, │ │ │ │ +0002b600: 2031 2c20 347d 2c20 7b31 2c20 332c 2034 1, 4}, {1, 3, 4 │ │ │ │ +0002b610: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c }, {0, 3, 4}}|.| │ │ │ │ 0002b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b640: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ -0002b650: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ -0002b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b670: 2020 2020 2020 2020 2020 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 7c0a 7c6f 3137 203a |.|o17 : │ │ │ │ +0002b670: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0002b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b690: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0002b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0002b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6e0: 2b0a 7c69 3138 203a 2058 203d 206e 6f72 +.|i18 : X = nor │ │ │ │ -0002b6f0: 6d61 6c54 6f72 6963 5661 7269 6574 7928 malToricVariety( │ │ │ │ -0002b700: 5268 6f2c 5369 676d 612c 436f 6566 6669 Rho,Sigma,Coeffi │ │ │ │ -0002b710: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33 cientRing =>ZZ/3 │ │ │ │ -0002b720: 3237 3439 2920 2020 2020 207c 0a7c 2020 2749) |.| │ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b700: 7c69 3138 203a 2058 203d 206e 6f72 6d61 |i18 : X = norma │ │ │ │ +0002b710: 6c54 6f72 6963 5661 7269 6574 7928 5268 lToricVariety(Rh │ │ │ │ +0002b720: 6f2c 5369 676d 612c 436f 6566 6669 6369 o,Sigma,Coeffici │ │ │ │ +0002b730: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237 entRing =>ZZ/327 │ │ │ │ +0002b740: 3439 2920 2020 2020 207c 0a7c 2020 2020 49) |.| │ │ │ │ 0002b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b770: 2020 2020 2020 7c0a 7c6f 3138 203d 2058 |.|o18 = X │ │ │ │ +0002b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b790: 2020 2020 7c0a 7c6f 3138 203d 2058 2020 |.|o18 = X │ │ │ │ 0002b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b7e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b800: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002b810: 3138 203a 204e 6f72 6d61 6c54 6f72 6963 18 : NormalToric │ │ │ │ -0002b820: 5661 7269 6574 7920 2020 2020 2020 2020 Variety │ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b850: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ +0002b830: 203a 204e 6f72 6d61 6c54 6f72 6963 5661 : NormalToricVa │ │ │ │ +0002b840: 7269 6574 7920 2020 2020 2020 2020 2020 riety │ │ │ │ +0002b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b870: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0002b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b8a0: 2d2d 2b0a 7c69 3139 203a 2043 6865 636b --+.|i19 : Check │ │ │ │ -0002b8b0: 546f 7269 6356 6172 6965 7479 5661 6c69 ToricVarietyVali │ │ │ │ -0002b8c0: 6428 5829 2020 2020 2020 2020 2020 2020 d(X) │ │ │ │ -0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8c0: 2b0a 7c69 3139 203a 2043 6865 636b 546f +.|i19 : CheckTo │ │ │ │ +0002b8d0: 7269 6356 6172 6965 7479 5661 6c69 6428 ricVarietyValid( │ │ │ │ +0002b8e0: 5829 2020 2020 2020 2020 2020 2020 2020 X) │ │ │ │ 0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b900: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b930: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -0002b940: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ -0002b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b960: 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: 2020 2020 2020 7c0a 7c6f 3139 203d 2074 |.|o19 = t │ │ │ │ +0002b960: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ 0002b970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b980: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0002b990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002b9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002b9d0: 7c69 3230 203a 2052 3d72 696e 6728 5829 |i20 : R=ring(X) │ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002b9f0: 3230 203a 2052 3d72 696e 6728 5829 2020 20 : R=ring(X) │ │ │ │ 0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba10: 2020 2020 2020 2020 207c 0a7c 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 │ │ │ │ +0002ba30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba60: 2020 2020 7c0a 7c6f 3230 203d 2052 2020 |.|o20 = R │ │ │ │ +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 │ │ │ │ +0002ba80: 2020 7c0a 7c6f 3230 203d 2052 2020 2020 |.|o20 = R │ │ │ │ 0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002baa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bab0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002bac0: 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 207c 0a7c |.| │ │ │ │ 0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002baf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3230 |.|o20 │ │ │ │ -0002bb00: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0002bb10: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb10: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ +0002bb20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ 0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb90: 2b0a 7c69 3231 203a 2049 3d69 6465 616c +.|i21 : I=ideal │ │ │ │ -0002bba0: 2852 5f30 5e34 2a52 5f31 2c52 5f30 2a52 (R_0^4*R_1,R_0*R │ │ │ │ -0002bbb0: 5f33 2a52 5f34 2a52 5f32 2d52 5f32 5e32 _3*R_4*R_2-R_2^2 │ │ │ │ -0002bbc0: 2a52 5f30 5e32 2920 2020 2020 2020 2020 *R_0^2) │ │ │ │ -0002bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002bbb0: 7c69 3231 203a 2049 3d69 6465 616c 2852 |i21 : I=ideal(R │ │ │ │ +0002bbc0: 5f30 5e34 2a52 5f31 2c52 5f30 2a52 5f33 _0^4*R_1,R_0*R_3 │ │ │ │ +0002bbd0: 2a52 5f34 2a52 5f32 2d52 5f32 5e32 2a52 *R_4*R_2-R_2^2*R │ │ │ │ +0002bbe0: 5f30 5e32 2920 2020 2020 2020 2020 2020 _0^2) │ │ │ │ +0002bbf0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002bc30: 2020 2020 2020 2034 2020 2020 2020 2032 4 2 │ │ │ │ -0002bc40: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002bc50: 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002bc50: 2020 2020 2034 2020 2020 2020 2032 2032 4 2 2 │ │ │ │ 0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc70: 207c 0a7c 6f32 3120 3d20 6964 6561 6c20 |.|o21 = ideal │ │ │ │ -0002bc80: 2878 2078 202c 202d 2078 2078 2020 2b20 (x x , - x x + │ │ │ │ -0002bc90: 7820 7820 7820 7820 2920 2020 2020 2020 x x x x ) │ │ │ │ -0002bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bcb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002bcc0: 2020 2020 2020 2020 2020 2020 2030 2031 0 1 │ │ │ │ -0002bcd0: 2020 2020 2030 2032 2020 2020 3020 3220 0 2 0 2 │ │ │ │ -0002bce0: 3320 3420 2020 2020 2020 2020 2020 2020 3 4 │ │ │ │ -0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bc90: 0a7c 6f32 3120 3d20 6964 6561 6c20 2878 .|o21 = ideal (x │ │ │ │ +0002bca0: 2078 202c 202d 2078 2078 2020 2b20 7820 x , - x x + x │ │ │ │ +0002bcb0: 7820 7820 7820 2920 2020 2020 2020 2020 x x x ) │ │ │ │ +0002bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bcd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bce0: 2020 2020 2020 2020 2020 2030 2031 2020 0 1 │ │ │ │ +0002bcf0: 2020 2030 2032 2020 2020 3020 3220 3320 0 2 0 2 3 │ │ │ │ +0002bd00: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0002bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd50: 2020 7c0a 7c6f 3231 203a 2049 6465 616c |.|o21 : Ideal │ │ │ │ -0002bd60: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -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 207c 0a2b |.+ │ │ │ │ -0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd70: 7c0a 7c6f 3231 203a 2049 6465 616c 206f |.|o21 : Ideal o │ │ │ │ +0002bd80: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ +0002bd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bdb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bde0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ -0002bdf0: 2053 6567 7265 2858 2c49 2920 2020 2020 Segre(X,I) │ │ │ │ -0002be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be00: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2053 ------+.|i22 : S │ │ │ │ +0002be10: 6567 7265 2858 2c49 2920 2020 2020 2020 egre(X,I) │ │ │ │ 0002be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be30: 2020 207c 0a7c 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 │ │ │ │ +0002be50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002be80: 7c20 2020 2020 2020 2020 2020 3220 2020 | 2 │ │ │ │ -0002be90: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bec0: 2020 2020 2020 2020 207c 0a7c 6f32 3220 |.|o22 │ │ │ │ -0002bed0: 3d20 2d20 3732 7820 7820 202b 2033 7820 = - 72x x + 3x │ │ │ │ -0002bee0: 202b 2038 7820 7820 202b 2078 2020 2020 + 8x x + x │ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bf20: 2020 3320 3420 2020 2020 3320 2020 2020 3 4 3 │ │ │ │ -0002bf30: 3320 3420 2020 2033 2020 2020 2020 2020 3 4 3 │ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bf60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bea0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002beb0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bee0: 2020 2020 2020 207c 0a7c 6f32 3220 3d20 |.|o22 = │ │ │ │ +0002bef0: 2d20 3732 7820 7820 202b 2033 7820 202b - 72x x + 3x + │ │ │ │ +0002bf00: 2038 7820 7820 202b 2078 2020 2020 2020 8x x + x │ │ │ │ +0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002bf40: 3320 3420 2020 2020 3320 2020 2020 3320 3 4 3 3 │ │ │ │ +0002bf50: 3420 2020 2033 2020 2020 2020 2020 2020 4 3 │ │ │ │ +0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfc0: 2020 205a 5a5b 7820 2e2e 7820 5d20 2020 ZZ[x ..x ] │ │ │ │ +0002bfc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bff0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002bfe0: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020 ZZ[x ..x ] │ │ │ │ +0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c010: 2020 3020 2020 3420 2020 2020 2020 2020 0 4 │ │ │ │ +0002c010: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c040: 7c0a 7c6f 3232 203a 202d 2d2d 2d2d 2d2d |.|o22 : ------- │ │ │ │ -0002c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c070: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ -0002c080: 2020 2020 2020 2020 2020 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Note that │ │ │ │ +0002c6b0: 2074 6865 2062 6572 7469 6e69 206f 7074 the bertini opt │ │ │ │ +0002c6c0: 696f 6e20 6973 206f 6e6c 7920 6176 6169 ion is only avai │ │ │ │ +0002c6d0: 6c61 626c 6520 666f 720a 7375 6273 6368 lable for.subsch │ │ │ │ +0002c6e0: 656d 6573 206f 6620 5c50 505e 6e2e 0a0a emes of \PP^n... │ │ │ │ +0002c6f0: 4f62 7365 7276 6520 7468 6174 2074 6865 Observe that the │ │ │ │ +0002c700: 2061 6c67 6f72 6974 686d 2069 7320 6120 algorithm is a │ │ │ │ +0002c710: 7072 6f62 6162 696c 6973 7469 6320 616c probabilistic al │ │ │ │ +0002c720: 676f 7269 7468 6d20 616e 6420 6d61 7920 gorithm and may │ │ │ │ +0002c730: 6769 7665 2061 2077 726f 6e67 0a61 6e73 give a wrong.ans │ │ │ │ +0002c740: 7765 7220 7769 7468 2061 2073 6d61 6c6c wer with a small │ │ │ │ +0002c750: 2062 7574 206e 6f6e 7a65 726f 2070 726f but nonzero pro │ │ │ │ +0002c760: 6261 6269 6c69 7479 2e20 5265 6164 206d bability. Read m │ │ │ │ +0002c770: 6f72 6520 756e 6465 7220 2a6e 6f74 650a ore under *note. │ │ │ │ +0002c780: 7072 6f62 6162 696c 6973 7469 6320 616c probabilistic al │ │ │ │ +0002c790: 676f 7269 7468 6d3a 2070 726f 6261 6269 gorithm: probabi │ │ │ │ +0002c7a0: 6c69 7374 6963 2061 6c67 6f72 6974 686d listic algorithm │ │ │ │ +0002c7b0: 2c2e 0a0a 5761 7973 2074 6f20 7573 6520 ,...Ways to use │ │ │ │ +0002c7c0: 5365 6772 653a 0a3d 3d3d 3d3d 3d3d 3d3d Segre:.========= │ │ │ │ +0002c7d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +0002c7e0: 5365 6772 6528 4964 6561 6c29 220a 2020 Segre(Ideal)". │ │ │ │ +0002c7f0: 2a20 2253 6567 7265 2849 6465 616c 2c53 * "Segre(Ideal,S │ │ │ │ +0002c800: 796d 626f 6c29 220a 2020 2a20 2253 6567 ymbol)". * "Seg │ │ │ │ +0002c810: 7265 2851 756f 7469 656e 7452 696e 672c re(QuotientRing, │ │ │ │ +0002c820: 4964 6561 6c29 220a 0a46 6f72 2074 6865 Ideal)"..For the │ │ │ │ +0002c830: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +0002c840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0002c850: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +0002c860: 2053 6567 7265 3a20 5365 6772 652c 2069 Segre: Segre, i │ │ │ │ +0002c870: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0002c880: 2066 756e 6374 696f 6e20 7769 7468 206f function with o │ │ │ │ +0002c890: 7074 696f 6e73 3a0a 284d 6163 6175 6c61 ptions:.(Macaula │ │ │ │ +0002c8a0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0002c8b0: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +0002c8c0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0002c8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8f0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0002c900: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0002c910: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0002c920: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0002c930: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -0002c940: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0002c950: 7932 2f70 6163 6b61 6765 732f 0a43 6861 y2/packages/.Cha │ │ │ │ -0002c960: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -0002c970: 6573 2e6d 323a 3137 3236 3a30 2e0a 1f0a es.m2:1726:0.... │ │ │ │ -0002c980: 4669 6c65 3a20 4368 6172 6163 7465 7269 File: Characteri │ │ │ │ -0002c990: 7374 6963 436c 6173 7365 732e 696e 666f sticClasses.info │ │ │ │ -0002c9a0: 2c20 4e6f 6465 3a20 546f 7269 6343 686f , Node: ToricCho │ │ │ │ -0002c9b0: 7752 696e 672c 2050 7265 763a 2053 6567 wRing, Prev: Seg │ │ │ │ -0002c9c0: 7265 2c20 5570 3a20 546f 700a 0a54 6f72 re, Up: Top..Tor │ │ │ │ -0002c9d0: 6963 4368 6f77 5269 6e67 202d 2d20 436f icChowRing -- Co │ │ │ │ -0002c9e0: 6d70 7574 6573 2074 6865 2043 686f 7720 mputes the Chow │ │ │ │ -0002c9f0: 7269 6e67 206f 6620 6120 6e6f 726d 616c ring of a normal │ │ │ │ -0002ca00: 2074 6f72 6963 2076 6172 6965 7479 0a2a toric variety.* │ │ │ │ -0002ca10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ca20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002c8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c910: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0002c920: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0002c930: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0002c940: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0002c950: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +0002c960: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0002c970: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +0002c980: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +0002c990: 2e6d 323a 3137 3236 3a30 2e0a 1f0a 4669 .m2:1726:0....Fi │ │ │ │ +0002c9a0: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ +0002c9b0: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ +0002c9c0: 4e6f 6465 3a20 546f 7269 6343 686f 7752 Node: ToricChowR │ │ │ │ +0002c9d0: 696e 672c 2050 7265 763a 2053 6567 7265 ing, Prev: Segre │ │ │ │ +0002c9e0: 2c20 5570 3a20 546f 700a 0a54 6f72 6963 , Up: Top..Toric │ │ │ │ +0002c9f0: 4368 6f77 5269 6e67 202d 2d20 436f 6d70 ChowRing -- Comp │ │ │ │ +0002ca00: 7574 6573 2074 6865 2043 686f 7720 7269 utes the Chow ri │ │ │ │ +0002ca10: 6e67 206f 6620 6120 6e6f 726d 616c 2074 ng of a normal t │ │ │ │ +0002ca20: 6f72 6963 2076 6172 6965 7479 0a2a 2a2a oric variety.*** │ │ │ │ 0002ca30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002ca40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ca50: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -0002ca60: 2020 2020 2020 546f 7269 6343 686f 7752 ToricChowR │ │ │ │ -0002ca70: 696e 6720 580a 2020 2a20 496e 7075 7473 ing X. * Inputs │ │ │ │ -0002ca80: 3a0a 2020 2020 2020 2a20 522c 2061 202a :. * R, a * │ │ │ │ -0002ca90: 6e6f 7465 206e 6f72 6d61 6c20 746f 7269 note normal tori │ │ │ │ -0002caa0: 6320 7661 7269 6574 793a 0a20 2020 2020 c variety:. │ │ │ │ -0002cab0: 2020 2028 4e6f 726d 616c 546f 7269 6356 (NormalToricV │ │ │ │ -0002cac0: 6172 6965 7469 6573 294e 6f72 6d61 6c54 arieties)NormalT │ │ │ │ -0002cad0: 6f72 6963 5661 7269 6574 792c 2c20 4120 oricVariety,, A │ │ │ │ -0002cae0: 6e6f 726d 616c 2074 6f72 6963 2076 6172 normal toric var │ │ │ │ -0002caf0: 6965 7479 0a20 202a 204f 7574 7075 7473 iety. * Outputs │ │ │ │ -0002cb00: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -0002cb10: 6520 7175 6f74 6965 6e74 2072 696e 673a e quotient ring: │ │ │ │ -0002cb20: 2028 4d61 6361 756c 6179 3244 6f63 2951 (Macaulay2Doc)Q │ │ │ │ -0002cb30: 756f 7469 656e 7452 696e 672c 2c20 0a0a uotientRing,, .. │ │ │ │ -0002cb40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0002cb50: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2058 2062 =======..Let X b │ │ │ │ -0002cb60: 6520 6120 746f 7269 6320 7661 7269 6574 e a toric variet │ │ │ │ -0002cb70: 7920 7769 7468 2074 6f74 616c 2063 6f6f y with total coo │ │ │ │ -0002cb80: 7264 696e 6174 6520 7269 6e67 2028 436f rdinate ring (Co │ │ │ │ -0002cb90: 7820 7269 6e67 2920 522e 2054 6869 7320 x ring) R. This │ │ │ │ -0002cba0: 6d65 7468 6f64 0a63 6f6d 7075 7465 7320 method.computes │ │ │ │ -0002cbb0: 7468 6520 4368 6f77 2072 696e 6720 2043 the Chow ring C │ │ │ │ -0002cbc0: 686f 7720 7269 6e67 2043 683d 522f 2853 how ring Ch=R/(S │ │ │ │ -0002cbd0: 522b 4c52 2920 6f66 2058 3b20 6865 7265 R+LR) of X; here │ │ │ │ -0002cbe0: 2053 5220 6973 2074 6865 0a53 7461 6e6c SR is the.Stanl │ │ │ │ -0002cbf0: 6579 2d52 6569 736e 6572 2069 6465 616c ey-Reisner ideal │ │ │ │ -0002cc00: 206f 6620 7468 6520 636f 7272 6573 706f of the correspo │ │ │ │ -0002cc10: 6e64 696e 6720 6661 6e20 616e 6420 4c52 nding fan and LR │ │ │ │ -0002cc20: 2069 7320 7468 6520 6964 6561 6c20 6f66 is the ideal of │ │ │ │ -0002cc30: 206c 696e 6561 720a 7265 6c61 7469 6f6e linear.relation │ │ │ │ -0002cc40: 7320 616d 6f75 6e74 2074 6865 2072 6179 s amount the ray │ │ │ │ -0002cc50: 732e 2049 7420 6973 206e 6565 6465 6420 s. It is needed │ │ │ │ -0002cc60: 666f 7220 696e 7075 7420 696e 746f 2074 for input into t │ │ │ │ -0002cc70: 6865 206d 6574 686f 6473 202a 6e6f 7465 he methods *note │ │ │ │ -0002cc80: 2053 6567 7265 3a0a 5365 6772 652c 2c20 Segre:.Segre,, │ │ │ │ -0002cc90: 2a6e 6f74 6520 4368 6572 6e3a 2043 6865 *note Chern: Che │ │ │ │ -0002cca0: 726e 2c20 616e 6420 2a6e 6f74 6520 4353 rn, and *note CS │ │ │ │ -0002ccb0: 4d3a 2043 534d 2c20 696e 2074 6865 2063 M: CSM, in the c │ │ │ │ -0002ccc0: 6173 6573 2077 6865 7265 2061 2074 6f72 ases where a tor │ │ │ │ -0002ccd0: 6963 0a76 6172 6965 7479 2069 7320 616c ic.variety is al │ │ │ │ -0002cce0: 736f 2069 6e70 7574 2074 6f20 656e 7375 so input to ensu │ │ │ │ -0002ccf0: 7265 2074 6861 7420 7468 6573 6520 6d65 re that these me │ │ │ │ -0002cd00: 7468 6f64 7320 7265 7475 726e 2072 6573 thods return res │ │ │ │ -0002cd10: 756c 7473 2069 6e20 7468 6520 7361 6d65 ults in the same │ │ │ │ -0002cd20: 0a72 696e 672e 2057 6520 6769 7665 2061 .ring. We give a │ │ │ │ -0002cd30: 6e20 6578 616d 706c 6520 6f66 2074 6865 n example of the │ │ │ │ -0002cd40: 2075 7365 206f 6620 7468 6973 206d 6574 use of this met │ │ │ │ -0002cd50: 686f 6420 746f 2077 6f72 6b20 7769 7468 hod to work with │ │ │ │ -0002cd60: 2065 6c65 6d65 6e74 7320 6f66 2074 6865 elements of the │ │ │ │ -0002cd70: 0a43 686f 7720 7269 6e67 206f 6620 6120 .Chow ring of a │ │ │ │ -0002cd80: 746f 7269 6320 7661 7269 6574 790a 0a2b toric variety..+ │ │ │ │ -0002cd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ca50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ca60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0002ca70: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0002ca80: 2020 2020 546f 7269 6343 686f 7752 696e ToricChowRin │ │ │ │ +0002ca90: 6720 580a 2020 2a20 496e 7075 7473 3a0a g X. * Inputs:. │ │ │ │ +0002caa0: 2020 2020 2020 2a20 522c 2061 202a 6e6f * R, a *no │ │ │ │ +0002cab0: 7465 206e 6f72 6d61 6c20 746f 7269 6320 te normal toric │ │ │ │ +0002cac0: 7661 7269 6574 793a 0a20 2020 2020 2020 variety:. │ │ │ │ +0002cad0: 2028 4e6f 726d 616c 546f 7269 6356 6172 (NormalToricVar │ │ │ │ +0002cae0: 6965 7469 6573 294e 6f72 6d61 6c54 6f72 ieties)NormalTor │ │ │ │ +0002caf0: 6963 5661 7269 6574 792c 2c20 4120 6e6f icVariety,, A no │ │ │ │ +0002cb00: 726d 616c 2074 6f72 6963 2076 6172 6965 rmal toric varie │ │ │ │ +0002cb10: 7479 0a20 202a 204f 7574 7075 7473 3a0a ty. * Outputs:. │ │ │ │ +0002cb20: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ +0002cb30: 7175 6f74 6965 6e74 2072 696e 673a 2028 quotient ring: ( │ │ │ │ +0002cb40: 4d61 6361 756c 6179 3244 6f63 2951 756f Macaulay2Doc)Quo │ │ │ │ +0002cb50: 7469 656e 7452 696e 672c 2c20 0a0a 4465 tientRing,, ..De │ │ │ │ +0002cb60: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0002cb70: 3d3d 3d3d 3d0a 0a4c 6574 2058 2062 6520 =====..Let X be │ │ │ │ +0002cb80: 6120 746f 7269 6320 7661 7269 6574 7920 a toric variety │ │ │ │ +0002cb90: 7769 7468 2074 6f74 616c 2063 6f6f 7264 with total coord │ │ │ │ +0002cba0: 696e 6174 6520 7269 6e67 2028 436f 7820 inate ring (Cox │ │ │ │ +0002cbb0: 7269 6e67 2920 522e 2054 6869 7320 6d65 ring) R. This me │ │ │ │ +0002cbc0: 7468 6f64 0a63 6f6d 7075 7465 7320 7468 thod.computes th │ │ │ │ +0002cbd0: 6520 4368 6f77 2072 696e 6720 2043 686f e Chow ring Cho │ │ │ │ +0002cbe0: 7720 7269 6e67 2043 683d 522f 2853 522b w ring Ch=R/(SR+ │ │ │ │ +0002cbf0: 4c52 2920 6f66 2058 3b20 6865 7265 2053 LR) of X; here S │ │ │ │ +0002cc00: 5220 6973 2074 6865 0a53 7461 6e6c 6579 R is the.Stanley │ │ │ │ +0002cc10: 2d52 6569 736e 6572 2069 6465 616c 206f -Reisner ideal o │ │ │ │ +0002cc20: 6620 7468 6520 636f 7272 6573 706f 6e64 f the correspond │ │ │ │ +0002cc30: 696e 6720 6661 6e20 616e 6420 4c52 2069 ing fan and LR i │ │ │ │ +0002cc40: 7320 7468 6520 6964 6561 6c20 6f66 206c s the ideal of l │ │ │ │ +0002cc50: 696e 6561 720a 7265 6c61 7469 6f6e 7320 inear.relations │ │ │ │ +0002cc60: 616d 6f75 6e74 2074 6865 2072 6179 732e amount the rays. │ │ │ │ +0002cc70: 2049 7420 6973 206e 6565 6465 6420 666f It is needed fo │ │ │ │ +0002cc80: 7220 696e 7075 7420 696e 746f 2074 6865 r input into the │ │ │ │ +0002cc90: 206d 6574 686f 6473 202a 6e6f 7465 2053 methods *note S │ │ │ │ +0002cca0: 6567 7265 3a0a 5365 6772 652c 2c20 2a6e egre:.Segre,, *n │ │ │ │ +0002ccb0: 6f74 6520 4368 6572 6e3a 2043 6865 726e ote Chern: Chern │ │ │ │ +0002ccc0: 2c20 616e 6420 2a6e 6f74 6520 4353 4d3a , and *note CSM: │ │ │ │ +0002ccd0: 2043 534d 2c20 696e 2074 6865 2063 6173 CSM, in the cas │ │ │ │ +0002cce0: 6573 2077 6865 7265 2061 2074 6f72 6963 es where a toric │ │ │ │ +0002ccf0: 0a76 6172 6965 7479 2069 7320 616c 736f .variety is also │ │ │ │ +0002cd00: 2069 6e70 7574 2074 6f20 656e 7375 7265 input to ensure │ │ │ │ +0002cd10: 2074 6861 7420 7468 6573 6520 6d65 7468 that these meth │ │ │ │ +0002cd20: 6f64 7320 7265 7475 726e 2072 6573 756c ods return resul │ │ │ │ +0002cd30: 7473 2069 6e20 7468 6520 7361 6d65 0a72 ts in the same.r │ │ │ │ +0002cd40: 696e 672e 2057 6520 6769 7665 2061 6e20 ing. We give an │ │ │ │ +0002cd50: 6578 616d 706c 6520 6f66 2074 6865 2075 example of the u │ │ │ │ +0002cd60: 7365 206f 6620 7468 6973 206d 6574 686f se of this metho │ │ │ │ +0002cd70: 6420 746f 2077 6f72 6b20 7769 7468 2065 d to work with e │ │ │ │ +0002cd80: 6c65 6d65 6e74 7320 6f66 2074 6865 0a43 lements of the.C │ │ │ │ +0002cd90: 686f 7720 7269 6e67 206f 6620 6120 746f how ring of a to │ │ │ │ +0002cda0: 7269 6320 7661 7269 6574 790a 0a2b 2d2d ric variety..+-- │ │ │ │ 0002cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002cde0: 6931 203a 206e 6565 6473 5061 636b 6167 i1 : needsPackag │ │ │ │ -0002cdf0: 6520 224e 6f72 6d61 6c54 6f72 6963 5661 e "NormalToricVa │ │ │ │ -0002ce00: 7269 6574 6965 7322 2020 2020 2020 2020 rieties" │ │ │ │ -0002ce10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ce20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002ce00: 203a 206e 6565 6473 5061 636b 6167 6520 : needsPackage │ │ │ │ +0002ce10: 224e 6f72 6d61 6c54 6f72 6963 5661 7269 "NormalToricVari │ │ │ │ +0002ce20: 6574 6965 7322 2020 2020 2020 2020 2020 eties" │ │ │ │ 0002ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ce40: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 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 207c 0a7c |.| │ │ │ │ -0002ce80: 6f31 203d 204e 6f72 6d61 6c54 6f72 6963 o1 = NormalToric │ │ │ │ -0002ce90: 5661 7269 6574 6965 7320 2020 2020 2020 Varieties │ │ │ │ -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 207c 0a7c |.| │ │ │ │ +0002ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ce90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002cea0: 203d 204e 6f72 6d61 6c54 6f72 6963 5661 = NormalToricVa │ │ │ │ +0002ceb0: 7269 6574 6965 7320 2020 2020 2020 2020 rieties │ │ │ │ +0002cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cee0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002cf20: 6f31 203a 2050 6163 6b61 6765 2020 2020 o1 : Package │ │ │ │ -0002cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf40: 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002cf40: 203a 2050 6163 6b61 6765 2020 2020 2020 : Package │ │ │ │ 0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cf80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002cfc0: 6932 203a 2052 686f 203d 207b 7b31 2c30 i2 : Rho = {{1,0 │ │ │ │ -0002cfd0: 2c30 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 ,0},{0,1,0},{0,0 │ │ │ │ -0002cfe0: 2c31 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 ,1},{-1,-1,0},{0 │ │ │ │ -0002cff0: 2c30 2c2d 317d 7d20 2020 2020 2020 2020 ,0,-1}} │ │ │ │ -0002d000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002cfe0: 203a 2052 686f 203d 207b 7b31 2c30 2c30 : Rho = {{1,0,0 │ │ │ │ +0002cff0: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31 },{0,1,0},{0,0,1 │ │ │ │ +0002d000: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30 },{-1,-1,0},{0,0 │ │ │ │ +0002d010: 2c2d 317d 7d20 2020 2020 2020 2020 2020 ,-1}} │ │ │ │ +0002d020: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d050: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d060: 6f32 203d 207b 7b31 2c20 302c 2030 7d2c o2 = {{1, 0, 0}, │ │ │ │ -0002d070: 207b 302c 2031 2c20 307d 2c20 7b30 2c20 {0, 1, 0}, {0, │ │ │ │ -0002d080: 302c 2031 7d2c 207b 2d31 2c20 2d31 2c20 0, 1}, {-1, -1, │ │ │ │ -0002d090: 307d 2c20 7b30 2c20 302c 202d 317d 7d20 0}, {0, 0, -1}} │ │ │ │ -0002d0a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d0c0: 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 207c 0a7c 6f32 |.|o2 │ │ │ │ +0002d080: 203d 207b 7b31 2c20 302c 2030 7d2c 207b = {{1, 0, 0}, { │ │ │ │ +0002d090: 302c 2031 2c20 307d 2c20 7b30 2c20 302c 0, 1, 0}, {0, 0, │ │ │ │ +0002d0a0: 2031 7d2c 207b 2d31 2c20 2d31 2c20 307d 1}, {-1, -1, 0} │ │ │ │ +0002d0b0: 2c20 7b30 2c20 302c 202d 317d 7d20 2020 , {0, 0, -1}} │ │ │ │ +0002d0c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d100: 6f32 203a 204c 6973 7420 2020 2020 2020 o2 : List │ │ │ │ -0002d110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d110: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0002d120: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 0002d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d140: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d160: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002d170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002d1a0: 6933 203a 2053 6967 6d61 203d 207b 7b30 i3 : Sigma = {{0 │ │ │ │ -0002d1b0: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30 ,1,2},{1,2,3},{0 │ │ │ │ -0002d1c0: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31 ,2,3},{0,1,4},{1 │ │ │ │ -0002d1d0: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020 ,3,4},{0,3,4}} │ │ │ │ -0002d1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +0002d1c0: 203a 2053 6967 6d61 203d 207b 7b30 2c31 : Sigma = {{0,1 │ │ │ │ +0002d1d0: 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30 2c32 ,2},{1,2,3},{0,2 │ │ │ │ +0002d1e0: 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31 2c33 ,3},{0,1,4},{1,3 │ │ │ │ +0002d1f0: 2c34 7d2c 7b30 2c33 2c34 7d7d 2020 2020 ,4},{0,3,4}} │ │ │ │ +0002d200: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d230: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d240: 6f33 203d 207b 7b30 2c20 312c 2032 7d2c o3 = {{0, 1, 2}, │ │ │ │ -0002d250: 207b 312c 2032 2c20 337d 2c20 7b30 2c20 {1, 2, 3}, {0, │ │ │ │ -0002d260: 322c 2033 7d2c 207b 302c 2031 2c20 347d 2, 3}, {0, 1, 4} │ │ │ │ -0002d270: 2c20 7b31 2c20 332c 2034 7d2c 207b 302c , {1, 3, 4}, {0, │ │ │ │ -0002d280: 2033 2c20 347d 7d20 2020 2020 207c 0a7c 3, 4}} |.| │ │ │ │ -0002d290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d250: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0002d260: 203d 207b 7b30 2c20 312c 2032 7d2c 207b = {{0, 1, 2}, { │ │ │ │ +0002d270: 312c 2032 2c20 337d 2c20 7b30 2c20 322c 1, 2, 3}, {0, 2, │ │ │ │ +0002d280: 2033 7d2c 207b 302c 2031 2c20 347d 2c20 3}, {0, 1, 4}, │ │ │ │ +0002d290: 7b31 2c20 332c 2034 7d2c 207b 302c 2033 {1, 3, 4}, {0, 3 │ │ │ │ +0002d2a0: 2c20 347d 7d20 2020 2020 207c 0a7c 2020 , 4}} |.| │ │ │ │ 0002d2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d2d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d2e0: 6f33 203a 204c 6973 7420 2020 2020 2020 o3 : List │ │ │ │ -0002d2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d2f0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0002d300: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 0002d310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d320: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d340: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2b 0a7c -------------+.| │ │ │ │ -0002d380: 6934 203a 2058 203d 206e 6f72 6d61 6c54 i4 : X = normalT │ │ │ │ -0002d390: 6f72 6963 5661 7269 6574 7928 5268 6f2c oricVariety(Rho, │ │ │ │ -0002d3a0: 5369 676d 612c 436f 6566 6669 6369 656e Sigma,Coefficien │ │ │ │ -0002d3b0: 7452 696e 6720 3d3e 5a5a 2f33 3237 3439 tRing =>ZZ/32749 │ │ │ │ -0002d3c0: 2920 2020 2020 2020 2020 2020 207c 0a7c ) |.| │ │ │ │ -0002d3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +0002d3a0: 203a 2058 203d 206e 6f72 6d61 6c54 6f72 : X = normalTor │ │ │ │ +0002d3b0: 6963 5661 7269 6574 7928 5268 6f2c 5369 icVariety(Rho,Si │ │ │ │ +0002d3c0: 676d 612c 436f 6566 6669 6369 656e 7452 gma,CoefficientR │ │ │ │ +0002d3d0: 696e 6720 3d3e 5a5a 2f33 3237 3439 2920 ing =>ZZ/32749) │ │ │ │ +0002d3e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d410: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d420: 6f34 203d 2058 2020 2020 2020 2020 2020 o4 = X │ │ │ │ -0002d430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d430: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0002d440: 203d 2058 2020 2020 2020 2020 2020 2020 = X │ │ │ │ 0002d450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d480: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d4b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d4c0: 6f34 203a 204e 6f72 6d61 6c54 6f72 6963 o4 : NormalToric │ │ │ │ -0002d4d0: 5661 7269 6574 7920 2020 2020 2020 2020 Variety │ │ │ │ -0002d4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d500: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d4d0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0002d4e0: 203a 204e 6f72 6d61 6c54 6f72 6963 5661 : NormalToricVa │ │ │ │ +0002d4f0: 7269 6574 7920 2020 2020 2020 2020 2020 riety │ │ │ │ +0002d500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002d530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002d560: 6935 203a 2052 3d72 696e 6720 5820 2020 i5 : R=ring X │ │ │ │ -0002d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +0002d580: 203a 2052 3d72 696e 6720 5820 2020 2020 : R=ring X │ │ │ │ 0002d590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d5c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d5f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d600: 6f35 203d 2052 2020 2020 2020 2020 2020 o5 = R │ │ │ │ -0002d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d610: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +0002d620: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0002d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d640: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d660: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d6a0: 6f35 203a 2050 6f6c 796e 6f6d 6961 6c52 o5 : PolynomialR │ │ │ │ -0002d6b0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0002d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d6e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d6b0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +0002d6c0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0002d6d0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0002d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d700: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002d710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002d740: 6936 203a 2043 683d 546f 7269 6343 686f i6 : Ch=ToricCho │ │ │ │ -0002d750: 7752 696e 6728 5829 2020 2020 2020 2020 wRing(X) │ │ │ │ -0002d760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d780: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +0002d760: 203a 2043 683d 546f 7269 6343 686f 7752 : Ch=ToricChowR │ │ │ │ +0002d770: 696e 6728 5829 2020 2020 2020 2020 2020 ing(X) │ │ │ │ +0002d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d7a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d7d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d7e0: 6f36 203d 2043 6820 2020 2020 2020 2020 o6 = Ch │ │ │ │ -0002d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d7f0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0002d800: 203d 2043 6820 2020 2020 2020 2020 2020 = Ch │ │ │ │ 0002d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d840: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d870: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002d880: 6f36 203a 2051 756f 7469 656e 7452 696e o6 : QuotientRin │ │ │ │ -0002d890: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -0002d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d890: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0002d8a0: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ 0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 0a2b 2d2d |.+-- │ │ │ │ 0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d910: 2d2d 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207c 0a7c |.| │ │ │ │ -0002da60: 6f37 203d 202d 2d2d 2d2d 2d2d 2d2d 2d2d o7 = ----------- │ │ │ │ -0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 -------------- │ │ │ │ -0002da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daa0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002dab0: 2020 2020 2028 7820 7820 2c20 7820 7820 (x x , x x │ │ │ │ -0002dac0: 7820 2c20 7820 202d 2078 202c 2078 2020 x , x - x , x │ │ │ │ -0002dad0: 2d20 7820 2c20 7820 202d 2078 2029 2020 - x , x - x ) │ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002db00: 2020 2020 2020 2032 2034 2020 2030 2031 2 4 0 1 │ │ │ │ -0002db10: 2033 2020 2030 2020 2020 3320 2020 3120 3 0 3 1 │ │ │ │ -0002db20: 2020 2033 2020 2032 2020 2020 3420 2020 3 2 4 │ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db40: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002db50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da40: 2020 2020 2020 2030 2020 2034 2020 2020 0 4 │ │ │ │ +0002da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da70: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ +0002da80: 203d 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d = ------------- │ │ │ │ +0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 ------------ │ │ │ │ +0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dac0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002dad0: 2020 2028 7820 7820 2c20 7820 7820 7820 (x x , x x x │ │ │ │ +0002dae0: 2c20 7820 202d 2078 202c 2078 2020 2d20 , x - x , x - │ │ │ │ +0002daf0: 7820 2c20 7820 202d 2078 2029 2020 2020 x , x - x ) │ │ │ │ +0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002db20: 2020 2020 2032 2034 2020 2030 2031 2033 2 4 0 1 3 │ │ │ │ +0002db30: 2020 2030 2020 2020 3320 2020 3120 2020 0 3 1 │ │ │ │ +0002db40: 2033 2020 2032 2020 2020 3420 2020 2020 3 2 4 │ │ │ │ +0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002dba0: 6938 203a 2072 3d67 656e 7320 5220 2020 i8 : r=gens R │ │ │ │ -0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002dc90: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ -0002dca0: 2020 2033 2020 2034 2020 2020 2020 2020 3 4 │ │ │ │ -0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc50: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ +0002dc60: 203d 207b 7820 2c20 7820 2c20 7820 2c20 = {x , x , x , │ │ │ │ +0002dc70: 7820 2c20 7820 7d20 2020 2020 2020 2020 x , x } │ │ │ │ +0002dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dca0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002dcb0: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ +0002dcc0: 2033 2020 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +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 │ │ │ │ +0002dcf0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002dd30: 6f38 203a 204c 6973 7420 2020 2020 2020 o8 : List │ │ │ │ -0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ +0002dd50: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd70: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002dd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002ddd0: 6939 203a 2049 3d69 6465 616c 2872 616e i9 : I=ideal(ran │ │ │ │ -0002dde0: 646f 6d28 7b31 2c30 7d2c 5229 2920 2020 dom({1,0},R)) │ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ +0002ddf0: 203a 2049 3d69 6465 616c 2872 616e 646f : I=ideal(rando │ │ │ │ +0002de00: 6d28 7b31 2c30 7d2c 5229 2920 2020 2020 m({1,0},R)) │ │ │ │ +0002de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002de40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002de70: 6f39 203d 2069 6465 616c 2831 3037 7820 o9 = ideal(107x │ │ │ │ -0002de80: 202b 2034 3337 3678 2020 2d20 3633 3136 + 4376x - 6316 │ │ │ │ -0002de90: 7820 2920 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002deb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002dec0: 2020 2020 2020 2020 2020 2020 2020 2030 0 │ │ │ │ -0002ded0: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ -0002dee0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de80: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ +0002de90: 203d 2069 6465 616c 2831 3037 7820 202b = ideal(107x + │ │ │ │ +0002dea0: 2034 3337 3678 2020 2d20 3633 3136 7820 4376x - 6316x │ │ │ │ +0002deb0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ded0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2030 2020 0 │ │ │ │ +0002def0: 2020 2020 2020 3120 2020 2020 2020 2033 1 3 │ │ │ │ +0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002df30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002df60: 6f39 203a 2049 6465 616c 206f 6620 5220 o9 : Ideal of R │ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df70: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ +0002df80: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ 0002df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfa0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dfc0: 2d2d 2d2d 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-----------+.|i1 │ │ │ │ +0002e020: 3020 3a20 4b3d 6964 6561 6c28 7261 6e64 0 : K=ideal(rand │ │ │ │ +0002e030: 6f6d 287b 312c 317d 2c52 2929 2020 2020 om({1,1},R)) │ │ │ │ +0002e040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e060: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e090: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e0a0: 6f31 3020 3d20 6964 6561 6c28 3331 3837 o10 = ideal(3187 │ │ │ │ -0002e0b0: 7820 7820 202d 2036 3035 3378 2078 2020 x x - 6053x x │ │ │ │ -0002e0c0: 2d20 3136 3039 3078 2078 2020 2b20 3337 - 16090x x + 37 │ │ │ │ -0002e0d0: 3833 7820 7820 202b 2038 3537 3078 2078 83x x + 8570x x │ │ │ │ -0002e0e0: 2020 2b20 3834 3434 7820 7820 297c 0a7c + 8444x x )|.| │ │ │ │ -0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e100: 2030 2032 2020 2020 2020 2020 3120 3220 0 2 1 2 │ │ │ │ -0002e110: 2020 2020 2020 2020 3220 3320 2020 2020 2 3 │ │ │ │ -0002e120: 2020 2030 2034 2020 2020 2020 2020 3120 0 4 1 │ │ │ │ -0002e130: 3420 2020 2020 2020 2033 2034 207c 0a7c 4 3 4 |.| │ │ │ │ -0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e150: 2020 2020 2020 2020 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 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002e0c0: 3020 3d20 6964 6561 6c28 3331 3837 7820 0 = ideal(3187x │ │ │ │ +0002e0d0: 7820 202d 2036 3035 3378 2078 2020 2d20 x - 6053x x - │ │ │ │ +0002e0e0: 3136 3039 3078 2078 2020 2b20 3337 3833 16090x x + 3783 │ │ │ │ +0002e0f0: 7820 7820 202b 2038 3537 3078 2078 2020 x x + 8570x x │ │ │ │ +0002e100: 2b20 3834 3434 7820 7820 297c 0a7c 2020 + 8444x x )|.| │ │ │ │ +0002e110: 2020 2020 2020 2020 2020 2020 2020 2030 0 │ │ │ │ +0002e120: 2032 2020 2020 2020 2020 3120 3220 2020 2 1 2 │ │ │ │ +0002e130: 2020 2020 2020 3220 3320 2020 2020 2020 2 3 │ │ │ │ +0002e140: 2030 2034 2020 2020 2020 2020 3120 3420 0 4 1 4 │ │ │ │ +0002e150: 2020 2020 2020 2033 2034 207c 0a7c 2020 3 4 |.| │ │ │ │ 0002e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e180: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e190: 6f31 3020 3a20 4964 6561 6c20 6f66 2052 o10 : Ideal of R │ │ │ │ -0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002e1b0: 3020 3a20 4964 6561 6c20 6f66 2052 2020 0 : Ideal of R │ │ │ │ 0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1d0: 2020 2020 2020 2020 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│ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e2f0: 2020 2020 2020 3220 2020 2020 2020 3220 2 2 │ │ │ │ 0002e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e310: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e320: 6f31 3120 3d20 3478 2078 2020 2b20 3278 o11 = 4x x + 2x │ │ │ │ -0002e330: 2020 2b20 3278 2078 2020 2b20 7820 2020 + 2x x + x │ │ │ │ -0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e370: 2020 2020 2020 2020 3320 3420 2020 2020 3 4 │ │ │ │ -0002e380: 3320 2020 2020 3320 3420 2020 2033 2020 3 3 4 3 │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e330: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002e340: 3120 3d20 3478 2078 2020 2b20 3278 2020 1 = 4x x + 2x │ │ │ │ +0002e350: 2b20 3278 2078 2020 2b20 7820 2020 2020 + 2x x + x │ │ │ │ +0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e380: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e390: 2020 2020 2020 3320 3420 2020 2020 3320 3 4 3 │ │ │ │ +0002e3a0: 2020 2020 3320 3420 2020 2033 2020 2020 3 4 3 │ │ │ │ +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 2020 │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e400: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e410: 6f31 3120 3a20 4368 2020 2020 2020 2020 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2 │ │ │ │ 0002e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e810: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e820: 6f31 3320 3d20 2d20 7820 7820 202d 2033 o13 = - x x - 3 │ │ │ │ -0002e830: 7820 202d 2034 7820 7820 202b 2078 2020 x - 4x x + x │ │ │ │ -0002e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e860: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e870: 2020 2020 2020 2020 2033 2034 2020 2020 3 4 │ │ │ │ -0002e880: 2033 2020 2020 2033 2034 2020 2020 3420 3 3 4 4 │ │ │ │ -0002e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e830: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002e840: 3320 3d20 2d20 7820 7820 202d 2033 7820 3 = - x x - 3x │ │ │ │ +0002e850: 202d 2034 7820 7820 202b 2078 2020 2020 - 4x x + x │ │ │ │ +0002e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e880: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e890: 2020 2020 2020 2033 2034 2020 2020 2033 3 4 3 │ │ │ │ +0002e8a0: 2020 2020 2033 2034 2020 2020 3420 2020 3 4 4 │ │ │ │ +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 │ │ │ │ +0002e8d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e900: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e910: 6f31 3320 3a20 4368 2020 2020 2020 2020 o13 : Ch │ │ │ │ -0002e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e920: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002e930: 3320 3a20 4368 2020 2020 2020 2020 2020 3 : Ch │ │ │ │ 0002e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e950: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002e960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e970: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002e9b0: 6931 3420 3a20 732a 6320 2020 2020 2020 i14 : s*c │ │ │ │ -0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002e9d0: 3420 3a20 732a 6320 2020 2020 2020 2020 4 : s*c │ │ │ │ 0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002ea50: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ -0002ea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ea70: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ 0002ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eaa0: 6f31 3420 3d20 3278 2078 2020 2b20 7820 o14 = 2x x + x │ │ │ │ -0002eab0: 202b 2078 2078 2020 2020 2020 2020 2020 + x x │ │ │ │ -0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eae0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eaf0: 2020 2020 2020 2020 3320 3420 2020 2033 3 4 3 │ │ │ │ -0002eb00: 2020 2020 3320 3420 2020 2020 2020 2020 3 4 │ │ │ │ -0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eab0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002eac0: 3420 3d20 3278 2078 2020 2b20 7820 202b 4 = 2x x + x + │ │ │ │ +0002ead0: 2078 2078 2020 2020 2020 2020 2020 2020 x x │ │ │ │ +0002eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002eb10: 2020 2020 2020 3320 3420 2020 2033 2020 3 4 3 │ │ │ │ +0002eb20: 2020 3320 3420 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eb90: 6f31 3420 3a20 4368 2020 2020 2020 2020 o14 : Ch │ │ │ │ -0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 6f31 |.|o1 │ │ │ │ +0002ebb0: 3420 3a20 4368 2020 2020 2020 2020 2020 4 : Ch │ │ │ │ 0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebd0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 0a2b 2d2d |.+-- │ │ │ │ 0002ec00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0002ec30: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0002ec40: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0002ec50: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0002ec60: 7420 2a6e 6f74 6520 546f 7269 6343 686f t *note ToricCho │ │ │ │ -0002ec70: 7752 696e 673a 2054 6f72 6963 4368 6f77 wRing: ToricChow │ │ │ │ -0002ec80: 5269 6e67 2c20 6973 2061 202a 6e6f 7465 Ring, is a *note │ │ │ │ -0002ec90: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -0002eca0: 3a0a 284d 6163 6175 6c61 7932 446f 6329 :.(Macaulay2Doc) │ │ │ │ -0002ecb0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ -0002ecc0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0002ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 466f -----------+..Fo │ │ │ │ +0002ec50: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +0002ec60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0002ec70: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +0002ec80: 2a6e 6f74 6520 546f 7269 6343 686f 7752 *note ToricChowR │ │ │ │ +0002ec90: 696e 673a 2054 6f72 6963 4368 6f77 5269 ing: ToricChowRi │ │ │ │ +0002eca0: 6e67 2c20 6973 2061 202a 6e6f 7465 206d ng, is a *note m │ │ │ │ +0002ecb0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ +0002ecc0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0002ecd0: 7468 6f64 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6465 3a20 546f Table:.Node: To │ │ │ │ -0002edb0: 707f 3333 360a 4e6f 6465 3a20 6265 7274 p.336.Node: bert │ │ │ │ -0002edc0: 696e 6943 6865 636b 7f31 3539 3738 0a4e iniCheck.15978.N │ │ │ │ -0002edd0: 6f64 653a 2043 6865 636b 536d 6f6f 7468 ode: CheckSmooth │ │ │ │ -0002ede0: 7f31 3731 3935 0a4e 6f64 653a 2043 6865 .17195.Node: Che │ │ │ │ -0002edf0: 636b 546f 7269 6356 6172 6965 7479 5661 ckToricVarietyVa │ │ │ │ -0002ee00: 6c69 647f 3231 3939 310a 4e6f 6465 3a20 lid.21991.Node: │ │ │ │ -0002ee10: 4368 6572 6e7f 3332 3938 320a 4e6f 6465 Chern.32982.Node │ │ │ │ -0002ee20: 3a20 4368 6f77 5269 6e67 7f35 3034 3930 : ChowRing.50490 │ │ │ │ -0002ee30: 0a4e 6f64 653a 2043 6c61 7373 496e 4368 .Node: ClassInCh │ │ │ │ -0002ee40: 6f77 5269 6e67 7f35 3832 3231 0a4e 6f64 owRing.58221.Nod │ │ │ │ -0002ee50: 653a 2043 6c61 7373 496e 546f 7269 6343 e: ClassInToricC │ │ │ │ -0002ee60: 686f 7752 696e 677f 3630 3538 310a 4e6f howRing.60581.No │ │ │ │ -0002ee70: 6465 3a20 436f 6d70 4d65 7468 6f64 7f36 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3938 320a 4e6f 6465 3a20 ern.32982.Node: │ │ │ │ +0002ee40: 4368 6f77 5269 6e67 7f35 3034 3930 0a4e ChowRing.50490.N │ │ │ │ +0002ee50: 6f64 653a 2043 6c61 7373 496e 4368 6f77 ode: ClassInChow │ │ │ │ +0002ee60: 5269 6e67 7f35 3832 3231 0a4e 6f64 653a Ring.58221.Node: │ │ │ │ +0002ee70: 2043 6c61 7373 496e 546f 7269 6343 686f ClassInToricCho │ │ │ │ +0002ee80: 7752 696e 677f 3630 3538 310a 4e6f 6465 wRing.60581.Node │ │ │ │ +0002ee90: 3a20 436f 6d70 4d65 7468 6f64 7f36 3538 : CompMethod.658 │ │ │ │ +0002eea0: 3234 0a4e 6f64 653a 2063 6f6e 6669 6775 24.Node: configu │ │ │ │ +0002eeb0: 7269 6e67 2042 6572 7469 6e69 7f37 3537 ring Bertini.757 │ │ │ │ +0002eec0: 3531 0a4e 6f64 653a 2043 534d 7f37 3733 51.Node: CSM.773 │ │ │ │ +0002eed0: 3836 0a4e 6f64 653a 2045 756c 6572 7f31 86.Node: Euler.1 │ │ │ │ +0002eee0: 3031 3032 340a 4e6f 6465 3a20 496e 6473 01024.Node: Inds │ │ │ │ +0002eef0: 4f66 536d 6f6f 7468 7f31 3138 3930 320a OfSmooth.118902. │ │ │ │ +0002ef00: 4e6f 6465 3a20 496e 7075 7449 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│ │ │ 0001b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b340: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ -0001b350: 3320 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ +0001b340: 2020 2020 2020 2020 3220 2020 3220 2020 2 2 │ │ │ │ +0001b350: 2020 2020 2020 3320 2020 2020 2020 3220 3 2 │ │ │ │ 0001b360: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001b370: 2020 2020 2020 2020 2020 2020 506f 6c79 Poly │ │ │ │ -0001b380: 6e6f 6d69 616c 5365 7420 3d3e 207b 312c nomialSet => {1, │ │ │ │ -0001b390: 2074 202c 2074 2074 202c 2074 202c 2074 t , t t , t , t │ │ │ │ -0001b3a0: 202c 2074 202c 2074 202c 2074 2074 207d , t , t , t t } │ │ │ │ +0001b380: 6e6f 6d69 616c 5365 7420 3d3e 207b 7420 nomialSet => {t │ │ │ │ +0001b390: 7420 2c20 312c 2074 202c 2074 2074 202c t , 1, t , t t , │ │ │ │ +0001b3a0: 2074 202c 2074 202c 2074 202c 2074 207d t , t , t , t } │ │ │ │ 0001b3b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001b3c0: 2020 2020 2020 2020 2020 2020 2020 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2d2d ---------------- │ │ │ │ 0001c4e0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7350,41 +7350,41 @@ │ │ │ │ 0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cb70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cbc0: 2020 2020 207c 0a7c 312c 2061 2c20 6120 |.|1, a, a │ │ │ │ -0001cbd0: 2b20 312c 2031 2c20 307d 2c20 7b30 2c20 + 1, 1, 0}, {0, │ │ │ │ -0001cbe0: 6120 2b20 312c 2061 202b 2031 2c20 6120 a + 1, a + 1, a │ │ │ │ -0001cbf0: 2b20 312c 2061 2c20 612c 2061 2c20 317d + 1, a, a, a, 1} │ │ │ │ -0001cc00: 2c20 7b30 2c20 6120 2b20 312c 2031 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-0006ca90: 2c20 302c 2031 2c20 302c 2031 2c20 302c , 0, 1, 0, 1, 0, │ │ │ │ +0006ca80: 2031 2c20 302c 2030 2c20 307d 2c20 7b30 1, 0, 0, 0}, {0 │ │ │ │ +0006ca90: 2c20 312c 2031 2c20 302c 2031 2c20 302c , 1, 1, 0, 1, 0, │ │ │ │ 0006caa0: 2030 7d2c 207c 0a7c 2020 2020 2020 2020 0}, |.| │ │ │ │ 0006cab0: 2020 2020 2020 2020 5061 7269 7479 4368 ParityCh │ │ │ │ 0006cac0: 6563 6b4d 6174 7269 7820 3d3e 207c 2031 eckMatrix => | 1 │ │ │ │ 0006cad0: 2031 2031 2031 2030 2030 2030 207c 2020 1 1 1 0 0 0 | │ │ │ │ 0006cae0: 2020 2020 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 207c 2030 | 0 │ │ │ │ 0006cb20: 2030 2031 2031 2031 2031 2030 207c 2020 0 1 1 1 1 0 | │ │ │ │ 0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb60: 2020 2020 2020 2020 2020 2020 207c 2030 | 0 │ │ │ │ -0006cb70: 2031 2030 2031 2030 2031 2031 207c 2020 1 0 1 0 1 1 | │ │ │ │ +0006cb70: 2031 2030 2031 2031 2030 2031 207c 2020 1 0 1 1 0 1 | │ │ │ │ 0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006cba0: 2020 2020 2020 2020 5061 7269 7479 4368 ParityCh │ │ │ │ 0006cbb0: 6563 6b52 6f77 7320 3d3e 207b 7b31 2c20 eckRows => {{1, │ │ │ │ 0006cbc0: 312c 2031 2c20 312c 2030 2c20 302c 2030 1, 1, 1, 0, 0, 0 │ │ │ │ 0006cbd0: 7d2c 207b 302c 2030 2c20 312c 2031 2c20 }, {0, 0, 1, 1, │ │ │ │ 0006cbe0: 312c 2031 2c7c 0a7c 2020 2020 2020 2020 1, 1,|.| │ │ │ │ @@ -27917,15 +27917,15 @@ │ │ │ │ 0006d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d0e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d130: 2020 2020 207c 0a7c 7b30 2c20 312c 2031 |.|{0, 1, 1 │ │ │ │ +0006d130: 2020 2020 207c 0a7c 7b31 2c20 302c 2031 |.|{1, 0, 1 │ │ │ │ 0006d140: 2c20 302c 2030 2c20 312c 2030 7d2c 207b , 0, 0, 1, 0}, { │ │ │ │ 0006d150: 312c 2031 2c20 302c 2030 2c20 302c 2030 1, 1, 0, 0, 0, 0 │ │ │ │ 0006d160: 2c20 317d 7d20 2020 2020 2020 2020 2020 , 1}} │ │ │ │ 0006d170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d180: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -27938,15 +27938,15 @@ │ │ │ │ 0006d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d220: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006d230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d270: 2020 2020 207c 0a7c 2030 7d2c 207b 302c |.| 0}, {0, │ │ │ │ -0006d280: 2031 2c20 302c 2031 2c20 302c 2031 2c20 1, 0, 1, 0, 1, │ │ │ │ +0006d280: 2031 2c20 302c 2031 2c20 312c 2030 2c20 1, 0, 1, 1, 0, │ │ │ │ 0006d290: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ 0006d2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d2c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0006d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/CohomCalg.info.gz │ │ │ ├── CohomCalg.info │ │ │ │ @@ -1042,15 +1042,15 @@ │ │ │ │ 00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469 |i20 : elapsedTi │ │ │ │ 00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d me hvecs = cohom │ │ │ │ 00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020 Calg(X, D2) │ │ │ │ 00004160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00004180: 7c20 2d2d 2032 2e35 3837 3432 7320 656c | -- 2.58742s el │ │ │ │ +00004180: 7c20 2d2d 2033 2e32 3339 3835 7320 656c | -- 3.23985s 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 3333 3633 3733 7320 656c | -- .336373s el │ │ │ │ +00006930: 7c20 2d2d 202e 3534 3136 3033 7320 656c | -- .541603s el │ │ │ │ 00006940: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000069a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ 00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469 |i24 : elapsedTi │ │ │ │ 00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066 me cohomvec2 = f │ │ │ │ 00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064 or j from 0 to d │ │ │ │ 00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048 im X list rank H │ │ │ │ 00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a H^j(X, |. │ │ │ │ -00006b60: 7c20 2d2d 2031 302e 3234 3331 7320 656c | -- 10.2431s el │ │ │ │ +00006b60: 7c20 2d2d 2039 2e35 3732 3938 7320 656c | -- 9.57298s 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 │ │ │ │ @@ -1797,16 +1797,16 @@ │ │ │ │ 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 3330 3539 3034 7320 656c | -- .305904s el │ │ │ │ -000070c0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +000070b0: 7c20 2d2d 202e 3530 3139 3973 2065 6c61 | -- .50199s ela │ │ │ │ +000070c0: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 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 │ │ │ │ 00007130: 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 3636 3538 3331 7320 656c | -- .665831s el │ │ │ │ +000072e0: 7c20 2d2d 202e 3438 3433 3436 7320 656c | -- .484346s el │ │ │ │ 000072f0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007330: 7c20 2d2d 202e 3636 3538 3831 7320 656c | -- .665881s el │ │ │ │ +00007330: 7c20 2d2d 202e 3438 3433 3934 7320 656c | -- .484394s 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 │ │ │ │ @@ -4343,17 +4343,17 @@ │ │ │ │ 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: 2e36 3038 3434 7320 2863 7075 293b 2035 .60844s (cpu); 5 │ │ │ │ -00010ff0: 2e30 3137 3939 7320 2874 6872 6561 6429 .01799s (thread) │ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2037 |.| -- used 7 │ │ │ │ +00010fe0: 2e32 3836 3037 7320 2863 7075 293b 2035 .28607s (cpu); 5 │ │ │ │ +00010ff0: 2e35 3833 3634 7320 2874 6872 6561 6429 .58364s (thread) │ │ │ │ 00011000: 3b20 3073 2028 6763 2920 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 │ │ │ │ @@ -4884,17 +4884,17 @@ │ │ │ │ 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 3833 3639 3373 2028 6370 7529 3b20 .183693s (cpu); │ │ │ │ -000131b0: 302e 3039 3936 3230 3273 2028 7468 7265 0.0996202s (thre │ │ │ │ -000131c0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +000131a0: 2e32 3138 3837 3973 2028 6370 7529 3b20 .218879s (cpu); │ │ │ │ +000131b0: 302e 3133 3931 3437 7320 2874 6872 6561 0.139147s (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 │ │ │ │ 00013230: 2020 3138 2020 2020 2020 2033 3820 2020 18 38 │ │ │ │ @@ -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 3632 3136 3973 2028 6370 ed 0.962169s (cp │ │ │ │ -00013440: 7529 3b20 302e 3733 3330 3736 7320 2874 u); 0.733076s (t │ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00013430: 6564 2031 2e30 3338 3534 7320 2863 7075 ed 1.03854s (cpu │ │ │ │ +00013440: 293b 2030 2e38 3334 3332 3473 2028 7468 ); 0.834324s (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 │ │ │ │ @@ -4977,24033 +4977,24034 @@ │ │ │ │ 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 3939 3834 3634 7320 2863 sed 0.998464s (c │ │ │ │ -000137d0: 7075 293b 2030 2e37 3639 3239 3473 2028 pu); 0.769294s ( │ │ │ │ -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 3034 3539 3373 2028 6370 7529 d 1.04593s (cpu) │ │ │ │ +000137d0: 3b20 302e 3832 3239 3133 7320 2874 6872 ; 0.822913s (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: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 | {- │ │ │ │ -00015850: 347d 207c 2020 2020 2020 2020 2020 207c 4} | | │ │ │ │ -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 │ │ │ │ +00015860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015890: 2020 3020 2020 2020 2020 207c 2020 7b2d 0 | {- │ │ │ │ -000158a0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -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 | {- │ │ │ │ -000158f0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015880: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00015890: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | {-5} | │ │ │ │ +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 |.| --- │ │ │ │ +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 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 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| │ │ │ │ -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 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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 │ │ │ │ -00017570: 6f74 6965 6e74 206f 6620 5362 6172 2020 otient of Sbar │ │ │ │ +00017540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017550: 6f31 3920 3a20 5362 6172 2d6d 6f64 756c o19 : Sbar-modul │ │ │ │ +00017560: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +00017570: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 00017580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000175a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ -00017600: 2069 6465 616c 2070 7265 7365 6e74 6174 ideal presentat │ │ │ │ -00017610: 696f 6e20 7072 756e 6520 4848 5f31 2063 ion prune HH_1 c │ │ │ │ -00017620: 6f6d 706c 6578 7b64 7561 6c20 2853 6261 omplex{dual (Sba │ │ │ │ -00017630: 722a 2a64 3129 2c20 6475 616c 2853 6261 r**d1), dual(Sba │ │ │ │ -00017640: 722a 2a64 3029 7d20 7c0a 7c20 2020 2020 r**d0)} |.| │ │ │ │ +000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000175f0: 6932 3020 3a20 6964 6561 6c20 7072 6573 i20 : ideal pres │ │ │ │ +00017600: 656e 7461 7469 6f6e 2070 7275 6e65 2048 entation prune H │ │ │ │ +00017610: 485f 3120 636f 6d70 6c65 787b 6475 616c H_1 complex{dual │ │ │ │ +00017620: 2028 5362 6172 2a2a 6431 292c 2064 7561 (Sbar**d1), dua │ │ │ │ +00017630: 6c28 5362 6172 2a2a 6430 297d 207c 0a7c l(Sbar**d0)} |.| │ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017690: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -000176a0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017690: 6f32 3020 3d20 7472 7565 2020 2020 2020 o20 = true │ │ │ │ +000176a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176e0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ +000176d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 7c0a 7c3d 3d20 6964 --------|.|== id │ │ │ │ -00017740: 6561 6c20 7072 6573 656e 7461 7469 6f6e eal presentation │ │ │ │ -00017750: 204d 6261 7227 2020 2020 2020 2020 2020 Mbar' │ │ │ │ +00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00017730: 3d3d 2069 6465 616c 2070 7265 7365 6e74 == ideal present │ │ │ │ +00017740: 6174 696f 6e20 4d62 6172 2720 2020 2020 ation Mbar' │ │ │ │ +00017750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017770: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -000177e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -000177f0: 2a20 2a6e 6f74 6520 4578 743a 2028 4d61 * *note Ext: (Ma │ │ │ │ -00017800: 6361 756c 6179 3244 6f63 2945 7874 2c20 caulay2Doc)Ext, │ │ │ │ -00017810: 2d2d 2063 6f6d 7075 7465 2061 6e20 4578 -- compute an Ex │ │ │ │ -00017820: 7420 6d6f 6475 6c65 0a20 202a 202a 6e6f t module. * *no │ │ │ │ -00017830: 7465 206e 6577 4578 743a 206e 6577 4578 te newExt: newEx │ │ │ │ -00017840: 742c 202d 2d20 476c 6f62 616c 2045 7874 t, -- Global Ext │ │ │ │ -00017850: 2066 6f72 206d 6f64 756c 6573 206f 7665 for modules ove │ │ │ │ -00017860: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00017870: 2049 6e74 6572 7365 6374 696f 6e0a 2020 Intersection. │ │ │ │ -00017880: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f * *note makeHomo │ │ │ │ -00017890: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f topies: makeHomo │ │ │ │ -000178a0: 746f 7069 6573 2c20 2d2d 2072 6574 7572 topies, -- retur │ │ │ │ -000178b0: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ -000178c0: 6967 6865 720a 2020 2020 686f 6d6f 746f igher. homoto │ │ │ │ -000178d0: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ -000178e0: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ -000178f0: 6854 6f74 616c 3a0a 3d3d 3d3d 3d3d 3d3d hTotal:.======== │ │ │ │ -00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00017920: 4569 7365 6e62 7564 5368 616d 6173 6854 EisenbudShamashT │ │ │ │ -00017930: 6f74 616c 284d 6f64 756c 6529 220a 0a46 otal(Module)"..F │ │ │ │ -00017940: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00017950: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00017960: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00017970: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00017980: 6861 6d61 7368 546f 7461 6c3a 2045 6973 hamashTotal: Eis │ │ │ │ -00017990: 656e 6275 6453 6861 6d61 7368 546f 7461 enbudShamashTota │ │ │ │ -000179a0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d65 l, is a *note me │ │ │ │ -000179b0: 7468 6f64 0a66 756e 6374 696f 6e20 7769 thod.function wi │ │ │ │ -000179c0: 7468 206f 7074 696f 6e73 3a20 284d 6163 th options: (Mac │ │ │ │ -000179d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -000179e0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -000179f0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000177d0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000177e0: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +000177f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00017800: 4578 742c 202d 2d20 636f 6d70 7574 6520 Ext, -- compute │ │ │ │ +00017810: 616e 2045 7874 206d 6f64 756c 650a 2020 an Ext module. │ │ │ │ +00017820: 2a20 2a6e 6f74 6520 6e65 7745 7874 3a20 * *note newExt: │ │ │ │ +00017830: 6e65 7745 7874 2c20 2d2d 2047 6c6f 6261 newExt, -- Globa │ │ │ │ +00017840: 6c20 4578 7420 666f 7220 6d6f 6475 6c65 l Ext for module │ │ │ │ +00017850: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00017860: 650a 2020 2020 496e 7465 7273 6563 7469 e. Intersecti │ │ │ │ +00017870: 6f6e 0a20 202a 202a 6e6f 7465 206d 616b on. * *note mak │ │ │ │ +00017880: 6548 6f6d 6f74 6f70 6965 733a 206d 616b eHomotopies: mak │ │ │ │ +00017890: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20 eHomotopies, -- │ │ │ │ +000178a0: 7265 7475 726e 7320 6120 7379 7374 656d returns a system │ │ │ │ +000178b0: 206f 6620 6869 6768 6572 0a20 2020 2068 of higher. h │ │ │ │ +000178c0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320 omotopies..Ways │ │ │ │ +000178d0: 746f 2075 7365 2045 6973 656e 6275 6453 to use EisenbudS │ │ │ │ +000178e0: 6861 6d61 7368 546f 7461 6c3a 0a3d 3d3d hamashTotal:.=== │ │ │ │ +000178f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00017910: 2020 2a20 2245 6973 656e 6275 6453 6861 * "EisenbudSha │ │ │ │ +00017920: 6d61 7368 546f 7461 6c28 4d6f 6475 6c65 mashTotal(Module │ │ │ │ +00017930: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00017940: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00017950: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00017960: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00017970: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00017980: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00017990: 6854 6f74 616c 2c20 6973 2061 202a 6e6f hTotal, is a *no │ │ │ │ +000179a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ +000179b0: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ +000179c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000179d0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +000179e0: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00017a50: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00017a60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00017a70: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00017a80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00017a90: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ -00017aa0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00017ab0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00017ac0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00017ad0: 323a 3531 3639 3a30 2e0a 1f0a 4669 6c65 2:5169:0....File │ │ │ │ -00017ae0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -00017af0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00017b00: 732e 696e 666f 2c20 4e6f 6465 3a20 6576 s.info, Node: ev │ │ │ │ -00017b10: 656e 4578 744d 6f64 756c 652c 204e 6578 enExtModule, Nex │ │ │ │ -00017b20: 743a 2065 7870 6f2c 2050 7265 763a 2045 t: expo, Prev: E │ │ │ │ -00017b30: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00017b40: 7461 6c2c 2055 703a 2054 6f70 0a0a 6576 tal, Up: Top..ev │ │ │ │ -00017b50: 656e 4578 744d 6f64 756c 6520 2d2d 2065 enExtModule -- e │ │ │ │ -00017b60: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ -00017b70: 2a28 4d2c 6b29 206f 7665 7220 6120 636f *(M,k) over a co │ │ │ │ -00017b80: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ -00017b90: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -00017ba0: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -00017bb0: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ing.************ │ │ │ │ +00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00017a40: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00017a50: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00017a60: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00017a70: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00017a80: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ +00017a90: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00017aa0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00017ab0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00017ac0: 6f6e 732e 6d32 3a35 3136 393a 302e 0a1f ons.m2:5169:0... │ │ │ │ +00017ad0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00017ae0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00017af0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00017b00: 653a 2065 7665 6e45 7874 4d6f 6475 6c65 e: evenExtModule │ │ │ │ +00017b10: 2c20 4e65 7874 3a20 6578 706f 2c20 5072 , Next: expo, Pr │ │ │ │ +00017b20: 6576 3a20 4569 7365 6e62 7564 5368 616d ev: EisenbudSham │ │ │ │ +00017b30: 6173 6854 6f74 616c 2c20 5570 3a20 546f ashTotal, Up: To │ │ │ │ +00017b40: 700a 0a65 7665 6e45 7874 4d6f 6475 6c65 p..evenExtModule │ │ │ │ +00017b50: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ +00017b60: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ +00017b70: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ +00017b80: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00017b90: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00017ba0: 746f 7220 7269 6e67 0a2a 2a2a 2a2a 2a2a tor ring.******* │ │ │ │ +00017bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00017c20: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -00017c30: 203d 2065 7665 6e45 7874 4d6f 6475 6c65 = evenExtModule │ │ │ │ -00017c40: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ -00017c50: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00017c60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00017c70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00017c80: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00017c90: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ -00017ca0: 2020 2020 2020 7269 6e67 0a20 202a 202a ring. * * │ │ │ │ -00017cb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -00017cc0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00017cd0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00017ce0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00017cf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00017d00: 202a 204f 7574 5269 6e67 203d 3e20 2e2e * OutRing => .. │ │ │ │ -00017d10: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00017d20: 2030 0a20 202a 204f 7574 7075 7473 3a0a 0. * Outputs:. │ │ │ │ -00017d30: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ -00017d40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00017d50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00017d60: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d , over a polynom │ │ │ │ -00017d70: 6961 6c20 7269 6e67 2077 6974 680a 2020 ial ring with. │ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465 gens in de │ │ │ │ -00017d90: 6772 6565 2031 0a0a 4465 7363 7269 7074 gree 1..Descript │ │ │ │ -00017da0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00017db0: 0a45 7874 7261 6374 7320 7468 6520 6576 .Extracts the ev │ │ │ │ -00017dc0: 656e 2064 6567 7265 6520 7061 7274 2066 en degree part f │ │ │ │ -00017dd0: 726f 6d20 4578 744d 6f64 756c 6520 4d20 rom ExtModule M │ │ │ │ -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, │ │ │ │ -00017e30: 7468 656e 2074 6865 206f 7574 7075 7420 then the output │ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65 will be a module │ │ │ │ -00017e50: 0a6f 7665 7220 542e 0a0a 2b2d 2d2d 2d2d .over T...+----- │ │ │ │ +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 │ │ │ │ +00017c30: 6f64 756c 6520 4d0a 2020 2a20 496e 7075 odule M. * Inpu │ │ │ │ +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 │ │ │ │ +00017c70: 756c 652c 2c20 6f76 6572 2061 2063 6f6d ule,, over a com │ │ │ │ +00017c80: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00017c90: 6f6e 0a20 2020 2020 2020 2072 696e 670a on. ring. │ │ │ │ +00017ca0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +00017cb0: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ +00017cc0: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ +00017cd0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +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 │ │ │ │ +00017d20: 7574 733a 0a20 2020 2020 202a 2045 2c20 uts:. * E, │ │ │ │ +00017d30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00017d40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00017d50: 6475 6c65 2c2c 206f 7665 7220 6120 706f dule,, over a po │ │ │ │ +00017d60: 6c79 6e6f 6d69 616c 2072 696e 6720 7769 lynomial ring wi │ │ │ │ +00017d70: 7468 0a20 2020 2020 2020 2067 656e 7320 th. gens │ │ │ │ +00017d80: 696e 2064 6567 7265 6520 310a 0a44 6573 in degree 1..Des │ │ │ │ +00017d90: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00017da0: 3d3d 3d3d 0a0a 4578 7472 6163 7473 2074 ====..Extracts t │ │ │ │ +00017db0: 6865 2065 7665 6e20 6465 6772 6565 2070 he even degree p │ │ │ │ +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 3036 2b64 732f 4d32 2f4d -1.25.06+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 │ │ │ │ -00019390: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000193a0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -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 2e30 362b 6473 ulay2-1.25.06+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 ---------------- │ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d60: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030 -------|.| 0 │ │ │ │ -00019d70: 2c20 317d 2c20 7b30 2c20 322c 2030 7d2c , 1}, {0, 2, 0}, │ │ │ │ -00019d80: 207b 302c 2031 2c20 317d 2c20 7b33 2c20 {0, 1, 1}, {3, │ │ │ │ -00019d90: 302c 2030 7d2c 207b 322c 2031 2c20 307d 0, 0}, {2, 1, 0} │ │ │ │ -00019da0: 2c20 7b32 2c20 302c 2031 7d2c 207b 312c , {2, 0, 1}, {1, │ │ │ │ -00019db0: 2032 2c20 307d 2c7c 0a7c 2020 2020 202d 2, 0},|.| - │ │ │ │ +00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019d60: 2020 2020 302c 2031 7d2c 207b 302c 2032 0, 1}, {0, 2 │ │ │ │ +00019d70: 2c20 307d 2c20 7b30 2c20 312c 2031 7d2c , 0}, {0, 1, 1}, │ │ │ │ +00019d80: 207b 332c 2030 2c20 307d 2c20 7b32 2c20 {3, 0, 0}, {2, │ │ │ │ +00019d90: 312c 2030 7d2c 207b 322c 2030 2c20 317d 1, 0}, {2, 0, 1} │ │ │ │ +00019da0: 2c20 7b31 2c20 322c 2030 7d2c 7c0a 7c20 , {1, 2, 0},|.| │ │ │ │ +00019db0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e00: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b -------|.| { │ │ │ │ -00019e10: 312c 2031 2c20 317d 2c20 7b30 2c20 322c 1, 1, 1}, {0, 2, │ │ │ │ -00019e20: 2031 7d2c 207b 332c 2031 2c20 307d 2c20 1}, {3, 1, 0}, │ │ │ │ -00019e30: 7b33 2c20 302c 2031 7d2c 207b 322c 2032 {3, 0, 1}, {2, 2 │ │ │ │ -00019e40: 2c20 307d 2c20 7b32 2c20 312c 2031 7d2c , 0}, {2, 1, 1}, │ │ │ │ -00019e50: 207b 312c 2032 2c7c 0a7c 2020 2020 202d {1, 2,|.| - │ │ │ │ +00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019e00: 2020 2020 7b31 2c20 312c 2031 7d2c 207b {1, 1, 1}, { │ │ │ │ +00019e10: 302c 2032 2c20 317d 2c20 7b33 2c20 312c 0, 2, 1}, {3, 1, │ │ │ │ +00019e20: 2030 7d2c 207b 332c 2030 2c20 317d 2c20 0}, {3, 0, 1}, │ │ │ │ +00019e30: 7b32 2c20 322c 2030 7d2c 207b 322c 2031 {2, 2, 0}, {2, 1 │ │ │ │ +00019e40: 2c20 317d 2c20 7b31 2c20 322c 7c0a 7c20 , 1}, {1, 2,|.| │ │ │ │ +00019e50: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ -00019eb0: 7d2c 207b 332c 2032 2c20 307d 2c20 7b33 }, {3, 2, 0}, {3 │ │ │ │ -00019ec0: 2c20 312c 2031 7d2c 207b 322c 2032 2c20 , 1, 1}, {2, 2, │ │ │ │ -00019ed0: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019ea0: 2020 2020 317d 2c20 7b33 2c20 322c 2030 1}, {3, 2, 0 │ │ │ │ +00019eb0: 7d2c 207b 332c 2031 2c20 317d 2c20 7b32 }, {3, 1, 1}, {2 │ │ │ │ +00019ec0: 2c20 322c 2031 7d7d 2020 2020 2020 2020 , 2, 1}} │ │ │ │ +00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f40: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ -00019f50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +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 │ │ │ │ -00019f90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00019f80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -00019fe0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00019ff0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -0001a000: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -0001a010: 6861 6d61 7368 3a20 4569 7365 6e62 7564 hamash: Eisenbud │ │ │ │ -0001a020: 5368 616d 6173 682c 202d 2d20 436f 6d70 Shamash, -- Comp │ │ │ │ -0001a030: 7574 6573 2074 6865 2045 6973 656e 6275 utes the Eisenbu │ │ │ │ -0001a040: 642d 5368 616d 6173 680a 2020 2020 436f d-Shamash. Co │ │ │ │ -0001a050: 6d70 6c65 780a 2020 2a20 2a6e 6f74 6520 mplex. * *note │ │ │ │ -0001a060: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -0001a070: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -0001a080: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ -0001a090: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ -0001a0a0: 2020 686f 6d6f 746f 7069 6573 0a0a 5761 homotopies..Wa │ │ │ │ -0001a0b0: 7973 2074 6f20 7573 6520 6578 706f 3a0a ys to use expo:. │ │ │ │ -0001a0c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001a0d0: 3d0a 0a20 202a 2022 6578 706f 285a 5a2c =.. * "expo(ZZ, │ │ │ │ -0001a0e0: 4c69 7374 2922 0a20 202a 2022 6578 706f List)". * "expo │ │ │ │ -0001a0f0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ -0001a100: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0001a110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0001a120: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0001a130: 6520 6578 706f 3a20 6578 706f 2c20 6973 e expo: expo, is │ │ │ │ -0001a140: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0001a150: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0001a160: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0001a170: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00019fe0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00019ff0: 0a0a 2020 2a20 2a6e 6f74 6520 4569 7365 .. * *note Eise │ │ │ │ +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. * * │ │ │ │ +0001a050: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +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 │ │ │ │ -0001a1d0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0001a1e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0001a1f0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0001a200: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0001a210: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ -0001a220: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0001a230: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0001a240: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -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 │ │ │ │ -0001a280: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0001a290: 2065 7874 6572 696f 7245 7874 4d6f 6475 exteriorExtModu │ │ │ │ -0001a2a0: 6c65 2c20 4e65 7874 3a20 6578 7465 7269 le, Next: exteri │ │ │ │ -0001a2b0: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ 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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 │ │ │ │ -0001a510: 756c 6520 6f76 6572 2061 6e20 6578 7465 ule over an exte │ │ │ │ -0001a520: 7269 6f72 0a20 2020 2020 2020 2061 6c67 rior. alg │ │ │ │ -0001a530: 6562 7261 2077 6974 6820 7661 7269 6162 ebra with variab │ │ │ │ -0001a540: 6c65 7320 636f 7272 6573 706f 6e64 696e les correspondin │ │ │ │ -0001a550: 6720 746f 2065 6c65 6d65 6e74 7320 6f66 g to elements of │ │ │ │ -0001a560: 2066 0a0a 4465 7363 7269 7074 696f 6e0a f..Description. │ │ │ │ -0001a570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ -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 = │ │ │ │ -0001a5c0: 2028 665f 312e 2e66 5f63 292c 0a61 6e64 (f_1..f_c),.and │ │ │ │ -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 │ │ │ │ -0001a5f0: 656e 2074 6865 2073 6372 6970 7420 6578 en the script ex │ │ │ │ -0001a600: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001a610: 662c 4d29 2072 6574 7572 6e73 0a45 7874 f,M) returns.Ext │ │ │ │ -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 │ │ │ │ -0001a650: 3d20 6b3c 655f 312c 2e2e 2e2c 655f 633e = k │ │ │ │ -0001a660: 2c20 7768 6572 6520 7468 650a 655f 6920 , where the.e_i │ │ │ │ -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 │ │ │ │ -0001a730: 6520 6465 6772 6565 7320 7b64 5f69 2c31 e degrees {d_i,1 │ │ │ │ -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 │ │ │ │ -0001a770: 7275 6374 7572 652c 2069 6e20 6569 7468 ructure, in eith │ │ │ │ -0001a780: 6572 2063 6173 652c 2069 7320 6465 6669 er case, is defi │ │ │ │ -0001a790: 6e65 6420 6279 2074 6865 0a68 6f6d 6f74 ned by the.homot │ │ │ │ -0001a7a0: 6f70 6965 7320 666f 7220 7468 6520 665f opies for the f_ │ │ │ │ -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 │ │ │ │ -0001a7d0: 6564 2062 7920 7468 6520 7363 7269 7074 ed by the script │ │ │ │ -0001a7e0: 0a6d 616b 6548 6f6d 6f74 6f70 6965 7331 .makeHomotopies1 │ │ │ │ -0001a7f0: 2e54 6865 2073 6372 6970 7420 6361 6c6c .The script call │ │ │ │ -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 │ │ │ │ +0001a4a0: 6174 6564 2062 7920 7468 6520 656c 656d ated by the elem │ │ │ │ +0001a4b0: 656e 7473 0a20 2020 2020 2020 206f 6620 ents. of │ │ │ │ +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 │ │ │ │ +0001a4f0: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0001a500: 2c20 4d6f 6475 6c65 206f 7665 7220 616e , Module over an │ │ │ │ +0001a510: 2065 7874 6572 696f 720a 2020 2020 2020 exterior. │ │ │ │ +0001a520: 2020 616c 6765 6272 6120 7769 7468 2076 algebra with v │ │ │ │ +0001a530: 6172 6961 626c 6573 2063 6f72 7265 7370 ariables corresp │ │ │ │ +0001a540: 6f6e 6469 6e67 2074 6f20 656c 656d 656e onding to elemen │ │ │ │ +0001a550: 7473 206f 6620 660a 0a44 6573 6372 6970 ts of f..Descrip │ │ │ │ +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 │ │ │ │ +0001a680: 6564 2061 7320 7468 6520 452d 6475 616c ed as the E-dual │ │ │ │ +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 │ │ │ │ +0001a6d0: 726e 7320 4578 745f 5328 4d2c 4e29 2061 rns Ext_S(M,N) a │ │ │ │ +0001a6e0: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ +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_ │ │ │ │ +0001a720: 6920 6861 7665 2064 6567 7265 6573 207b i have degrees { │ │ │ │ +0001a730: 645f 692c 317d 2c20 7768 6572 6520 645f d_i,1}, where d_ │ │ │ │ +0001a740: 690a 6973 2074 6865 2064 6567 7265 6520 i.is the degree │ │ │ │ +0001a750: 6f66 2066 5f69 2e20 5468 6520 6d6f 6475 of f_i. The modu │ │ │ │ +0001a760: 6c65 2073 7472 7563 7475 7265 2c20 696e le structure, in │ │ │ │ +0001a770: 2065 6974 6865 7220 6361 7365 2c20 6973 either case, is │ │ │ │ +0001a780: 2064 6566 696e 6564 2062 7920 7468 650a defined by the. │ │ │ │ +0001a790: 686f 6d6f 746f 7069 6573 2066 6f72 2074 homotopies for t │ │ │ │ +0001a7a0: 6865 2066 5f69 206f 6e20 7468 6520 7265 he f_i on the re │ │ │ │ +0001a7b0: 736f 6c75 7469 6f6e 206f 6620 4d2c 2063 solution of M, c │ │ │ │ +0001a7c0: 6f6d 7075 7465 6420 6279 2074 6865 2073 omputed by the s │ │ │ │ +0001a7d0: 6372 6970 740a 6d61 6b65 486f 6d6f 746f cript.makeHomoto │ │ │ │ +0001a7e0: 7069 6573 312e 5468 6520 7363 7269 7074 pies1.The script │ │ │ │ +0001a7f0: 2063 616c 6c73 206d 616b 654d 6f64 756c calls makeModul │ │ │ │ +0001a800: 6520 746f 2063 6f6d 7075 7465 2061 2028 e to compute a ( │ │ │ │ +0001a810: 6e6f 6e2d 6d69 6e69 6d61 6c29 0a70 7265 non-minimal).pre │ │ │ │ +0001a820: 7365 6e74 6174 696f 6e20 6f66 2074 6869 sentation of thi │ │ │ │ +0001a830: 7320 6d6f 6475 6c65 2e0a 0a2b 2d2d 2d2d s module...+---- │ │ │ │ +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 ---------------- │ │ │ │ -0001a880: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a --+.|i1 : kk = Z │ │ │ │ -0001a890: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0001a870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ +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 | │ │ │ │ +0001a970: 0a7c 6f31 203a 2051 756f 7469 656e 7452 .|o1 : QuotientR │ │ │ │ +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 ; |.+-------- │ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ -0001c860: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ -0001c870: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -0001c880: 6520 7661 7269 6162 6c65 2873 2920 207c e variable(s) | │ │ │ │ -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 │ │ │ │ +0001cab0: 696e 6773 0a6d 7573 7420 636f 6e74 6169 ings.must contai │ │ │ │ +0001cac0: 6e20 6120 4465 6772 6565 4d61 7020 6f70 n a DegreeMap op │ │ │ │ +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 |. │ │ │ │ -0001cf60: 7c6f 3234 203d 2074 7275 6520 2020 2020 |o24 = true │ │ │ │ +0001cf50: 2020 207c 0a7c 6f32 3420 3d20 7472 7565 |.|o24 = true │ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cfa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001cfa0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0001cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -0001d000: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0001d010: 2a20 2a6e 6f74 6520 6578 7465 7269 6f72 * *note exterior │ │ │ │ -0001d020: 546f 724d 6f64 756c 653a 2065 7874 6572 TorModule: exter │ │ │ │ -0001d030: 696f 7254 6f72 4d6f 6475 6c65 2c20 2d2d iorTorModule, -- │ │ │ │ -0001d040: 2054 6f72 2061 7320 6120 6d6f 6475 6c65 Tor as a module │ │ │ │ -0001d050: 206f 7665 7220 616e 0a20 2020 2065 7874 over an. ext │ │ │ │ -0001d060: 6572 696f 7220 616c 6765 6272 6120 6f72 erior algebra or │ │ │ │ -0001d070: 2062 6967 7261 6465 6420 616c 6765 6272 bigraded algebr │ │ │ │ -0001d080: 610a 2020 2a20 2a6e 6f74 6520 6d61 6b65 a. * *note make │ │ │ │ -0001d090: 4d6f 6475 6c65 3a20 6d61 6b65 4d6f 6475 Module: makeModu │ │ │ │ -0001d0a0: 6c65 2c20 2d2d 206d 616b 6573 2061 204d le, -- makes a M │ │ │ │ -0001d0b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063 odule out of a c │ │ │ │ -0001d0c0: 6f6c 6c65 6374 696f 6e20 6f66 0a20 2020 ollection of. │ │ │ │ -0001d0d0: 206d 6f64 756c 6573 2061 6e64 206d 6170 modules and map │ │ │ │ -0001d0e0: 730a 0a57 6179 7320 746f 2075 7365 2065 s..Ways to use e │ │ │ │ -0001d0f0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ -0001d100: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -0001d110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d120: 0a0a 2020 2a20 2265 7874 6572 696f 7245 .. * "exteriorE │ │ │ │ -0001d130: 7874 4d6f 6475 6c65 284d 6174 7269 782c xtModule(Matrix, │ │ │ │ -0001d140: 4d6f 6475 6c65 2922 0a20 202a 2022 6578 Module)". * "ex │ │ │ │ -0001d150: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001d160: 4d61 7472 6978 2c4d 6f64 756c 652c 4d6f Matrix,Module,Mo │ │ │ │ -0001d170: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ -0001d180: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0001d1a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0001d1b0: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d1c0: 653a 2065 7874 6572 696f 7245 7874 4d6f e: exteriorExtMo │ │ │ │ -0001d1d0: 6475 6c65 2c20 6973 2061 202a 6e6f 7465 dule, is a *note │ │ │ │ -0001d1e0: 206d 6574 686f 640a 6675 6e63 7469 6f6e method.function │ │ │ │ -0001d1f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001d200: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ -0001d210: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0001cff0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0001d000: 3d0a 0a20 202a 202a 6e6f 7465 2065 7874 =.. * *note ext │ │ │ │ +0001d010: 6572 696f 7254 6f72 4d6f 6475 6c65 3a20 eriorTorModule: │ │ │ │ +0001d020: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001d030: 652c 202d 2d20 546f 7220 6173 2061 206d e, -- Tor as a m │ │ │ │ +0001d040: 6f64 756c 6520 6f76 6572 2061 6e0a 2020 odule over an. │ │ │ │ +0001d050: 2020 6578 7465 7269 6f72 2061 6c67 6562 exterior algeb │ │ │ │ +0001d060: 7261 206f 7220 6269 6772 6164 6564 2061 ra or bigraded a │ │ │ │ +0001d070: 6c67 6562 7261 0a20 202a 202a 6e6f 7465 lgebra. * *note │ │ │ │ +0001d080: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ +0001d090: 654d 6f64 756c 652c 202d 2d20 6d61 6b65 eModule, -- make │ │ │ │ +0001d0a0: 7320 6120 4d6f 6475 6c65 206f 7574 206f s a Module out o │ │ │ │ +0001d0b0: 6620 6120 636f 6c6c 6563 7469 6f6e 206f f a collection o │ │ │ │ +0001d0c0: 660a 2020 2020 6d6f 6475 6c65 7320 616e f. modules an │ │ │ │ +0001d0d0: 6420 6d61 7073 0a0a 5761 7973 2074 6f20 d maps..Ways to │ │ │ │ +0001d0e0: 7573 6520 6578 7465 7269 6f72 4578 744d use exteriorExtM │ │ │ │ +0001d0f0: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d odule:.========= │ │ │ │ +0001d100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d110: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d120: 7269 6f72 4578 744d 6f64 756c 6528 4d61 riorExtModule(Ma │ │ │ │ +0001d130: 7472 6978 2c4d 6f64 756c 6529 220a 2020 trix,Module)". │ │ │ │ +0001d140: 2a20 2265 7874 6572 696f 7245 7874 4d6f * "exteriorExtMo │ │ │ │ +0001d150: 6475 6c65 284d 6174 7269 782c 4d6f 6475 dule(Matrix,Modu │ │ │ │ +0001d160: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ +0001d170: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0001d180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d190: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0001d1a0: 6e6f 7465 2065 7874 6572 696f 7245 7874 note exteriorExt │ │ │ │ +0001d1b0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d1c0: 4578 744d 6f64 756c 652c 2069 7320 6120 ExtModule, is a │ │ │ │ +0001d1d0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ +0001d1e0: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ +0001d1f0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0001d200: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0001d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d260: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0001d270: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0001d280: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0001d290: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0001d2a0: 6361 756c 6179 322d 312e 3235 2e30 362b caulay2-1.25.06+ │ │ │ │ -0001d2b0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0001d2c0: 7061 636b 6167 6573 2f0a 436f 6d70 6c65 packages/.Comple │ │ │ │ -0001d2d0: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -0001d2e0: 736f 6c75 7469 6f6e 732e 6d32 3a34 3236 solutions.m2:426 │ │ │ │ -0001d2f0: 333a 302e 0a1f 0a46 696c 653a 2043 6f6d 3:0....File: Com │ │ │ │ -0001d300: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -0001d310: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66 nResolutions.inf │ │ │ │ -0001d320: 6f2c 204e 6f64 653a 2065 7874 6572 696f o, Node: exterio │ │ │ │ -0001d330: 7248 6f6d 6f6c 6f67 794d 6f64 756c 652c rHomologyModule, │ │ │ │ -0001d340: 204e 6578 743a 2065 7874 6572 696f 7254 Next: exteriorT │ │ │ │ -0001d350: 6f72 4d6f 6475 6c65 2c20 5072 6576 3a20 orModule, Prev: │ │ │ │ -0001d360: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d370: 652c 2055 703a 2054 6f70 0a0a 6578 7465 e, Up: Top..exte │ │ │ │ -0001d380: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ -0001d390: 6c65 202d 2d20 4d61 6b65 2074 6865 2068 le -- Make the h │ │ │ │ -0001d3a0: 6f6d 6f6c 6f67 7920 6f66 2061 2063 6f6d omology of a com │ │ │ │ -0001d3b0: 706c 6578 2069 6e74 6f20 6120 6d6f 6475 plex into a modu │ │ │ │ -0001d3c0: 6c65 206f 7665 7220 616e 2065 7874 6572 le over an exter │ │ │ │ -0001d3d0: 696f 7220 616c 6765 6272 610a 2a2a 2a2a ior algebra.**** │ │ │ │ +0001d250: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0001d260: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0001d270: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0001d280: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0001d290: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0001d2a0: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ +0001d2b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ +0001d2c0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0001d2d0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ +0001d2e0: 323a 3432 3633 3a30 2e0a 1f0a 4669 6c65 2:4263:0....File │ │ │ │ +0001d2f0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ +0001d300: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +0001d310: 732e 696e 666f 2c20 4e6f 6465 3a20 6578 s.info, Node: ex │ │ │ │ +0001d320: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f teriorHomologyMo │ │ │ │ +0001d330: 6475 6c65 2c20 4e65 7874 3a20 6578 7465 dule, Next: exte │ │ │ │ +0001d340: 7269 6f72 546f 724d 6f64 756c 652c 2050 riorTorModule, P │ │ │ │ +0001d350: 7265 763a 2065 7874 6572 696f 7245 7874 rev: exteriorExt │ │ │ │ +0001d360: 4d6f 6475 6c65 2c20 5570 3a20 546f 700a Module, Up: Top. │ │ │ │ +0001d370: 0a65 7874 6572 696f 7248 6f6d 6f6c 6f67 .exteriorHomolog │ │ │ │ +0001d380: 794d 6f64 756c 6520 2d2d 204d 616b 6520 yModule -- Make │ │ │ │ +0001d390: 7468 6520 686f 6d6f 6c6f 6779 206f 6620 the homology of │ │ │ │ +0001d3a0: 6120 636f 6d70 6c65 7820 696e 746f 2061 a complex into a │ │ │ │ +0001d3b0: 206d 6f64 756c 6520 6f76 6572 2061 6e20 module over an │ │ │ │ +0001d3c0: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ +0001d3d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 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 **************** │ │ │ │ 0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001d440: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001d450: 204d 203d 2065 7874 6572 696f 7248 6f6d M = exteriorHom │ │ │ │ -0001d460: 6f6c 6f67 794d 6f64 756c 6528 6666 2c20 ologyModule(ff, │ │ │ │ -0001d470: 4329 0a20 202a 2049 6e70 7574 733a 0a20 C). * Inputs:. │ │ │ │ -0001d480: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -0001d490: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -0001d4a0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -0001d4b0: 2c20 4d61 7472 6978 206f 6620 656c 656d , Matrix of elem │ │ │ │ -0001d4c0: 656e 7473 2074 6861 7420 6172 650a 2020 ents that are. │ │ │ │ -0001d4d0: 2020 2020 2020 686f 6d6f 746f 7069 6320 homotopic │ │ │ │ -0001d4e0: 746f 2030 206f 6e20 430a 2020 2020 2020 to 0 on C. │ │ │ │ -0001d4f0: 2a20 432c 2061 202a 6e6f 7465 2063 6f6d * C, a *note com │ │ │ │ -0001d500: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0001d510: 2943 6f6d 706c 6578 2c2c 200a 2020 2a20 )Complex,, . * │ │ │ │ -0001d520: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0001d530: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0001d540: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0001d550: 6329 4d6f 6475 6c65 2c2c 200a 0a44 6573 c)Module,, ..Des │ │ │ │ -0001d560: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0001d570: 3d3d 3d3d 0a0a 4173 7375 6d69 6e67 2074 ====..Assuming t │ │ │ │ -0001d580: 6861 7420 7468 6520 656c 656d 656e 7473 hat the elements │ │ │ │ -0001d590: 206f 6620 7468 6520 3178 6320 6d61 7472 of the 1xc matr │ │ │ │ -0001d5a0: 6978 2066 6620 6172 6520 6e75 6c6c 2d68 ix ff are null-h │ │ │ │ -0001d5b0: 6f6d 6f74 6f70 6963 206f 6e20 432c 2074 omotopic on C, t │ │ │ │ -0001d5c0: 6865 0a73 6372 6970 7420 7265 7475 726e he.script return │ │ │ │ -0001d5d0: 7320 7468 6520 6469 7265 6374 2073 756d s the direct sum │ │ │ │ -0001d5e0: 206f 6620 7468 6520 686f 6d6f 6c6f 6779 of the homology │ │ │ │ -0001d5f0: 206f 6620 4320 6173 2061 206d 6f64 756c of C as a modul │ │ │ │ -0001d600: 6520 6f76 6572 2061 206e 6577 2072 696e e over a new rin │ │ │ │ -0001d610: 672c 0a63 6f6e 7369 7374 696e 6720 6f66 g,.consisting of │ │ │ │ -0001d620: 2072 696e 6720 4320 7769 7468 2063 2065 ring C with c e │ │ │ │ -0001d630: 7874 6572 696f 7220 7661 7269 6162 6c65 xterior variable │ │ │ │ -0001d640: 7320 6164 6a6f 696e 6564 2e20 5468 6520 s adjoined. The │ │ │ │ -0001d650: 7363 7269 7074 2069 7320 7468 6520 6d61 script is the ma │ │ │ │ -0001d660: 696e 0a63 6f6d 706f 6e65 6e74 206f 6620 in.component of │ │ │ │ -0001d670: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d680: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d e..See also.==== │ │ │ │ -0001d690: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0001d6a0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d6b0: 653a 2065 7874 6572 696f 7254 6f72 4d6f e: exteriorTorMo │ │ │ │ -0001d6c0: 6475 6c65 2c20 2d2d 2054 6f72 2061 7320 dule, -- Tor as │ │ │ │ -0001d6d0: 6120 6d6f 6475 6c65 206f 7665 7220 616e a module over an │ │ │ │ -0001d6e0: 0a20 2020 2065 7874 6572 696f 7220 616c . exterior al │ │ │ │ -0001d6f0: 6765 6272 6120 6f72 2062 6967 7261 6465 gebra or bigrade │ │ │ │ -0001d700: 6420 616c 6765 6272 610a 2020 2a20 2a6e d algebra. * *n │ │ │ │ -0001d710: 6f74 6520 6d61 6b65 486f 6d6f 746f 7069 ote makeHomotopi │ │ │ │ -0001d720: 6573 4f6e 486f 6d6f 6c6f 6779 3a20 6d61 esOnHomology: ma │ │ │ │ -0001d730: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f keHomotopiesOnHo │ │ │ │ -0001d740: 6d6f 6c6f 6779 2c20 2d2d 2048 6f6d 6f6c mology, -- Homol │ │ │ │ -0001d750: 6f67 7920 6f66 2061 0a20 2020 2063 6f6d ogy of a. com │ │ │ │ -0001d760: 706c 6578 2061 7320 6578 7465 7269 6f72 plex as exterior │ │ │ │ -0001d770: 206d 6f64 756c 650a 0a57 6179 7320 746f module..Ways to │ │ │ │ -0001d780: 2075 7365 2065 7874 6572 696f 7248 6f6d use exteriorHom │ │ │ │ -0001d790: 6f6c 6f67 794d 6f64 756c 653a 0a3d 3d3d ologyModule:.=== │ │ │ │ +0001d430: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0001d440: 2020 2020 2020 4d20 3d20 6578 7465 7269 M = exteri │ │ │ │ +0001d450: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ +0001d460: 2866 662c 2043 290a 2020 2a20 496e 7075 (ff, C). * Inpu │ │ │ │ +0001d470: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ +0001d480: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +0001d490: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +0001d4a0: 7472 6978 2c2c 204d 6174 7269 7820 6f66 trix,, Matrix of │ │ │ │ +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 │ │ │ │ +0001d500: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +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 │ │ │ │ +0001d580: 6d65 6e74 7320 6f66 2074 6865 2031 7863 ments of the 1xc │ │ │ │ +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 │ │ │ │ +0001d5e0: 6f6c 6f67 7920 6f66 2043 2061 7320 6120 ology of C as a │ │ │ │ +0001d5f0: 6d6f 6475 6c65 206f 7665 7220 6120 6e65 module over a ne │ │ │ │ +0001d600: 7720 7269 6e67 2c0a 636f 6e73 6973 7469 w ring,.consisti │ │ │ │ +0001d610: 6e67 206f 6620 7269 6e67 2043 2077 6974 ng of ring C wit │ │ │ │ +0001d620: 6820 6320 6578 7465 7269 6f72 2076 6172 h c exterior var │ │ │ │ +0001d630: 6961 626c 6573 2061 646a 6f69 6e65 642e iables adjoined. │ │ │ │ +0001d640: 2054 6865 2073 6372 6970 7420 6973 2074 The script is t │ │ │ │ +0001d650: 6865 206d 6169 6e0a 636f 6d70 6f6e 656e he main.componen │ │ │ │ +0001d660: 7420 6f66 2065 7874 6572 696f 7254 6f72 t of exteriorTor │ │ │ │ +0001d670: 4d6f 6475 6c65 0a0a 5365 6520 616c 736f Module..See also │ │ │ │ +0001d680: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0001d690: 6e6f 7465 2065 7874 6572 696f 7254 6f72 note exteriorTor │ │ │ │ +0001d6a0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d6b0: 546f 724d 6f64 756c 652c 202d 2d20 546f TorModule, -- To │ │ │ │ +0001d6c0: 7220 6173 2061 206d 6f64 756c 6520 6f76 r as a module ov │ │ │ │ +0001d6d0: 6572 2061 6e0a 2020 2020 6578 7465 7269 er an. exteri │ │ │ │ +0001d6e0: 6f72 2061 6c67 6562 7261 206f 7220 6269 or algebra or bi │ │ │ │ +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 │ │ │ │ +0001d770: 7973 2074 6f20 7573 6520 6578 7465 7269 ys to use exteri │ │ │ │ +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 ================ │ │ │ │ -0001d7c0: 0a0a 2020 2a20 2265 7874 6572 696f 7248 .. * "exteriorH │ │ │ │ -0001d7d0: 6f6d 6f6c 6f67 794d 6f64 756c 6528 4d61 omologyModule(Ma │ │ │ │ -0001d7e0: 7472 6978 2c43 6f6d 706c 6578 2922 0a0a trix,Complex)".. │ │ │ │ -0001d7f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0001d800: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -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 │ │ │ │ -0001d860: 7465 0a6d 6574 686f 6420 6675 6e63 7469 te.method functi │ │ │ │ -0001d870: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -0001d880: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0001d890: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0001d7b0: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d7c0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ +0001d7d0: 6c65 284d 6174 7269 782c 436f 6d70 6c65 le(Matrix,Comple │ │ │ │ +0001d7e0: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ +0001d7f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0001d800: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001d810: 6f62 6a65 6374 202a 6e6f 7465 2065 7874 object *note ext │ │ │ │ +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 2e30 macaulay2-1.25.0 │ │ │ │ -0001d930: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+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 3036 2b64 732f 4d32 2f4d 6163 .25.06+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 │ │ │ │ -0001df30: 6973 0a6d 6f64 756c 652e 0a0a 4672 6f6d is.module...From │ │ │ │ -0001df40: 2074 6865 2064 6573 6372 6970 7469 6f6e the description │ │ │ │ -0001df50: 2062 7920 6d61 7472 6978 2066 6163 746f by matrix facto │ │ │ │ -0001df60: 7269 7a61 7469 6f6e 7320 616e 6420 7468 rizations and th │ │ │ │ -0001df70: 6520 7061 7065 7220 2254 6f72 2061 7320 e paper "Tor as │ │ │ │ -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 │ │ │ │ -0001e150: 7320 6765 6e65 7261 7465 6420 696e 2064 s generated in d │ │ │ │ -0001e160: 6567 7265 6573 2030 2061 6e64 2031 2c20 egrees 0 and 1, │ │ │ │ -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 │ │ │ │ +0001de40: 7265 6520 6f66 2066 5f69 2e20 5468 6520 ree of f_i. The │ │ │ │ +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 │ │ │ │ +0001de90: 6f72 2074 6865 2066 5f69 206f 6e20 7468 or the f_i on th │ │ │ │ +0001dea0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +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 │ │ │ │ +0001df00: 7465 2061 2028 6e6f 6e2d 6d69 6e69 6d61 te a (non-minima │ │ │ │ +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 │ │ │ │ +0001dfd0: 7320 6120 6869 6768 2073 797a 7967 7920 s a high syzygy │ │ │ │ +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 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+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 3036 2b64 732f 4d32 y2-1.25.06+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 │ │ │ │ +00021050: 6f20 7573 6520 6578 7449 734f 6e65 506f o use extIsOnePo │ │ │ │ +00021060: 6c79 6e6f 6d69 616c 3a0a 3d3d 3d3d 3d3d lynomial:.====== │ │ │ │ +00021070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021080: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00021090: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69 extIsOnePolynomi │ │ │ │ +000210a0: 616c 284d 6f64 756c 6529 220a 0a46 6f72 al(Module)"..For │ │ │ │ +000210b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +000210c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000210d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +000210e0: 6e6f 7465 2065 7874 4973 4f6e 6550 6f6c note extIsOnePol │ │ │ │ +000210f0: 796e 6f6d 6961 6c3a 2065 7874 4973 4f6e ynomial: extIsOn │ │ │ │ +00021100: 6550 6f6c 796e 6f6d 6961 6c2c 2069 7320 ePolynomial, is │ │ │ │ +00021110: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00021120: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +00021130: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00021140: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -000211b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -000211c0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -000211d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -000211e0: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ -000211f0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00021200: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -00021210: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -00021220: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ -00021230: 3933 313a 302e 0a1f 0a46 696c 653a 2043 931:0....File: C │ │ │ │ -00021240: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00021250: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -00021260: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f nfo, Node: ExtMo │ │ │ │ -00021270: 6475 6c65 2c20 4e65 7874 3a20 4578 744d dule, Next: ExtM │ │ │ │ -00021280: 6f64 756c 6544 6174 612c 2050 7265 763a oduleData, Prev: │ │ │ │ -00021290: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -000212a0: 6961 6c2c 2055 703a 2054 6f70 0a0a 4578 ial, Up: Top..Ex │ │ │ │ -000212b0: 744d 6f64 756c 6520 2d2d 2045 7874 5e2a tModule -- Ext^* │ │ │ │ -000212c0: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d (M,k) over a com │ │ │ │ -000212d0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -000212e0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -000212f0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -00021300: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +00021190: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000211a0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000211b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000211c0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +000211d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +000211e0: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ +000211f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00021200: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +00021210: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +00021220: 2e6d 323a 3439 3331 3a30 2e0a 1f0a 4669 .m2:4931:0....Fi │ │ │ │ +00021230: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +00021240: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00021250: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +00021260: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ +00021270: 2045 7874 4d6f 6475 6c65 4461 7461 2c20 ExtModuleData, │ │ │ │ +00021280: 5072 6576 3a20 6578 7449 734f 6e65 506f Prev: extIsOnePo │ │ │ │ +00021290: 6c79 6e6f 6d69 616c 2c20 5570 3a20 546f lynomial, Up: To │ │ │ │ +000212a0: 700a 0a45 7874 4d6f 6475 6c65 202d 2d20 p..ExtModule -- │ │ │ │ +000212b0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ +000212c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ +000212d0: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +000212e0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +000212f0: 6f72 2072 696e 670a 2a2a 2a2a 2a2a 2a2a or ring.******** │ │ │ │ +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 │ │ │ │ -00021360: 6765 3a20 0a20 2020 2020 2020 2045 203d ge: . E = │ │ │ │ -00021370: 2045 7874 4d6f 6475 6c65 204d 0a20 202a ExtModule M. * │ │ │ │ -00021380: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00021390: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -000213a0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -000213b0: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220 c)Module,, over │ │ │ │ -000213c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ -000213d0: 7365 6374 696f 6e0a 2020 2020 2020 2020 section. │ │ │ │ -000213e0: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ -000213f0: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -00021400: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00021410: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00021420: 652c 2c20 6f76 6572 2061 2070 6f6c 796e e,, over a polyn │ │ │ │ -00021430: 6f6d 6961 6c20 7269 6e67 2077 6974 680a omial ring with. │ │ │ │ -00021440: 2020 2020 2020 2020 6765 6e73 2069 6e20 gens in │ │ │ │ -00021450: 6576 656e 2064 6567 7265 650a 0a44 6573 even degree..Des │ │ │ │ -00021460: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00021470: 3d3d 3d3d 0a0a 5573 6573 2063 6f64 6520 ====..Uses code │ │ │ │ -00021480: 6f66 2041 7672 616d 6f76 2d47 7261 7973 of Avramov-Grays │ │ │ │ -00021490: 6f6e 2064 6573 6372 6962 6564 2069 6e20 on described in │ │ │ │ -000214a0: 4d61 6361 756c 6179 3220 626f 6f6b 0a0a Macaulay2 book.. │ │ │ │ -000214b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00021350: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00021360: 2020 4520 3d20 4578 744d 6f64 756c 6520 E = ExtModule │ │ │ │ +00021370: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ +00021380: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00021390: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +000213a0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +000213b0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ +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 │ │ │ │ +00021400: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021410: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120 Module,, over a │ │ │ │ +00021420: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +00021430: 7769 7468 0a20 2020 2020 2020 2067 656e with. gen │ │ │ │ +00021440: 7320 696e 2065 7665 6e20 6465 6772 6565 s in even degree │ │ │ │ +00021450: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00021460: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320 =========..Uses │ │ │ │ +00021470: 636f 6465 206f 6620 4176 7261 6d6f 762d code of Avramov- │ │ │ │ +00021480: 4772 6179 736f 6e20 6465 7363 7269 6265 Grayson describe │ │ │ │ +00021490: 6420 696e 204d 6163 6175 6c61 7932 2062 d in Macaulay2 b │ │ │ │ +000214a0: 6f6f 6b0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ook..+---------- │ │ │ │ +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: 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--------------+. │ │ │ │ +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 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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 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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 │ │ │ │ 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│ -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 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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 3036 2b64 732f lay2-1.25.06+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 │ │ │ │ -00024150: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -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 2e30 macaulay2-1.25.0 │ │ │ │ +000240e0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+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 │ │ │ │ -00024440: 6567 7265 6520 3e3d 2030 206f 7665 7220 egree >= 0 over │ │ │ │ -00024450: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ -00024460: 6720 5420 7769 7468 2067 656e 6572 6174 g T with generat │ │ │ │ -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 │ │ │ │ -00024570: 756c 6172 6974 7920 6576 656e 4578 744d ularity evenExtM │ │ │ │ -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 │ │ │ │ -00024610: 6865 2072 6567 756c 6172 6974 6965 7320 he regularities │ │ │ │ -00024620: 6f66 2074 6865 2065 7665 6e20 616e 6420 of the even and │ │ │ │ -00024630: 6f64 6420 4578 7420 6d6f 6475 6c65 7320 odd Ext modules │ │ │ │ -00024640: 2a63 616e 2a20 6469 6666 6572 2062 7920 *can* differ by │ │ │ │ -00024650: 6d6f 7265 2074 6861 6e20 312e 0a41 6e20 more than 1..An │ │ │ │ -00024660: 6578 616d 706c 6520 6361 6e20 6265 2070 example can be p │ │ │ │ -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 │ │ │ │ +00024470: 6565 2031 2e0a 0a54 6865 2073 6372 6970 ee 1...The scrip │ │ │ │ +00024480: 7420 7265 7475 726e 730a 0a4c 203d 205c t returns..L = \ │ │ │ │ +00024490: 7b45 302c 4531 2c20 7265 6775 6c61 7269 {E0,E1, regulari │ │ │ │ +000244a0: 7479 2045 302c 2072 6567 756c 6172 6974 ty E0, regularit │ │ │ │ +000244b0: 7920 4531 5c7d 0a0a 616e 6420 7072 696e y E1\}..and prin │ │ │ │ +000244c0: 7473 2061 206d 6573 7361 6765 2069 6620 ts a message if │ │ │ │ +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 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+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 │ │ │ │ -00025a70: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -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 │ │ │ │ -00025b00: 2065 7874 6572 696f 7220 6d6f 6475 6c65 exterior module │ │ │ │ -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 2e30 362b 6473 2f4d 322f 2-1.25.06+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 │ │ │ │ +00025b10: 6162 6c65 206f 6620 7368 6561 6620 4578 able of sheaf Ex │ │ │ │ +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 │ │ │ │ -00025d20: 6f72 2061 7320 6578 7465 7269 6f72 0a20 or as exterior. │ │ │ │ -00025d30: 2020 2020 2020 206d 6f64 756c 6573 0a0a modules.. │ │ │ │ -00025d40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00025d50: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2061 =======..Given a │ │ │ │ -00025d60: 206d 6174 7269 7820 6666 2063 6f6e 7461 matrix ff conta │ │ │ │ -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 │ │ │ │ -00025d90: 6c79 6e6f 6d69 616c 2072 696e 6720 5320 lynomial ring S │ │ │ │ -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 3036 2b64 732f 4d32 2f4d 6163 .25.06+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 2e30 362b 6473 2f4d ay2-1.25.06+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 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-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 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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 │ │ │ │ -00028650: 7465 4265 7474 694e 756d 6265 7273 284c teBettiNumbers(L │ │ │ │ -00028660: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ -00028670: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00028680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00028690: 6520 6f62 6a65 6374 202a 6e6f 7465 2066 e object *note f │ │ │ │ -000286a0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -000286b0: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ -000286c0: 6d62 6572 732c 2069 7320 6120 2a6e 6f74 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│ │ │ +00028580: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +00028590: 6572 733a 2069 6e66 696e 6974 6542 6574 ers: infiniteBet │ │ │ │ +000285a0: 7469 4e75 6d62 6572 732c 202d 2d20 6265 tiNumbers, -- be │ │ │ │ +000285b0: 7474 6920 6e75 6d62 6572 7320 6f66 0a20 tti numbers of. │ │ │ │ +000285c0: 2020 2066 696e 6974 6520 7265 736f 6c75 finite resolu │ │ │ │ +000285d0: 7469 6f6e 2063 6f6d 7075 7465 6420 6672 tion computed fr │ │ │ │ +000285e0: 6f6d 2061 206d 6174 7269 7820 6661 6374 om a matrix fact │ │ │ │ +000285f0: 6f72 697a 6174 696f 6e0a 0a57 6179 7320 orization..Ways │ │ │ │ +00028600: 746f 2075 7365 2066 696e 6974 6542 6574 to use finiteBet │ │ │ │ +00028610: 7469 4e75 6d62 6572 733a 0a3d 3d3d 3d3d tiNumbers:.===== │ │ │ │ +00028620: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00028640: 2266 696e 6974 6542 6574 7469 4e75 6d62 "finiteBettiNumb │ │ │ │ +00028650: 6572 7328 4c69 7374 2922 0a0a 466f 7220 ers(List)"..For │ │ │ │ +00028660: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00028670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028680: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00028690: 6f74 6520 6669 6e69 7465 4265 7474 694e ote finiteBettiN │ │ │ │ +000286a0: 756d 6265 7273 3a20 6669 6e69 7465 4265 umbers: finiteBe │ │ │ │ +000286b0: 7474 694e 756d 6265 7273 2c20 6973 2061 ttiNumbers, is a │ │ │ │ +000286c0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +000286d0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ +000286e0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +000286f0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028720: 2d2d 2d2d 2d2d 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-------..The sou │ │ │ │ +00028750: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00028760: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00028770: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00028780: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00028790: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+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 │ │ │ │ +00028800: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2066 ns.info, Node: f │ │ │ │ +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 │ │ │ │ +00028880: 6e64 7320 6f66 2061 206d 6f64 756c 6520 nds of a module │ │ │ │ +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 **************** │ 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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 │ │ │ │ -000289c0: 6f63 294d 6174 7269 782c 2c20 4d61 7020 oc)Matrix,, Map │ │ │ │ -000289d0: 6672 6f6d 2061 2066 7265 6520 6d6f 6475 from a free modu │ │ │ │ -000289e0: 6c65 2074 6f20 4d2e 0a20 2020 2020 2020 le to M.. │ │ │ │ -000289f0: 2049 6d61 6765 2069 7320 7468 6520 6c61 Image is the la │ │ │ │ -00028a00: 7267 6573 7420 6672 6565 2073 756d 6d61 rgest free summa │ │ │ │ -00028a10: 6e64 0a0a 4465 7363 7269 7074 696f 6e0a nd..Description. │ │ │ │ -00028a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ +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 │ │ │ │ +00028920: 7253 756d 6d61 6e64 204d 0a20 202a 2049 rSummand M. * I │ │ │ │ +00028930: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +00028940: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00028950: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00028960: 4d6f 6475 6c65 2c2c 206f 7665 7220 616e Module,, over an │ │ │ │ +00028970: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ +00028980: 610a 2020 2a20 4f75 7470 7574 733a 0a20 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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 = │ │ │ │ -00028bb0: 206b 6b5b 652c 662c 672c 2053 6b65 7743 kk[e,f,g, SkewC │ │ │ │ -00028bc0: 6f6d 6d75 7461 7469 7665 203d 3e20 7472 ommutative => tr │ │ │ │ -00028bd0: 7565 5d20 2020 2020 2020 207c 0a7c 2020 ue] |.| │ │ │ │ +00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00028ba0: 3a20 4520 3d20 6b6b 5b65 2c66 2c67 2c20 : E = kk[e,f,g, │ │ │ │ +00028bb0: 536b 6577 436f 6d6d 7574 6174 6976 6520 SkewCommutative │ │ │ │ +00028bc0: 3d3e 2074 7275 655d 2020 2020 2020 2020 => true] │ │ │ │ +00028bd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c10: 207c 0a7c 6f32 203d 2045 2020 2020 2020 |.|o2 = E │ │ │ │ +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 |.| │ │ │ │ -00028c80: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52 o2 : PolynomialR │ │ │ │ -00028c90: 696e 672c 2033 2073 6b65 7720 636f 6d6d ing, 3 skew comm │ │ │ │ -00028ca0: 7574 6174 6976 6520 7661 7269 6162 6c65 utative variable │ │ │ │ -00028cb0: 2873 297c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d (s)|.+---------- │ │ │ │ +00028c70: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ +00028c80: 6d69 616c 5269 6e67 2c20 3320 736b 6577 mialRing, 3 skew │ │ │ │ +00028c90: 2063 6f6d 6d75 7461 7469 7665 2076 6172 commutative var │ │ │ │ +00028ca0: 6961 626c 6528 7329 7c0a 2b2d 2d2d 2d2d iable(s)|.+----- │ │ │ │ +00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ce0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -00028cf0: 204d 203d 2045 5e31 2b2b 6d6f 6475 6c65 M = E^1++module │ │ │ │ -00028d00: 2069 6465 616c 2076 6172 7320 452b 2b45 ideal vars E++E │ │ │ │ -00028d10: 5e7b 2d31 7d20 2020 2020 2020 2020 207c ^{-1} | │ │ │ │ -00028d20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00028ce0: 7c69 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6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -000291a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -000291b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -000291c0: 6520 6672 6565 4578 7465 7269 6f72 5375 e freeExteriorSu │ │ │ │ -000291d0: 6d6d 616e 643a 2066 7265 6545 7874 6572 mmand: freeExter │ │ │ │ -000291e0: 696f 7253 756d 6d61 6e64 2c20 6973 2061 iorSummand, is a │ │ │ │ -000291f0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00029200: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -00029210: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00029220: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00029110: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ +00029120: 2075 7365 2066 7265 6545 7874 6572 696f use freeExterio │ │ │ │ +00029130: 7253 756d 6d61 6e64 3a0a 3d3d 3d3d 3d3d rSummand:.====== │ │ │ │ +00029140: 3d3d 3d3d 3d3d 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CompleteIntersec │ │ │ │ -000292f0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00029300: 6d32 3a34 3238 353a 302e 0a1f 0a46 696c m2:4285:0....Fil │ │ │ │ -00029310: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00029320: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00029330: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2047 ns.info, Node: G │ │ │ │ -00029340: 7261 6469 6e67 2c20 4e65 7874 3a20 6866 rading, Next: hf │ │ │ │ -00029350: 2c20 5072 6576 3a20 6672 6565 4578 7465 , Prev: freeExte │ │ │ │ -00029360: 7269 6f72 5375 6d6d 616e 642c 2055 703a riorSummand, Up: │ │ │ │ -00029370: 2054 6f70 0a0a 4772 6164 696e 6720 2d2d Top..Grading -- │ │ │ │ -00029380: 204f 7074 696f 6e20 666f 7220 4569 7365 Option for Eise │ │ │ │ -00029390: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ -000293a0: 2c20 6e65 7745 7874 0a2a 2a2a 2a2a 2a2a , newExt.******* │ │ │ │ +00029260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00029270: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00029280: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00029290: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +000292a0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +000292b0: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ +000292c0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +000292d0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +000292e0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +000292f0: 696f 6e73 2e6d 323a 3432 3835 3a30 2e0a ions.m2:4285:0.. │ │ │ │ +00029300: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00029310: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00029320: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00029330: 6465 3a20 4772 6164 696e 672c 204e 6578 de: Grading, Nex │ │ │ │ +00029340: 743a 2068 662c 2050 7265 763a 2066 7265 t: hf, Prev: fre │ │ │ │ +00029350: 6545 7874 6572 696f 7253 756d 6d61 6e64 eExteriorSummand │ │ │ │ +00029360: 2c20 5570 3a20 546f 700a 0a47 7261 6469 , Up: Top..Gradi │ │ │ │ +00029370: 6e67 202d 2d20 4f70 7469 6f6e 2066 6f72 ng -- Option for │ │ │ │ +00029380: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ +00029390: 546f 7461 6c2c 206e 6577 4578 740a 2a2a Total, newExt.** │ │ │ │ +000293a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000293b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000293c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000293d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -000293e0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -000293f0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00029400: 546f 7461 6c28 4d62 6172 2c47 7261 6469 Total(Mbar,Gradi │ │ │ │ -00029410: 6e67 203d 3e20 3229 0a20 202a 2049 6e70 ng => 2). * Inp │ │ │ │ -00029420: 7574 733a 0a20 2020 2020 202a 2043 6865 uts:. * Che │ │ │ │ -00029430: 636b 2c20 616e 202a 6e6f 7465 2069 6e74 ck, an *note int │ │ │ │ -00029440: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -00029450: 446f 6329 5a5a 2c2c 200a 0a44 6573 6372 Doc)ZZ,, ..Descr │ │ │ │ -00029460: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00029470: 3d3d 0a0a 6966 2047 7261 6469 6e67 203d ==..if Grading = │ │ │ │ -00029480: 3e31 2c20 7468 656e 2074 6865 206f 7574 >1, then the out │ │ │ │ -00029490: 7075 7420 6973 2063 6f6e 7665 7274 6564 put is converted │ │ │ │ -000294a0: 2074 6f20 7369 6e67 6c65 2d67 7261 6469 to single-gradi │ │ │ │ -000294b0: 6e67 2c20 7573 6566 756c 2069 6e20 7468 ng, useful in th │ │ │ │ -000294c0: 650a 7061 636b 6167 6520 436c 6966 666f e.package Cliffo │ │ │ │ -000294d0: 7264 0a0a 5365 6520 616c 736f 0a3d 3d3d rd..See also.=== │ │ │ │ -000294e0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000294f0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00029500: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00029510: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00029520: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00029530: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -00029540: 7874 0a20 202a 202a 6e6f 7465 206e 6577 xt. * *note new │ │ │ │ -00029550: 4578 743a 206e 6577 4578 742c 202d 2d20 Ext: newExt, -- │ │ │ │ -00029560: 476c 6f62 616c 2045 7874 2066 6f72 206d Global Ext for m │ │ │ │ -00029570: 6f64 756c 6573 206f 7665 7220 6120 636f odules over a co │ │ │ │ -00029580: 6d70 6c65 7465 0a20 2020 2049 6e74 6572 mplete. Inter │ │ │ │ -00029590: 7365 6374 696f 6e0a 0a46 756e 6374 696f section..Functio │ │ │ │ -000295a0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -000295b0: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ -000295c0: 4772 6164 696e 673a 0a3d 3d3d 3d3d 3d3d Grading:.======= │ │ │ │ +000293d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +000293e0: 2020 2020 2020 4569 7365 6e62 7564 5368 EisenbudSh │ │ │ │ +000293f0: 616d 6173 6854 6f74 616c 284d 6261 722c amashTotal(Mbar, │ │ │ │ +00029400: 4772 6164 696e 6720 3d3e 2032 290a 2020 Grading => 2). │ │ │ │ +00029410: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00029420: 2a20 4368 6563 6b2c 2061 6e20 2a6e 6f74 * Check, an *not │ │ │ │ +00029430: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00029440: 756c 6179 3244 6f63 295a 5a2c 2c20 0a0a ulay2Doc)ZZ,, .. │ │ │ │ +00029450: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00029460: 3d3d 3d3d 3d3d 3d0a 0a69 6620 4772 6164 =======..if Grad │ │ │ │ +00029470: 696e 6720 3d3e 312c 2074 6865 6e20 7468 ing =>1, then th │ │ │ │ +00029480: 6520 6f75 7470 7574 2069 7320 636f 6e76 e output is conv │ │ │ │ +00029490: 6572 7465 6420 746f 2073 696e 676c 652d erted to single- │ │ │ │ +000294a0: 6772 6164 696e 672c 2075 7365 6675 6c20 grading, useful │ │ │ │ +000294b0: 696e 2074 6865 0a70 6163 6b61 6765 2043 in the.package C │ │ │ │ +000294c0: 6c69 6666 6f72 640a 0a53 6565 2061 6c73 lifford..See als │ │ │ │ +000294d0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +000294e0: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +000294f0: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00029500: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00029510: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00029520: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00029530: 7461 6c20 4578 740a 2020 2a20 2a6e 6f74 tal Ext. * *not │ │ │ │ +00029540: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ +00029550: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420 , -- Global Ext │ │ │ │ +00029560: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ +00029570: 2061 2063 6f6d 706c 6574 650a 2020 2020 a complete. │ │ │ │ +00029580: 496e 7465 7273 6563 7469 6f6e 0a0a 4675 Intersection..Fu │ │ │ │ +00029590: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +000295a0: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +000295b0: 616d 6564 2047 7261 6469 6e67 3a0a 3d3d amed Grading:.== │ │ │ │ +000295c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000295d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 ========.. * "E │ │ │ │ -00029600: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00029610: 7461 6c28 2e2e 2e2c 4772 6164 696e 673d tal(...,Grading= │ │ │ │ -00029620: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ -00029630: 6f74 6520 4569 7365 6e62 7564 5368 616d ote EisenbudSham │ │ │ │ -00029640: 6173 6854 6f74 616c 3a0a 2020 2020 4569 ashTotal:. 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-00029720: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00029730: 2a6e 6f74 6520 4772 6164 696e 673a 2047 *note Grading: G │ │ │ │ -00029740: 7261 6469 6e67 2c20 6973 2061 202a 6e6f rading, is a *no │ │ │ │ -00029750: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ -00029760: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -00029770: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000295f0: 202a 2022 4569 7365 6e62 7564 5368 616d * "EisenbudSham │ │ │ │ +00029600: 6173 6854 6f74 616c 282e 2e2e 2c47 7261 ashTotal(...,Gra │ │ │ │ +00029610: 6469 6e67 3d3e 2e2e 2e29 2220 2d2d 2073 ding=>...)" -- s │ │ │ │ +00029620: 6565 202a 6e6f 7465 2045 6973 656e 6275 ee *note Eisenbu │ │ │ │ +00029630: 6453 6861 6d61 7368 546f 7461 6c3a 0a20 dShamashTotal:. │ │ │ │ +00029640: 2020 2045 6973 656e 6275 6453 6861 6d61 EisenbudShama │ │ │ │ +00029650: 7368 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2047 7261 6469 ject *note Gradi │ │ │ │ +00029730: 6e67 3a20 4772 6164 696e 672c 2069 7320 ng: Grading, is │ │ │ │ +00029740: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +00029750: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00029760: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ +00029770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000297a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -000297d0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -000297e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -000297f0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00029800: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ -00029810: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00029820: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -00029830: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -00029840: 6573 6f6c 7574 696f 6e73 2e6d 323a 3332 esolutions.m2:32 │ │ │ │ -00029850: 3136 3a30 2e0a 1f0a 4669 6c65 3a20 436f 16:0....File: Co │ │ │ │ -00029860: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00029870: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -00029880: 666f 2c20 4e6f 6465 3a20 6866 2c20 4e65 fo, Node: hf, Ne │ │ │ │ -00029890: 7874 3a20 6866 4d6f 6475 6c65 4173 4578 xt: hfModuleAsEx │ │ │ │ -000298a0: 742c 2050 7265 763a 2047 7261 6469 6e67 t, Prev: Grading │ │ │ │ -000298b0: 2c20 5570 3a20 546f 700a 0a68 6620 2d2d , Up: Top..hf -- │ │ │ │ -000298c0: 2043 6f6d 7075 7465 7320 7468 6520 6869 Computes the hi │ │ │ │ -000298d0: 6c62 6572 7420 6675 6e63 7469 6f6e 2069 lbert function i │ │ │ │ -000298e0: 6e20 6120 7261 6e67 6520 6f66 2064 6567 n a range of deg │ │ │ │ -000298f0: 7265 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a rees.*********** │ │ │ │ +000297b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +000297c0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +000297d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +000297e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +000297f0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00029800: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ +00029810: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00029820: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00029830: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +00029840: 6d32 3a33 3231 363a 302e 0a1f 0a46 696c m2:3216:0....Fil │ │ │ │ +00029850: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +00029860: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00029870: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2068 ns.info, Node: h │ │ │ │ +00029880: 662c 204e 6578 743a 2068 664d 6f64 756c f, Next: hfModul │ │ │ │ +00029890: 6541 7345 7874 2c20 5072 6576 3a20 4772 eAsExt, Prev: Gr │ │ │ │ +000298a0: 6164 696e 672c 2055 703a 2054 6f70 0a0a ading, Up: Top.. │ │ │ │ +000298b0: 6866 202d 2d20 436f 6d70 7574 6573 2074 hf -- Computes t │ │ │ │ +000298c0: 6865 2068 696c 6265 7274 2066 756e 6374 he hilbert funct │ │ │ │ +000298d0: 696f 6e20 696e 2061 2072 616e 6765 206f ion in a range o │ │ │ │ +000298e0: 6620 6465 6772 6565 730a 2a2a 2a2a 2a2a f degrees.****** │ │ │ │ +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 **************.. │ │ │ │ -00029930: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00029940: 2020 2020 4820 3d20 6866 2873 2c50 290a H = hf(s,P). │ │ │ │ -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, │ │ │ │ -00029990: 2c20 6f72 204c 6973 740a 2020 2020 2020 , or List. │ │ │ │ -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 │ │ │ │ -00029a20: 7365 2068 663a 0a3d 3d3d 3d3d 3d3d 3d3d se hf:.========= │ │ │ │ -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 │ │ │ │ -00029a60: 6f64 756c 6529 220a 0a46 6f72 2074 6865 odule)"..For the │ │ │ │ -00029a70: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00029a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00029a90: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00029aa0: 2068 663a 2068 662c 2069 7320 6120 2a6e hf: hf, is a *n │ │ │ │ -00029ab0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00029ac0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00029ad0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00029ae0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00029920: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00029930: 0a20 2020 2020 2020 2048 203d 2068 6628 . H = hf( │ │ │ │ +00029940: 732c 5029 0a20 202a 2049 6e70 7574 733a s,P). * Inputs: │ │ │ │ +00029950: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ +00029960: 6f74 6520 7365 7175 656e 6365 3a20 284d ote sequence: (M │ │ │ │ +00029970: 6163 6175 6c61 7932 446f 6329 5365 7175 acaulay2Doc)Sequ │ │ │ │ +00029980: 656e 6365 2c2c 206f 7220 4c69 7374 0a20 ence,, or List. │ │ │ │ +00029990: 2020 2020 202a 2050 2c20 6120 2a6e 6f74 * P, a *not │ │ │ │ +000299a0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +000299b0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +000299c0: 2067 7261 6465 6420 6d6f 6475 6c65 0a20 graded module. │ │ │ │ +000299d0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +000299e0: 2020 2a20 482c 2061 202a 6e6f 7465 206c * H, a *note l │ │ │ │ +000299f0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00029a00: 6f63 294c 6973 742c 2c20 0a0a 5761 7973 oc)List,, ..Ways │ │ │ │ +00029a10: 2074 6f20 7573 6520 6866 3a0a 3d3d 3d3d to use hf:.==== │ │ │ │ +00029a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00029a30: 2022 6866 284c 6973 742c 4d6f 6475 6c65 "hf(List,Module │ │ │ │ +00029a40: 2922 0a20 202a 2022 6866 2853 6571 7565 )". * "hf(Seque │ │ │ │ +00029a50: 6e63 652c 4d6f 6475 6c65 2922 0a0a 466f nce,Module)"..Fo │ │ │ │ +00029a60: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00029a70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00029a80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00029a90: 2a6e 6f74 6520 6866 3a20 6866 2c20 6973 *note hf: hf, is │ │ │ │ +00029aa0: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00029ab0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00029ac0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00029ad0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b30: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00029b40: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -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: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 ================ │ │ │ │ -0002a9d0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002a9e0: 6f74 6520 6866 4d6f 6475 6c65 4173 4578 ote hfModuleAsEx │ │ │ │ -0002a9f0: 743a 2068 664d 6f64 756c 6541 7345 7874 t: hfModuleAsExt │ │ │ │ -0002aa00: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0002aa10: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0002aa20: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0002aa30: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0002a8d0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +0002a8e0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0002a8f0: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ +0002a900: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ +0002a910: 2d2d 2046 696e 6420 6120 6d6f 6475 6c65 -- Find a module │ │ │ │ +0002a920: 2077 6974 6820 6769 7665 6e20 6173 796d with given asym │ │ │ │ +0002a930: 7074 6f74 6963 0a20 2020 2072 6573 6f6c ptotic. resol │ │ │ │ +0002a940: 7574 696f 6e0a 0a57 6179 7320 746f 2075 ution..Ways to u │ │ │ │ +0002a950: 7365 2068 664d 6f64 756c 6541 7345 7874 se hfModuleAsExt │ │ │ │ +0002a960: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0002a980: 2a20 2268 664d 6f64 756c 6541 7345 7874 * "hfModuleAsExt │ │ │ │ +0002a990: 285a 5a2c 4d6f 6475 6c65 2c5a 5a29 220a (ZZ,Module,ZZ)". │ │ │ │ +0002a9a0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0002a9b0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0002a9c0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0002a9d0: 6374 202a 6e6f 7465 2068 664d 6f64 756c ct *note hfModul │ │ │ │ +0002a9e0: 6541 7345 7874 3a20 6866 4d6f 6475 6c65 eAsExt: hfModule │ │ │ │ +0002a9f0: 4173 4578 742c 2069 7320 6120 2a6e 6f74 AsExt, is a *not │ │ │ │ +0002aa00: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0002aa10: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0002aa20: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0002aa30: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0002aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0002aa90: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0002aaa0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0002aab0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0002aac0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0002aad0: 6179 322d 312e 3235 2e30 362b 6473 2f4d ay2-1.25.06+ds/M │ │ │ │ -0002aae0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0002aaf0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0002ab00: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0002ab10: 7469 6f6e 732e 6d32 3a33 3134 313a 302e tions.m2:3141:0. │ │ │ │ -0002ab20: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0002ab30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0002ab40: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0002ab50: 6f64 653a 2068 6967 6853 797a 7967 792c ode: highSyzygy, │ │ │ │ -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 │ │ │ │ -0002abc0: 2072 6567 756c 6172 6974 7920 6f66 2045 regularity of E │ │ │ │ -0002abd0: 7874 284d 2c6b 290a 2a2a 2a2a 2a2a 2a2a xt(M,k).******** │ │ │ │ +0002aa80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0002aa90: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0002aaa0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0002aab0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0002aac0: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +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 │ │ │ │ +0002ab30: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +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 │ │ │ │ +0002ac20: 6167 653a 200a 2020 2020 2020 2020 4d20 age: . M │ │ │ │ +0002ac30: 3d20 6869 6768 5379 7a79 6779 204d 300a = highSyzygy M0. │ │ │ │ +0002ac40: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002ac50: 2020 2a20 4d30 2c20 6120 2a6e 6f74 6520 * M0, a *note │ │ │ │ +0002ac60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0002ac70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +0002ac80: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +0002ac90: 6e74 6572 7365 6374 696f 6e0a 2020 2020 ntersection. │ │ │ │ +0002aca0: 2020 2020 7269 6e67 0a20 202a 202a 6e6f ring. * *no │ │ │ │ +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 │ │ │ │ +0002ad80: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002ad90: 0a41 2022 6869 6768 2073 797a 7967 7922 .A "high syzygy" │ │ │ │ +0002ada0: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0002adb0: 2069 6e74 6572 7365 6374 696f 6e20 6973 intersection is │ │ │ │ +0002adc0: 206f 6e65 2073 7563 6820 7468 6174 2067 one such that g │ │ │ │ +0002add0: 656e 6572 616c 0a63 692d 6f70 6572 6174 eneral.ci-operat │ │ │ │ +0002ade0: 6f72 7320 6861 7665 2073 706c 6974 206b ors have split k │ │ │ │ +0002adf0: 6572 6e65 6c73 2077 6865 6e20 6170 706c ernels when appl │ │ │ │ +0002ae00: 6965 6420 7265 6375 7273 6976 656c 7920 ied recursively │ │ │ │ +0002ae10: 6f6e 2063 6f73 797a 7967 7920 6368 6169 on cosyzygy chai │ │ │ │ +0002ae20: 6e73 206f 660a 7072 6576 696f 7573 206b ns of.previous k │ │ │ │ +0002ae30: 6572 6e65 6c73 2e0a 0a49 6620 7020 3d20 ernels...If p = │ │ │ │ +0002ae40: 6d66 426f 756e 6420 4d30 2c20 7468 656e mfBound M0, then │ │ │ │ +0002ae50: 2068 6967 6853 797a 7967 7920 4d30 2072 highSyzygy M0 r │ │ │ │ +0002ae60: 6574 7572 6e73 2074 6865 2070 2d74 6820 eturns the p-th │ │ │ │ +0002ae70: 7379 7a79 6779 206f 6620 4d30 2e20 2869 syzygy of M0. (i │ │ │ │ +0002ae80: 6620 4620 6973 2061 0a72 6573 6f6c 7574 f F is a.resolut │ │ │ │ +0002ae90: 696f 6e20 6f66 204d 2074 6869 7320 6973 ion of M this is │ │ │ │ +0002aea0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66 the cokernel of │ │ │ │ +0002aeb0: 2046 2e64 645f 7b70 2b31 7d29 2e20 4f70 F.dd_{p+1}). Op │ │ │ │ +0002aec0: 7469 6d69 736d 203d 3e20 7220 6173 206f timism => r as o │ │ │ │ +0002aed0: 7074 696f 6e61 6c0a 6172 6775 6d65 6e74 ptional.argument │ │ │ │ +0002aee0: 2c20 6869 6768 5379 7a79 6779 284d 302c , highSyzygy(M0, │ │ │ │ +0002aef0: 4f70 7469 6d69 736d 3d3e 7229 2072 6574 Optimism=>r) ret │ │ │ │ +0002af00: 7572 6e73 2074 6865 2028 702d 7229 2d74 urns the (p-r)-t │ │ │ │ +0002af10: 6820 7379 7a79 6779 2e20 5468 6520 7363 h syzygy. The sc │ │ │ │ +0002af20: 7269 7074 2069 730a 7573 6566 756c 2077 ript is.useful w │ │ │ │ +0002af30: 6974 6820 6d61 7472 6978 4661 6374 6f72 ith matrixFactor │ │ │ │ +0002af40: 697a 6174 696f 6e28 6666 2c20 6869 6768 ization(ff, high │ │ │ │ +0002af50: 5379 7a79 6779 204d 3029 2e0a 0a2b 2d2d Syzygy M0)...+-- │ │ │ │ +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 │ │ │ │ +0002c3d0: 7846 6163 746f 7269 7a61 7469 6f6e 2866 xFactorization(f │ │ │ │ +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 │ │ │ │ +0002c460: 6c65 2069 7320 736f 6d65 7469 6d65 7320 le is sometimes │ │ │ │ +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 │ │ │ │ -0002c7e0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002c7f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 3036 2b64 732f lay2-1.25.06+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 │ │ │ │ -0002ce40: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0002ce50: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ -0002ce60: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+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 │ │ │ │ +0002ce10: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002ce20: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002ce30: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002ce40: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002ce50: 2e32 352e 3036 2b64 732f 4d32 2f4d 6163 .25.06+ds/M2/Mac │ │ │ │ +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 │ │ │ │ +0002cf40: 6e67 2064 6972 6563 7420 7375 6d20 696e ng direct sum in │ │ │ │ +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 │ │ │ │ -0002d2a0: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ -0002d2b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -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, │ │ │ │ +0002d070: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002d080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ +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 │ │ │ │ +0002d0e0: 756d 206f 6620 7468 6520 486f 6d73 2062 um of the Homs b │ │ │ │ +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 │ │ │ │ -0002d3a0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0002d3b0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -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 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+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 │ │ │ │ +0002d390: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0002d3a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +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 2e30 362b 6473 2f4d 322f 2-1.25.06+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 │ │ │ │ -0002e900: 4265 7474 694e 756d 6265 7273 3a20 6669 BettiNumbers: fi │ │ │ │ -0002e910: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e920: 2c20 2d2d 2062 6574 7469 206e 756d 6265 , -- betti numbe │ │ │ │ -0002e930: 7273 206f 6620 6669 6e69 7465 0a20 2020 rs of finite. │ │ │ │ -0002e940: 2072 6573 6f6c 7574 696f 6e20 636f 6d70 resolution comp │ │ │ │ -0002e950: 7574 6564 2066 726f 6d20 6120 6d61 7472 uted from a matr │ │ │ │ -0002e960: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -0002e970: 0a0a 5761 7973 2074 6f20 7573 6520 696e ..Ways to use in │ │ │ │ -0002e980: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -0002e990: 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d rs:.============ │ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002e9b0: 3d3d 3d3d 3d0a 0a20 202a 2022 696e 6669 =====.. * "infi │ │ │ │ -0002e9c0: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e9d0: 284c 6973 742c 5a5a 2922 0a0a 466f 7220 (List,ZZ)"..For │ │ │ │ -0002e9e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0002e9f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002ea00: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002ea10: 6f74 6520 696e 6669 6e69 7465 4265 7474 ote infiniteBett │ │ │ │ -0002ea20: 694e 756d 6265 7273 3a20 696e 6669 6e69 iNumbers: infini │ │ │ │ -0002ea30: 7465 4265 7474 694e 756d 6265 7273 2c20 teBettiNumbers, │ │ │ │ -0002ea40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0002ea50: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ -0002ea60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0002ea70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0002e860: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +0002e870: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0002e880: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ +0002e890: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ +0002e8a0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ +0002e8b0: 6170 7320 696e 2061 2068 6967 6865 720a aps in a higher. │ │ │ │ +0002e8c0: 2020 2020 636f 6469 6d65 6e73 696f 6e20 codimension │ │ │ │ +0002e8d0: 6d61 7472 6978 2066 6163 746f 7269 7a61 matrix factoriza │ │ │ │ +0002e8e0: 7469 6f6e 0a20 202a 202a 6e6f 7465 2066 tion. * *note f │ │ │ │ +0002e8f0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ +0002e900: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ +0002e910: 6d62 6572 732c 202d 2d20 6265 7474 6920 mbers, -- betti │ │ │ │ +0002e920: 6e75 6d62 6572 7320 6f66 2066 696e 6974 numbers of finit │ │ │ │ +0002e930: 650a 2020 2020 7265 736f 6c75 7469 6f6e e. resolution │ │ │ │ +0002e940: 2063 6f6d 7075 7465 6420 6672 6f6d 2061 computed from a │ │ │ │ +0002e950: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +0002e960: 6174 696f 6e0a 0a57 6179 7320 746f 2075 ation..Ways to u │ │ │ │ +0002e970: 7365 2069 6e66 696e 6974 6542 6574 7469 se infiniteBetti │ │ │ │ +0002e980: 4e75 6d62 6572 733a 0a3d 3d3d 3d3d 3d3d Numbers:.======= │ │ │ │ +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 │ │ │ │ +0002e9c0: 6d62 6572 7328 4c69 7374 2c5a 5a29 220a mbers(List,ZZ)". │ │ │ │ +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 │ │ │ │ +0002ea20: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +0002ea30: 6572 732c 2069 7320 6120 2a6e 6f74 6520 ers, is a *note │ │ │ │ +0002ea40: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ +0002ea50: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0002ea60: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0002ea70: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 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 │ │ │ │ -0002ead0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0002eae0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0002eaf0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0002eb00: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0002eb10: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -0002eb20: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0002eb30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0002eb40: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0002eb50: 6f6e 732e 6d32 3a34 3131 333a 302e 0a1f ons.m2:4113:0... │ │ │ │ -0002eb60: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0002eb70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0002eb80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -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 │ │ │ │ -0002ebc0: 6542 6574 7469 4e75 6d62 6572 732c 2055 eBettiNumbers, U │ │ │ │ -0002ebd0: 703a 2054 6f70 0a0a 6973 4c69 6e65 6172 p: Top..isLinear │ │ │ │ -0002ebe0: 202d 2d20 6368 6563 6b20 7768 6574 6865 -- check whethe │ │ │ │ -0002ebf0: 7220 6d61 7472 6978 2065 6e74 7269 6573 r matrix entries │ │ │ │ -0002ec00: 2068 6176 6520 6465 6772 6565 2031 0a2a have degree 1.* │ │ │ │ +0002eac0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0002ead0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0002eae0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0002eaf0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0002eb00: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +0002eb10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0002eb20: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0002eb30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0002eb40: 6f6c 7574 696f 6e73 2e6d 323a 3431 3133 olutions.m2:4113 │ │ │ │ +0002eb50: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0002eb60: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002eb70: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0002eb80: 2c20 4e6f 6465 3a20 6973 4c69 6e65 6172 , Node: isLinear │ │ │ │ +0002eb90: 2c20 4e65 7874 3a20 6973 5175 6173 6952 , Next: isQuasiR │ │ │ │ +0002eba0: 6567 756c 6172 2c20 5072 6576 3a20 696e egular, Prev: in │ │ │ │ +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 │ │ │ │ +0002ebf0: 7472 6965 7320 6861 7665 2064 6567 7265 tries have degre │ │ │ │ +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 │ │ │ │ -0002eca0: 6174 7269 782c 2c20 0a20 202a 204f 7574 atrix,, . * Out │ │ │ │ -0002ecb0: 7075 7473 3a0a 2020 2020 2020 2a20 622c puts:. * b, │ │ │ │ -0002ecc0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e a *note Boolean │ │ │ │ -0002ecd0: 2076 616c 7565 3a20 284d 6163 6175 6c61 value: (Macaula │ │ │ │ -0002ece0: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20 y2Doc)Boolean,, │ │ │ │ -0002ecf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0002ed00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4e 6f74 6520 =========..Note │ │ │ │ -0002ed10: 7468 6174 2061 206c 696e 6561 7220 6d61 that a linear ma │ │ │ │ -0002ed20: 7472 6978 2c20 696e 2074 6869 7320 7365 trix, in this se │ │ │ │ -0002ed30: 6e73 652c 2063 616e 2073 7469 6c6c 2068 nse, can still h │ │ │ │ -0002ed40: 6176 6520 6469 6666 6572 656e 7420 7461 ave different ta │ │ │ │ -0002ed50: 7267 6574 0a64 6567 7265 6573 2028 696e rget.degrees (in │ │ │ │ -0002ed60: 2077 6869 6368 2063 6173 6520 7468 6520 which case the │ │ │ │ -0002ed70: 636f 6b65 726e 656c 2064 6563 6f6d 706f cokernel decompo │ │ │ │ -0002ed80: 7365 7320 696e 746f 2061 2064 6972 6563 ses into a direc │ │ │ │ -0002ed90: 7420 7375 6d20 6279 2067 656e 6572 6174 t sum by generat │ │ │ │ -0002eda0: 6f72 0a64 6567 7265 652e 290a 0a57 6179 or.degree.)..Way │ │ │ │ -0002edb0: 7320 746f 2075 7365 2069 734c 696e 6561 s to use isLinea │ │ │ │ -0002edc0: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r:.============= │ │ │ │ -0002edd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 ========.. * "i │ │ │ │ -0002ede0: 734c 696e 6561 7228 4d61 7472 6978 2922 sLinear(Matrix)" │ │ │ │ -0002edf0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002ee00: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002ee10: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002ee20: 6563 7420 2a6e 6f74 6520 6973 4c69 6e65 ect *note isLine │ │ │ │ -0002ee30: 6172 3a20 6973 4c69 6e65 6172 2c20 6973 ar: isLinear, is │ │ │ │ -0002ee40: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0002ee50: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0002ee60: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0002ee70: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0002ec40: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0002ec50: 6220 3d20 6973 4c69 6e65 6172 204d 0a20 b = isLinear M. │ │ │ │ +0002ec60: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +0002ec70: 202a 204d 2c20 6120 2a6e 6f74 6520 6d61 * M, a *note ma │ │ │ │ +0002ec80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +0002ec90: 446f 6329 4d61 7472 6978 2c2c 200a 2020 Doc)Matrix,, . │ │ │ │ +0002eca0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0002ecb0: 202a 2062 2c20 6120 2a6e 6f74 6520 426f * b, a *note Bo │ │ │ │ +0002ecc0: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ +0002ecd0: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ +0002ece0: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ +0002ecf0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002ed00: 4e6f 7465 2074 6861 7420 6120 6c69 6e65 Note that a line │ │ │ │ +0002ed10: 6172 206d 6174 7269 782c 2069 6e20 7468 ar matrix, in th │ │ │ │ +0002ed20: 6973 2073 656e 7365 2c20 6361 6e20 7374 is sense, can st │ │ │ │ +0002ed30: 696c 6c20 6861 7665 2064 6966 6665 7265 ill have differe │ │ │ │ +0002ed40: 6e74 2074 6172 6765 740a 6465 6772 6565 nt target.degree │ │ │ │ +0002ed50: 7320 2869 6e20 7768 6963 6820 6361 7365 s (in which case │ │ │ │ +0002ed60: 2074 6865 2063 6f6b 6572 6e65 6c20 6465 the cokernel de │ │ │ │ +0002ed70: 636f 6d70 6f73 6573 2069 6e74 6f20 6120 composes into a │ │ │ │ +0002ed80: 6469 7265 6374 2073 756d 2062 7920 6765 direct sum by ge │ │ │ │ +0002ed90: 6e65 7261 746f 720a 6465 6772 6565 2e29 nerator.degree.) │ │ │ │ +0002eda0: 0a0a 5761 7973 2074 6f20 7573 6520 6973 ..Ways to use is │ │ │ │ +0002edb0: 4c69 6e65 6172 3a0a 3d3d 3d3d 3d3d 3d3d Linear:.======== │ │ │ │ +0002edc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +0002edd0: 202a 2022 6973 4c69 6e65 6172 284d 6174 * "isLinear(Mat │ │ │ │ +0002ede0: 7269 7829 220a 0a46 6f72 2074 6865 2070 rix)"..For the p │ │ │ │ +0002edf0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002ee00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002ee10: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ +0002ee20: 734c 696e 6561 723a 2069 734c 696e 6561 sLinear: isLinea │ │ │ │ +0002ee30: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ +0002ee40: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0002ee50: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0002ee60: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +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 ---------------- │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0002eed0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002eee0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0002eef0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0002ef00: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0002ef10: 312e 3235 2e30 362b 6473 2f4d 322f 4d61 1.25.06+ds/M2/Ma │ │ │ │ -0002ef20: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0002ef30: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0002ef40: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002ef50: 732e 6d32 3a33 3436 333a 302e 0a1f 0a46 s.m2:3463:0....F │ │ │ │ -0002ef60: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0002ef70: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0002ef80: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0002ef90: 2069 7351 7561 7369 5265 6775 6c61 722c isQuasiRegular, │ │ │ │ -0002efa0: 204e 6578 743a 2069 7353 7461 626c 7954 Next: isStablyT │ │ │ │ -0002efb0: 7269 7669 616c 2c20 5072 6576 3a20 6973 rivial, Prev: is │ │ │ │ -0002efc0: 4c69 6e65 6172 2c20 5570 3a20 546f 700a Linear, Up: Top. │ │ │ │ -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 3036 2b64 732f lay2-1.25.06+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"; │ │ │ │ -0002f670: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +0002f640: 203a 2066 3420 3d20 6d61 7472 6978 2261 : f4 = matrix"a │ │ │ │ +0002f650: 2b62 2c20 6132 2b62 223b 2020 2020 2020 +b, a2+b"; │ │ │ │ +0002f660: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f6a0: 7c20 2020 2020 2020 2020 2020 2020 3120 | 1 │ │ │ │ -0002f6b0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0002f6d0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ -0002f6e0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -0002f6f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f690: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f6a0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f6c0: 0a7c 6f37 203a 204d 6174 7269 7820 5320 .|o7 : Matrix S │ │ │ │ +0002f6d0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0002f6e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f720: 2d2d 2b0a 7c69 3820 3a20 6973 5175 6173 --+.|i8 : isQuas │ │ │ │ -0002f730: 6952 6567 756c 6172 2866 312c 4529 2020 iRegular(f1,E) │ │ │ │ -0002f740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f750: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f770: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -0002f780: 3d20 6661 6c73 6520 2020 2020 2020 2020 = false │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f710: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2069 -------+.|i8 : i │ │ │ │ +0002f720: 7351 7561 7369 5265 6775 6c61 7228 6631 sQuasiRegular(f1 │ │ │ │ +0002f730: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f740: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f770: 0a7c 6f38 203d 2066 616c 7365 2020 2020 .|o8 = false │ │ │ │ +0002f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7d0: 2d2d 2b0a 7c69 3920 3a20 6973 5175 6173 --+.|i9 : isQuas │ │ │ │ -0002f7e0: 6952 6567 756c 6172 2866 322c 4529 2020 iRegular(f2,E) │ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f800: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f820: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ -0002f830: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f850: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f7c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069 -------+.|i9 : i │ │ │ │ +0002f7d0: 7351 7561 7369 5265 6775 6c61 7228 6632 sQuasiRegular(f2 │ │ │ │ +0002f7e0: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f7f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f820: 0a7c 6f39 203d 2074 7275 6520 2020 2020 .|o9 = true │ │ │ │ +0002f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f880: 2d2d 2b0a 7c69 3130 203a 2069 7351 7561 --+.|i10 : isQua │ │ │ │ -0002f890: 7369 5265 6775 6c61 7228 6633 2c45 2920 siRegular(f3,E) │ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f8b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ -0002f8e0: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f900: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ +0002f880: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f890: 332c 4529 2020 2020 2020 2020 2020 2020 3,E) │ │ │ │ +0002f8a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f8d0: 0a7c 6f31 3020 3d20 7472 7565 2020 2020 .|o10 = true │ │ │ │ +0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f930: 2d2d 2b0a 7c69 3131 203a 2069 7351 7561 --+.|i11 : isQua │ │ │ │ -0002f940: 7369 5265 6775 6c61 7228 6634 2c45 2920 siRegular(f4,E) │ │ │ │ -0002f950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f960: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f980: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -0002f990: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ -0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f920: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ +0002f930: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f940: 342c 4529 2020 2020 2020 2020 2020 2020 4,E) │ │ │ │ +0002f950: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f970: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f980: 0a7c 6f31 3120 3d20 6661 6c73 6520 2020 .|o11 = false │ │ │ │ +0002f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f9a0: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9e0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -0002f9f0: 2069 7351 7561 7369 5265 6775 6c61 723a isQuasiRegular: │ │ │ │ -0002fa00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0002fa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0002fa20: 2a20 2269 7351 7561 7369 5265 6775 6c61 * "isQuasiRegula │ │ │ │ -0002fa30: 7228 4c69 7374 2c4d 6f64 756c 6529 220a r(List,Module)". │ │ │ │ -0002fa40: 2020 2a20 2269 7351 7561 7369 5265 6775 * "isQuasiRegu │ │ │ │ -0002fa50: 6c61 7228 4d61 7472 6978 2c4d 6f64 756c lar(Matrix,Modul │ │ │ │ -0002fa60: 6529 220a 2020 2a20 2269 7351 7561 7369 e)". * "isQuasi │ │ │ │ -0002fa70: 5265 6775 6c61 7228 5365 7175 656e 6365 Regular(Sequence │ │ │ │ -0002fa80: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074 ,Module)"..For t │ │ │ │ -0002fa90: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -0002faa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002fab0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -0002fac0: 7465 2069 7351 7561 7369 5265 6775 6c61 te isQuasiRegula │ │ │ │ -0002fad0: 723a 2069 7351 7561 7369 5265 6775 6c61 r: isQuasiRegula │ │ │ │ -0002fae0: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ -0002faf0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -0002fb00: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0002fb10: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +0002f9e0: 6f20 7573 6520 6973 5175 6173 6952 6567 o use isQuasiReg │ │ │ │ +0002f9f0: 756c 6172 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ular:.========== │ │ │ │ +0002fa00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002fa10: 3d0a 0a20 202a 2022 6973 5175 6173 6952 =.. * "isQuasiR │ │ │ │ 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6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +0002fb00: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0002fb10: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 0002fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0002fb70: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0002fb80: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0002fb90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0002fba0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0002fbb0: 6c61 7932 2d31 2e32 352e 3036 2b64 732f lay2-1.25.06+ds/ │ │ │ │ -0002fbc0: 4d32 2f4d 6163 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2069 6620 returns true if │ │ │ │ -0002fca0: 7468 6520 6d61 7020 676f 6573 2074 6f20 the map goes to │ │ │ │ -0002fcb0: 3020 756e 6465 7220 7374 6162 6c65 486f 0 under stableHo │ │ │ │ -0002fcc0: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ +0002fb60: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +0002fb70: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0002fb80: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0002fb90: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0002fba0: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ +0002fbb0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +0002fbc0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +0002fbd0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002fbe0: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ +0002fbf0: 3632 373a 302e 0a1f 0a46 696c 653a 2043 627:0....File: C │ │ │ │ +0002fc00: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0002fc10: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +0002fc20: 6e66 6f2c 204e 6f64 653a 2069 7353 7461 nfo, Node: isSta │ │ │ │ +0002fc30: 626c 7954 7269 7669 616c 2c20 4e65 7874 blyTrivial, Next │ │ │ │ +0002fc40: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ +0002fc50: 6e2c 2050 7265 763a 2069 7351 7561 7369 n, Prev: isQuasi │ │ │ │ +0002fc60: 5265 6775 6c61 722c 2055 703a 2054 6f70 Regular, Up: Top │ │ │ │ +0002fc70: 0a0a 6973 5374 6162 6c79 5472 6976 6961 ..isStablyTrivia │ │ │ │ +0002fc80: 6c20 2d2d 2072 6574 7572 6e73 2074 7275 l -- returns tru │ │ │ │ +0002fc90: 6520 6966 2074 6865 206d 6170 2067 6f65 e if the map goe │ │ │ │ +0002fca0: 7320 746f 2030 2075 6e64 6572 2073 7461 s to 0 under sta │ │ │ │ +0002fcb0: 626c 6548 6f6d 0a2a 2a2a 2a2a 2a2a 2a2a bleHom.********* │ │ │ │ +0002fcc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fcd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fd00: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0002fd10: 653a 200a 2020 2020 2020 2020 6220 3d20 e: . b = │ │ │ │ -0002fd20: 6973 5374 6162 6c79 5472 6976 6961 6c20 isStablyTrivial │ │ │ │ -0002fd30: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ -0002fd40: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ -0002fd50: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ -0002fd60: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ -0002fd70: 6d61 7020 4d20 746f 204e 0a20 202a 204f map M to N. * O │ │ │ │ -0002fd80: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0002fd90: 622c 2061 202a 6e6f 7465 2042 6f6f 6c65 b, a *note Boole │ │ │ │ -0002fda0: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -0002fdb0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -0002fdc0: 2c20 7472 7565 2069 6666 2066 2066 6163 , true iff f fac │ │ │ │ -0002fdd0: 746f 7273 0a20 2020 2020 2020 2074 6872 tors. thr │ │ │ │ -0002fde0: 6f75 6768 2061 2070 726f 6a65 6374 6976 ough a projectiv │ │ │ │ -0002fdf0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d e..Description.= │ │ │ │ -0002fe00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 706f ==========..A po │ │ │ │ -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 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│ │ │ -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 │ │ │ │ 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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 │ │ │ │ -00031140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +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 ---------------- │ │ │ │ -00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031190: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -000311a0: 7432 203d 206d 616b 6554 2866 662c 462c t2 = makeT(ff,F, │ │ │ │ -000311b0: 6b2b 3229 3b20 2020 2020 2020 2020 2020 k+2); │ │ │ │ +00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031190: 3131 203a 2074 3220 3d20 6d61 6b65 5428 11 : t2 = makeT( │ │ │ │ +000311a0: 6666 2c46 2c6b 2b32 293b 2020 2020 2020 ff,F,k+2); │ │ │ │ +000311b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000311d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +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 ---------------- │ │ │ │ -00031230: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -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 │ │ │ │ -00031270: 7220 6a20 6672 6f6d 2069 2b31 2074 6f20 r j from i+1 to │ │ │ │ -00031280: 3220 6c69 7374 207c 0a7c 2d2d 2d2d 2d2d 2 list |.|------ │ │ │ │ +00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031230: 3132 203a 2054 3243 6f6d 706f 6e65 6e74 12 : T2Component │ │ │ │ +00031240: 7320 3d20 666c 6174 7465 6e20 666f 7220 s = flatten for │ │ │ │ +00031250: 6920 6672 6f6d 2030 2074 6f20 3120 6c69 i from 0 to 1 li │ │ │ │ +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 ---------------- │ │ │ │ 00031290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -000312d0: 2d2d 2d2d 2d2d 2d7c 0a7c 6d61 7028 465f -------|.|map(F_ │ │ │ │ -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) │ │ │ │ -00031300: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -00031310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031320: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +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 |.| │ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +00031470: 2030 2030 2030 2020 3020 3020 3020 3020 0 0 0 0 0 0 0 │ │ │ │ +00031480: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00031490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000314a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000314b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031500: 2020 2020 2020 207c 0a7c 6f31 3320 3a20 |.|o13 : │ │ │ │ -00031510: 4d61 7472 6978 2020 2020 2020 2020 2020 Matrix │ │ │ │ +000314f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031500: 3133 203a 204d 6174 7269 7820 2020 2020 13 : Matrix │ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ -000315e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ 00031600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031640: 2020 2020 2020 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 │ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031680: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +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 │ │ │ │ -000316f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00031700: 202a 6e6f 7465 2073 7461 626c 6548 6f6d *note stableHom │ │ │ │ -00031710: 3a20 7374 6162 6c65 486f 6d2c 202d 2d20 : stableHom, -- │ │ │ │ -00031720: 6d61 7020 6672 6f6d 2048 6f6d 284d 2c4e map from Hom(M,N │ │ │ │ -00031730: 2920 746f 2074 6865 2073 7461 626c 6520 ) to the stable │ │ │ │ -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 │ │ │ │ -00031790: 6162 6c79 5472 6976 6961 6c28 4d61 7472 ablyTrivial(Matr │ │ │ │ -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 │ │ │ │ -000317d0: 206f 626a 6563 7420 2a6e 6f74 6520 6973 object *note is │ │ │ │ -000317e0: 5374 6162 6c79 5472 6976 6961 6c3a 2069 StablyTrivial: i │ │ │ │ -000317f0: 7353 7461 626c 7954 7269 7669 616c 2c20 sStablyTrivial, │ │ │ │ -00031800: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00031810: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00031820: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -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 │ │ │ │ +00031790: 284d 6174 7269 7829 220a 0a46 6f72 2074 (Matrix)"..For t │ │ │ │ +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 │ │ │ │ -00031890: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -000318a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -000318b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -000318c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -000318d0: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -000318e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -000318f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -00031900: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00031910: 6f6e 732e 6d32 3a34 3639 393a 302e 0a1f ons.m2:4699:0... │ │ │ │ -00031920: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00031930: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -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 │ │ │ │ -00031990: 700a 0a6b 6f73 7a75 6c45 7874 656e 7369 p..koszulExtensi │ │ │ │ -000319a0: 6f6e 202d 2d20 6372 6561 7465 7320 7468 on -- creates th │ │ │ │ -000319b0: 6520 4b6f 737a 756c 2065 7874 656e 7369 e Koszul extensi │ │ │ │ -000319c0: 6f6e 2063 6f6d 706c 6578 206f 6620 6120 on complex of a │ │ │ │ -000319d0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a map.************ │ │ │ │ +00031880: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00031890: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000318a0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000318b0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +000318c0: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +000318d0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +000318e0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +000318f0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00031900: 6f6c 7574 696f 6e73 2e6d 323a 3436 3939 olutions.m2:4699 │ │ │ │ +00031910: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00031920: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +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 │ │ │ │ +00031960: 6179 6572 6564 2c20 5072 6576 3a20 6973 ayered, Prev: is │ │ │ │ +00031970: 5374 6162 6c79 5472 6976 6961 6c2c 2055 StablyTrivial, U │ │ │ │ +00031980: 703a 2054 6f70 0a0a 6b6f 737a 756c 4578 p: Top..koszulEx │ │ │ │ +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 │ │ │ │ -00031c40: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00031c50: 3d0a 0a49 6d70 6c65 6d65 6e74 7320 7468 =..Implements th │ │ │ │ -00031c60: 6520 636f 6e73 7472 7563 7469 6f6e 2069 e construction i │ │ │ │ -00031c70: 6e20 7468 6520 7061 7065 7220 224d 6174 n the paper "Mat │ │ │ │ -00031c80: 7269 7820 4661 6374 6f72 697a 6174 696f rix Factorizatio │ │ │ │ -00031c90: 6e73 2069 6e20 4869 6768 6572 0a43 6f64 ns in Higher.Cod │ │ │ │ -00031ca0: 696d 656e 7369 6f6e 2220 6279 2045 6973 imension" by Eis │ │ │ │ -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 │ │ │ │ -00031e00: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ -00031e10: 6e2c 2069 7320 6120 2a6e 6f74 6520 6d65 n, is a *note me │ │ │ │ -00031e20: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -00031e30: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00031e40: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +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 │ │ │ │ +00031b60: 6c61 7220 7365 7175 656e 6365 0a20 2020 lar sequence. │ │ │ │ +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 │ │ │ │ +00031bf0: 6865 0a20 2020 2020 2020 2069 6e64 7563 he. induc │ │ │ │ +00031c00: 6564 206d 6170 2042 5b2d 315d 5c6f 7469 ed map B[-1]\oti │ │ │ │ +00031c10: 6d65 7320 4b4b 2866 6629 2074 6f20 5720 mes KK(ff) to W │ │ │ │ +00031c20: 6578 7465 6e64 696e 6720 7073 690a 0a44 extending psi..D │ │ │ │ +00031c30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00031c40: 3d3d 3d3d 3d3d 0a0a 496d 706c 656d 656e ======..Implemen │ │ │ │ +00031c50: 7473 2074 6865 2063 6f6e 7374 7275 6374 ts the construct │ │ │ │ +00031c60: 696f 6e20 696e 2074 6865 2070 6170 6572 ion in the paper │ │ │ │ +00031c70: 2022 4d61 7472 6978 2046 6163 746f 7269 "Matrix Factori │ │ │ │ +00031c80: 7a61 7469 6f6e 7320 696e 2048 6967 6865 zations in Highe │ │ │ │ +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 │ │ │ │ +00031d50: 7874 656e 7369 6f6e 3a0a 3d3d 3d3d 3d3d xtension:.====== │ │ │ │ +00031d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031d70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226b 6f73 ======.. * "kos │ │ │ │ +00031d80: 7a75 6c45 7874 656e 7369 6f6e 2843 6f6d zulExtension(Com │ │ │ │ +00031d90: 706c 6578 2c43 6f6d 706c 6578 2c4d 6174 plex,Complex,Mat │ │ │ │ +00031da0: 7269 782c 4d61 7472 6978 2922 0a0a 466f rix,Matrix)"..Fo │ │ │ │ +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 │ │ │ │ +00031de0: 2a6e 6f74 6520 6b6f 737a 756c 4578 7465 *note koszulExte │ │ │ │ +00031df0: 6e73 696f 6e3a 206b 6f73 7a75 6c45 7874 nsion: koszulExt │ │ │ │ +00031e00: 656e 7369 6f6e 2c20 6973 2061 202a 6e6f ension, is a *no │ │ │ │ +00031e10: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00031e20: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +00031e30: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00031e40: 2c2e 0a0a 2d2d 2d2d 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6f6c ntersectionResol │ │ │ │ -00031f20: 7574 696f 6e73 2e6d 323a 3330 3038 3a30 utions.m2:3008:0 │ │ │ │ -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 2e30 macaulay2-1.25.0 │ │ │ │ +00031ee0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+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 │ │ │ │ +00031f10: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +00031f20: 3030 383a 302e 0a1f 0a46 696c 653a 2043 008:0....File: C │ │ │ │ +00031f30: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +00031f40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +00031f50: 6e66 6f2c 204e 6f64 653a 204c 6179 6572 nfo, Node: Layer │ │ │ │ +00031f60: 6564 2c20 4e65 7874 3a20 6c61 7965 7265 ed, Next: layere │ │ │ │ +00031f70: 6452 6573 6f6c 7574 696f 6e2c 2050 7265 dResolution, Pre │ │ │ │ +00031f80: 763a 206b 6f73 7a75 6c45 7874 656e 7369 v: koszulExtensi │ │ │ │ +00031f90: 6f6e 2c20 5570 3a20 546f 700a 0a4c 6179 on, Up: Top..Lay │ │ │ │ +00031fa0: 6572 6564 202d 2d20 4f70 7469 6f6e 2066 ered -- Option f │ │ │ │ +00031fb0: 6f72 206d 6174 7269 7846 6163 746f 7269 or matrixFactori │ │ │ │ +00031fc0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a zation.********* │ │ │ │ +00031fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00031fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00032000: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00032010: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -00032020: 7469 6f6e 2866 662c 6d2c 4c61 7965 7265 tion(ff,m,Layere │ │ │ │ -00032030: 6420 3d3e 2074 7275 6529 0a20 202a 2049 d => true). * I │ │ │ │ -00032040: 6e70 7574 733a 0a20 2020 2020 202a 2043 nputs:. * C │ │ │ │ -00032050: 6865 636b 2c20 6120 2a6e 6f74 6520 426f heck, a *note Bo │ │ │ │ -00032060: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ -00032070: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ -00032080: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ -00032090: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -000320a0: 4d61 6b65 7320 6d61 7472 6978 4661 6374 Makes matrixFact │ │ │ │ -000320b0: 6f72 697a 6174 696f 6e20 7573 6520 7468 orization use th │ │ │ │ -000320c0: 6520 226c 6179 6572 6564 2220 616c 676f e "layered" algo │ │ │ │ -000320d0: 7269 7468 6d2c 2077 6869 6368 2077 6f72 rithm, which wor │ │ │ │ -000320e0: 6b73 2066 6f72 2061 6e79 204d 434d 0a6d ks for any MCM.m │ │ │ │ -000320f0: 6f64 756c 652c 2062 7574 2072 6574 7572 odule, but retur │ │ │ │ -00032100: 6e73 2073 6f6d 6574 6869 6e67 206e 6f6e ns something non │ │ │ │ -00032110: 2d6d 696e 696d 616c 2069 6620 7468 6520 -minimal if the │ │ │ │ -00032120: 6d6f 6475 6c65 2069 7320 6e6f 7420 6120 module is not a │ │ │ │ -00032130: 2268 6967 6820 7379 7a79 6779 220a 696e "high syzygy".in │ │ │ │ -00032140: 2061 2073 7569 7461 626c 6520 7365 6e73 a suitable sens │ │ │ │ -00032150: 652e 2044 6566 6175 6c74 2069 7320 2274 e. Default is "t │ │ │ │ -00032160: 7275 6522 2e20 4e6f 7465 2074 6861 7420 rue". Note that │ │ │ │ -00032170: 7768 656e 2074 6865 206d 6f64 756c 6520 when the module │ │ │ │ -00032180: 6973 2061 2068 6967 680a 7379 7a79 6779 is a high.syzygy │ │ │ │ -00032190: 2c20 4c61 7965 7265 643d 3e20 6661 6c73 , Layered=> fals │ │ │ │ -000321a0: 6520 6973 206d 7563 6820 6661 7374 6572 e is much faster │ │ │ │ -000321b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -000321c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000321d0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -000321e0: 696f 6e3a 206d 6174 7269 7846 6163 746f ion: matrixFacto │ │ │ │ -000321f0: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170 rization, -- Map │ │ │ │ -00032200: 7320 696e 2061 2068 6967 6865 720a 2020 s in a higher. │ │ │ │ -00032210: 2020 636f 6469 6d65 6e73 696f 6e20 6d61 codimension ma │ │ │ │ -00032220: 7472 6978 2066 6163 746f 7269 7a61 7469 trix factorizati │ │ │ │ -00032230: 6f6e 0a0a 4675 6e63 7469 6f6e 7320 7769 on..Functions wi │ │ │ │ -00032240: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -00032250: 6d65 6e74 206e 616d 6564 204c 6179 6572 ment named Layer │ │ │ │ -00032260: 6564 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ed:.============ │ │ │ │ +00031ff0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00032000: 2020 2020 2020 6d61 7472 6978 4661 6374 matrixFact │ │ │ │ +00032010: 6f72 697a 6174 696f 6e28 6666 2c6d 2c4c orization(ff,m,L │ │ │ │ +00032020: 6179 6572 6564 203d 3e20 7472 7565 290a ayered => true). │ │ │ │ +00032030: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00032040: 2020 2a20 4368 6563 6b2c 2061 202a 6e6f * Check, a *no │ │ │ │ +00032050: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00032060: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00032070: 426f 6f6c 6561 6e2c 2c20 0a0a 4465 7363 Boolean,, ..Desc │ │ │ │ +00032080: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00032090: 3d3d 3d0a 0a4d 616b 6573 206d 6174 7269 ===..Makes matri │ │ │ │ +000320a0: 7846 6163 746f 7269 7a61 7469 6f6e 2075 xFactorization u │ │ │ │ +000320b0: 7365 2074 6865 2022 6c61 7965 7265 6422 se the "layered" │ │ │ │ +000320c0: 2061 6c67 6f72 6974 686d 2c20 7768 6963 algorithm, whic │ │ │ │ +000320d0: 6820 776f 726b 7320 666f 7220 616e 7920 h works for any │ │ │ │ +000320e0: 4d43 4d0a 6d6f 6475 6c65 2c20 6275 7420 MCM.module, but │ │ │ │ +000320f0: 7265 7475 726e 7320 736f 6d65 7468 696e returns somethin │ │ │ │ +00032100: 6720 6e6f 6e2d 6d69 6e69 6d61 6c20 6966 g non-minimal if │ │ │ │ +00032110: 2074 6865 206d 6f64 756c 6520 6973 206e the module is n │ │ │ │ +00032120: 6f74 2061 2022 6869 6768 2073 797a 7967 ot a "high syzyg │ │ │ │ +00032130: 7922 0a69 6e20 6120 7375 6974 6162 6c65 y".in a suitable │ │ │ │ +00032140: 2073 656e 7365 2e20 4465 6661 756c 7420 sense. Default │ │ │ │ +00032150: 6973 2022 7472 7565 222e 204e 6f74 6520 is "true". Note │ │ │ │ +00032160: 7468 6174 2077 6865 6e20 7468 6520 6d6f that when the mo │ │ │ │ +00032170: 6475 6c65 2069 7320 6120 6869 6768 0a73 dule is a high.s │ │ │ │ +00032180: 797a 7967 792c 204c 6179 6572 6564 3d3e yzygy, Layered=> │ │ │ │ +00032190: 2066 616c 7365 2069 7320 6d75 6368 2066 false is much f │ │ │ │ +000321a0: 6173 7465 722e 0a0a 5365 6520 616c 736f aster...See also │ │ │ │ +000321b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000321c0: 6e6f 7465 206d 6174 7269 7846 6163 746f note matrixFacto │ │ │ │ +000321d0: 7269 7a61 7469 6f6e 3a20 6d61 7472 6978 rization: matrix │ │ │ │ +000321e0: 4661 6374 6f72 697a 6174 696f 6e2c 202d Factorization, - │ │ │ │ +000321f0: 2d20 4d61 7073 2069 6e20 6120 6869 6768 - Maps in a high │ │ │ │ +00032200: 6572 0a20 2020 2063 6f64 696d 656e 7369 er. codimensi │ │ │ │ +00032210: 6f6e 206d 6174 7269 7820 6661 6374 6f72 on matrix factor │ │ │ │ +00032220: 697a 6174 696f 6e0a 0a46 756e 6374 696f ization..Functio │ │ │ │ +00032230: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00032240: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ +00032250: 4c61 7965 7265 643a 0a3d 3d3d 3d3d 3d3d Layered:.======= │ │ │ │ +00032260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00032270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032290: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978 ===.. * "matrix │ │ │ │ -000322a0: 4661 6374 6f72 697a 6174 696f 6e28 2e2e Factorization(.. │ │ │ │ -000322b0: 2e2c 4c61 7965 7265 643d 3e2e 2e2e 2922 .,Layered=>...)" │ │ │ │ -000322c0: 202d 2d20 7365 6520 2a6e 6f74 6520 6d61 -- see *note ma │ │ │ │ -000322d0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -000322e0: 6e3a 0a20 2020 206d 6174 7269 7846 6163 n:. matrixFac │ │ │ │ -000322f0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ -00032300: 6170 7320 696e 2061 2068 6967 6865 7220 aps in a higher │ │ │ │ -00032310: 636f 6469 6d65 6e73 696f 6e20 6d61 7472 codimension matr │ │ │ │ -00032320: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -00032330: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -00032340: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -00032350: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -00032360: 6563 7420 2a6e 6f74 6520 4c61 7965 7265 ect *note Layere │ │ │ │ -00032370: 643a 204c 6179 6572 6564 2c20 6973 2061 d: Layered, is a │ │ │ │ -00032380: 202a 6e6f 7465 2073 796d 626f 6c3a 2028 *note symbol: ( │ │ │ │ -00032390: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ -000323a0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ +00032280: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ +00032290: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +000322a0: 6f6e 282e 2e2e 2c4c 6179 6572 6564 3d3e on(...,Layered=> │ │ │ │ +000322b0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ +000322c0: 7465 206d 6174 7269 7846 6163 746f 7269 te matrixFactori │ │ │ │ +000322d0: 7a61 7469 6f6e 3a0a 2020 2020 6d61 7472 zation:. matr │ │ │ │ +000322e0: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ +000322f0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ +00032300: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ +00032310: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +00032320: 6174 696f 6e0a 0a46 6f72 2074 6865 2070 ation..For the p │ │ │ │ +00032330: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00032340: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00032350: 6520 6f62 6a65 6374 202a 6e6f 7465 204c e object *note L │ │ │ │ +00032360: 6179 6572 6564 3a20 4c61 7965 7265 642c ayered: Layered, │ │ │ │ +00032370: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +00032380: 6f6c 3a20 284d 6163 6175 6c61 7932 446f ol: (Macaulay2Do │ │ │ │ +00032390: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ +000323a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323f0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00032400: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00032410: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00032420: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00032430: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00032440: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ -00032450: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00032460: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00032470: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00032480: 323a 3332 3531 3a30 2e0a 1f0a 4669 6c65 2:3251:0....File │ │ │ │ -00032490: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -000324a0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -000324b0: 732e 696e 666f 2c20 4e6f 6465 3a20 6c61 s.info, Node: la │ │ │ │ -000324c0: 7965 7265 6452 6573 6f6c 7574 696f 6e2c yeredResolution, │ │ │ │ -000324d0: 204e 6578 743a 204c 6966 742c 2050 7265 Next: Lift, Pre │ │ │ │ -000324e0: 763a 204c 6179 6572 6564 2c20 5570 3a20 v: Layered, Up: │ │ │ │ -000324f0: 546f 700a 0a6c 6179 6572 6564 5265 736f Top..layeredReso │ │ │ │ -00032500: 6c75 7469 6f6e 202d 2d20 6c61 7965 7265 lution -- layere │ │ │ │ -00032510: 6420 6669 6e69 7465 2061 6e64 2069 6e66 d finite and inf │ │ │ │ -00032520: 696e 6974 6520 6c61 7965 7265 6420 7265 inite layered re │ │ │ │ -00032530: 736f 6c75 7469 6f6e 7320 6f66 2043 4d20 solutions of CM │ │ │ │ -00032540: 6d6f 6475 6c65 730a 2a2a 2a2a 2a2a 2a2a modules.******** │ │ │ │ +000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +000323f0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00032400: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00032410: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00032420: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00032430: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ +00032440: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00032450: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00032460: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00032470: 6f6e 732e 6d32 3a33 3235 313a 302e 0a1f ons.m2:3251:0... │ │ │ │ +00032480: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00032490: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000324a0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +000324b0: 653a 206c 6179 6572 6564 5265 736f 6c75 e: layeredResolu │ │ │ │ +000324c0: 7469 6f6e 2c20 4e65 7874 3a20 4c69 6674 tion, Next: Lift │ │ │ │ +000324d0: 2c20 5072 6576 3a20 4c61 7965 7265 642c , Prev: Layered, │ │ │ │ +000324e0: 2055 703a 2054 6f70 0a0a 6c61 7965 7265 Up: Top..layere │ │ │ │ +000324f0: 6452 6573 6f6c 7574 696f 6e20 2d2d 206c dResolution -- l │ │ │ │ +00032500: 6179 6572 6564 2066 696e 6974 6520 616e ayered finite an │ │ │ │ +00032510: 6420 696e 6669 6e69 7465 206c 6179 6572 d infinite layer │ │ │ │ +00032520: 6564 2072 6573 6f6c 7574 696f 6e73 206f ed resolutions o │ │ │ │ +00032530: 6620 434d 206d 6f64 756c 6573 0a2a 2a2a f CM modules.*** │ │ │ │ +00032540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032590: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000325a0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000325b0: 2846 462c 2061 7567 2920 3d20 6c61 7965 (FF, aug) = laye │ │ │ │ -000325c0: 7265 6452 6573 6f6c 7574 696f 6e28 6666 redResolution(ff │ │ │ │ -000325d0: 2c4d 290a 2020 2020 2020 2020 2846 462c ,M). (FF, │ │ │ │ -000325e0: 2061 7567 2920 3d20 6c61 7965 7265 6452 aug) = layeredR │ │ │ │ -000325f0: 6573 6f6c 7574 696f 6e28 6666 2c4d 2c6c esolution(ff,M,l │ │ │ │ -00032600: 656e 290a 2020 2a20 496e 7075 7473 3a0a en). * Inputs:. │ │ │ │ -00032610: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ -00032620: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00032630: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00032640: 2c2c 2031 2078 2063 206d 6174 7269 7820 ,, 1 x c matrix │ │ │ │ -00032650: 7768 6f73 6520 656e 7472 6965 730a 2020 whose entries. │ │ │ │ -00032660: 2020 2020 2020 6172 6520 6120 7265 6775 are a regu │ │ │ │ -00032670: 6c61 7220 7365 7175 656e 6365 2069 6e20 lar sequence in │ │ │ │ -00032680: 7468 6520 476f 7265 6e73 7465 696e 2072 the Gorenstein r │ │ │ │ -00032690: 696e 6720 530a 2020 2020 2020 2a20 4d2c ing S. * M, │ │ │ │ -000326a0: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ -000326b0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -000326c0: 6f64 756c 652c 2c20 4d43 4d20 6d6f 6475 odule,, MCM modu │ │ │ │ -000326d0: 6c65 206f 7665 7220 522c 0a20 2020 2020 le over R,. │ │ │ │ -000326e0: 2020 2072 6570 7265 7365 6e74 6564 2061 represented a │ │ │ │ -000326f0: 7320 616e 2053 2d6d 6f64 756c 6520 696e s an S-module in │ │ │ │ -00032700: 2074 6865 2066 6972 7374 2063 6173 6520 the first case │ │ │ │ -00032710: 616e 6420 6173 2061 6e20 522d 6d6f 6475 and as an R-modu │ │ │ │ -00032720: 6c65 2069 6e20 7468 650a 2020 2020 2020 le in the. │ │ │ │ -00032730: 2020 7365 636f 6e64 0a20 2020 2020 202a second. * │ │ │ │ -00032740: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -00032750: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -00032760: 7932 446f 6329 5a5a 2c2c 206c 656e 6774 y2Doc)ZZ,, lengt │ │ │ │ -00032770: 6820 6f66 2074 6865 2073 6567 6d65 6e74 h of the segment │ │ │ │ -00032780: 206f 6620 7468 650a 2020 2020 2020 2020 of the. │ │ │ │ -00032790: 7265 736f 6c75 7469 6f6e 2074 6f20 6265 resolution to be │ │ │ │ -000327a0: 2063 6f6d 7075 7465 6420 6f76 6572 2052 computed over R │ │ │ │ -000327b0: 2c20 696e 2074 6865 2073 6563 6f6e 6420 , in the second │ │ │ │ -000327c0: 666f 726d 2e0a 2020 2a20 2a6e 6f74 6520 form.. * *note │ │ │ │ -000327d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ -000327e0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ -000327f0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ -00032800: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ -00032810: 7574 732c 3a0a 2020 2020 2020 2a20 4368 uts,:. * Ch │ │ │ │ -00032820: 6563 6b20 3d3e 202e 2e2e 2c20 6465 6661 eck => ..., defa │ │ │ │ -00032830: 756c 7420 7661 6c75 6520 6661 6c73 650a ult value false. │ │ │ │ -00032840: 2020 2020 2020 2a20 5665 7262 6f73 6520 * Verbose │ │ │ │ -00032850: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00032860: 7661 6c75 6520 6661 6c73 650a 2020 2a20 value false. * │ │ │ │ -00032870: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00032880: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -00032890: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -000328a0: 2943 6f6d 706c 6578 2c2c 2072 6573 6f6c )Complex,, resol │ │ │ │ -000328b0: 7574 696f 6e20 6f66 204d 206f 7665 7220 ution of M over │ │ │ │ -000328c0: 5320 696e 2074 6865 0a20 2020 2020 2020 S in the. │ │ │ │ -000328d0: 2066 6972 7374 2063 6173 653b 206c 656e first case; len │ │ │ │ -000328e0: 6774 6820 6c65 6e20 7365 676d 656e 7420 gth len segment │ │ │ │ -000328f0: 6f66 2074 6865 2072 6573 6f6c 7574 696f of the resolutio │ │ │ │ -00032900: 6e20 6f76 6572 2052 2069 6e20 7468 6520 n over R in the │ │ │ │ -00032910: 7365 636f 6e64 2e0a 0a44 6573 6372 6970 second...Descrip │ │ │ │ -00032920: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00032930: 0a0a 5468 6520 7265 736f 6c75 7469 6f6e ..The resolution │ │ │ │ -00032940: 7320 636f 6d70 7574 6564 2061 7265 2074 s computed are t │ │ │ │ -00032950: 686f 7365 2064 6573 6372 6962 6564 2069 hose described i │ │ │ │ -00032960: 6e20 7468 6520 7061 7065 7220 224c 6179 n the paper "Lay │ │ │ │ -00032970: 6572 6564 2052 6573 6f6c 7574 696f 6e73 ered Resolutions │ │ │ │ -00032980: 0a6f 6620 436f 6865 6e2d 4d61 6361 756c .of Cohen-Macaul │ │ │ │ -00032990: 6179 206d 6f64 756c 6573 2220 6279 2045 ay modules" by E │ │ │ │ -000329a0: 6973 656e 6275 6420 616e 6420 5065 6576 isenbud and Peev │ │ │ │ -000329b0: 612e 2054 6865 7920 6172 6520 626f 7468 a. They are both │ │ │ │ -000329c0: 206d 696e 696d 616c 2077 6865 6e20 4d0a minimal when M. │ │ │ │ -000329d0: 6973 2061 2073 7566 6669 6369 656e 746c is a sufficientl │ │ │ │ -000329e0: 7920 6869 6768 2073 797a 7967 7920 6f66 y high syzygy of │ │ │ │ -000329f0: 2061 206d 6f64 756c 6520 4e2e 2049 6620 a module N. If │ │ │ │ -00032a00: 7468 6520 6f70 7469 6f6e 2056 6572 626f the option Verbo │ │ │ │ -00032a10: 7365 3d3e 7472 7565 2069 730a 7365 742c se=>true is.set, │ │ │ │ -00032a20: 2074 6865 6e20 2869 6e20 7468 6520 6361 then (in the ca │ │ │ │ -00032a30: 7365 206f 6620 7468 6520 7265 736f 6c75 se of the resolu │ │ │ │ -00032a40: 7469 6f6e 206f 7665 7220 5329 2074 6865 tion over S) the │ │ │ │ -00032a50: 2072 616e 6b73 206f 6620 7468 6520 6d6f ranks of the mo │ │ │ │ -00032a60: 6475 6c65 7320 425f 730a 696e 2074 6865 dules B_s.in the │ │ │ │ -00032a70: 2072 6573 6f6c 7574 696f 6e20 6172 6520 resolution are │ │ │ │ -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 │ │ │ │ -00032ac0: 6f6c 7574 696f 6e3a 0a0a 2b2d 2d2d 2d2d olution:..+----- │ │ │ │ +00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00032590: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000325a0: 2020 2020 2028 4646 2c20 6175 6729 203d (FF, aug) = │ │ │ │ +000325b0: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ +000325c0: 6f6e 2866 662c 4d29 0a20 2020 2020 2020 on(ff,M). │ │ │ │ +000325d0: 2028 4646 2c20 6175 6729 203d 206c 6179 (FF, aug) = lay │ │ │ │ +000325e0: 6572 6564 5265 736f 6c75 7469 6f6e 2866 eredResolution(f │ │ │ │ +000325f0: 662c 4d2c 6c65 6e29 0a20 202a 2049 6e70 f,M,len). * Inp │ │ │ │ +00032600: 7574 733a 0a20 2020 2020 202a 2066 662c uts:. * ff, │ │ │ │ +00032610: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00032620: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +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 │ │ │ │ +00032660: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ +00032670: 6520 696e 2074 6865 2047 6f72 656e 7374 e in the Gorenst │ │ │ │ +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 │ │ │ │ +000326e0: 7465 6420 6173 2061 6e20 532d 6d6f 6475 ted as an S-modu │ │ │ │ +000326f0: 6c65 2069 6e20 7468 6520 6669 7273 7420 le in the first │ │ │ │ +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 │ │ │ │ +00032740: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +00032750: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +00032760: 6c65 6e67 7468 206f 6620 7468 6520 7365 length of the se │ │ │ │ +00032770: 676d 656e 7420 6f66 2074 6865 0a20 2020 gment of the. │ │ │ │ +00032780: 2020 2020 2072 6573 6f6c 7574 696f 6e20 resolution │ │ │ │ +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 │ │ │ │ +000327b0: 636f 6e64 2066 6f72 6d2e 0a20 202a 202a cond form.. * * │ │ │ │ +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 │ │ │ │ +000327f0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +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 │ │ │ │ +00032830: 616c 7365 0a20 2020 2020 202a 2056 6572 alse. * Ver │ │ │ │ +00032840: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566 bose => ..., def │ │ │ │ +00032850: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ +00032860: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00032870: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +00032880: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +00032890: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +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. │ │ │ │ +000328c0: 2020 2020 2020 6669 7273 7420 6361 7365 first case │ │ │ │ +000328d0: 3b20 6c65 6e67 7468 206c 656e 2073 6567 ; length len seg │ │ │ │ +000328e0: 6d65 6e74 206f 6620 7468 6520 7265 736f ment of the reso │ │ │ │ +000328f0: 6c75 7469 6f6e 206f 7665 7220 5220 696e lution over R in │ │ │ │ +00032900: 2074 6865 2073 6563 6f6e 642e 0a0a 4465 the second...De │ │ │ │ +00032910: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00032920: 3d3d 3d3d 3d0a 0a54 6865 2072 6573 6f6c =====..The resol │ │ │ │ +00032930: 7574 696f 6e73 2063 6f6d 7075 7465 6420 utions computed │ │ │ │ +00032940: 6172 6520 7468 6f73 6520 6465 7363 7269 are those descri │ │ │ │ +00032950: 6265 6420 696e 2074 6865 2070 6170 6572 bed in the paper │ │ │ │ +00032960: 2022 4c61 7965 7265 6420 5265 736f 6c75 "Layered Resolu │ │ │ │ +00032970: 7469 6f6e 730a 6f66 2043 6f68 656e 2d4d tions.of Cohen-M │ │ │ │ +00032980: 6163 6175 6c61 7920 6d6f 6475 6c65 7322 acaulay modules" │ │ │ │ +00032990: 2062 7920 4569 7365 6e62 7564 2061 6e64 by Eisenbud and │ │ │ │ +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 │ │ │ │ +000329d0: 6965 6e74 6c79 2068 6967 6820 7379 7a79 iently high syzy │ │ │ │ +000329e0: 6779 206f 6620 6120 6d6f 6475 6c65 204e gy of a module N │ │ │ │ +000329f0: 2e20 4966 2074 6865 206f 7074 696f 6e20 . If the option │ │ │ │ +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 │ │ │ │ +00032a50: 6865 206d 6f64 756c 6573 2042 5f73 0a69 he modules B_s.i │ │ │ │ +00032a60: 6e20 7468 6520 7265 736f 6c75 7469 6f6e n the resolution │ │ │ │ +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 │ │ │ │ +00032aa0: 6d73 206f 6620 616e 2069 6e66 696e 6974 ms of an infinit │ │ │ │ +00032ab0: 6520 7265 736f 6c75 7469 6f6e 3a0a 0a2b e resolution:..+ │ │ │ │ +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 : │ │ │ │ -00032c60: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +00032c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032c50: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00032c60: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 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 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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 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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 │ │ │ │ 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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 -------------+.| 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+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 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2020 2020 2020 2020 │ │ │ │ -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 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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 │ │ │ │ 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|.| │ │ │ │ +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 │ │ │ │ -00037bf0: 6e20 7769 7468 206f 7074 696f 6e73 3a20 n with options: │ │ │ │ -00037c00: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00037c10: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00037c20: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 ----------+..Way │ │ │ │ +00037ae0: 7320 746f 2075 7365 206c 6179 6572 6564 s to use layered │ │ │ │ +00037af0: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ +00037b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00037b20: 226c 6179 6572 6564 5265 736f 6c75 7469 "layeredResoluti │ │ │ │ +00037b30: 6f6e 284d 6174 7269 782c 4d6f 6475 6c65 on(Matrix,Module │ │ │ │ +00037b40: 2922 0a20 202a 2022 6c61 7965 7265 6452 )". * "layeredR │ │ │ │ +00037b50: 6573 6f6c 7574 696f 6e28 4d61 7472 6978 esolution(Matrix │ │ │ │ +00037b60: 2c4d 6f64 756c 652c 5a5a 2922 0a0a 466f ,Module,ZZ)"..Fo │ │ │ │ +00037b70: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00037b80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00037b90: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00037ba0: 2a6e 6f74 6520 6c61 7965 7265 6452 6573 *note layeredRes │ │ │ │ +00037bb0: 6f6c 7574 696f 6e3a 206c 6179 6572 6564 olution: layered │ │ │ │ +00037bc0: 5265 736f 6c75 7469 6f6e 2c20 6973 2061 Resolution, is a │ │ │ │ +00037bd0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +00037be0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ +00037bf0: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00037c00: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00037c10: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00037c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c70: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00037c80: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00037c90: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00037ca0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00037cb0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00037cc0: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ -00037cd0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00037ce0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00037cf0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00037d00: 6e73 2e6d 323a 3438 3935 3a30 2e0a 1f0a ns.m2:4895:0.... │ │ │ │ -00037d10: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00037d20: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00037d30: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00037d40: 3a20 4c69 6674 2c20 4e65 7874 3a20 6d61 : Lift, Next: ma │ │ │ │ -00037d50: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -00037d60: 6f6e 2c20 5072 6576 3a20 6c61 7965 7265 on, Prev: layere │ │ │ │ -00037d70: 6452 6573 6f6c 7574 696f 6e2c 2055 703a dResolution, Up: │ │ │ │ -00037d80: 2054 6f70 0a0a 4c69 6674 202d 2d20 4f70 Top..Lift -- Op │ │ │ │ -00037d90: 7469 6f6e 2066 6f72 206e 6577 4578 740a tion for newExt. │ │ │ │ -00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00037db0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00037dc0: 7361 6765 3a20 0a20 2020 2020 2020 206e sage: . n │ │ │ │ -00037dd0: 6577 4578 7428 4d2c 4e2c 4368 6563 6b20 ewExt(M,N,Check │ │ │ │ -00037de0: 3d3e 7472 7565 290a 2020 2a20 496e 7075 =>true). * Inpu │ │ │ │ -00037df0: 7473 3a0a 2020 2020 2020 2a20 4368 6563 ts:. * Chec │ │ │ │ -00037e00: 6b2c 2061 202a 6e6f 7465 2042 6f6f 6c65 k, a *note Boole │ │ │ │ -00037e10: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -00037e20: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -00037e30: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -00037e40: 3d3d 3d3d 3d3d 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Int │ │ │ │ -00037ee0: 6572 7365 6374 696f 6e0a 0a46 756e 6374 ersection..Funct │ │ │ │ -00037ef0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -00037f00: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ -00037f10: 6420 4c69 6674 3a0a 3d3d 3d3d 3d3d 3d3d d Lift:.======== │ │ │ │ +00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00037c70: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00037c80: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00037c90: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00037ca0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00037cb0: 756c 6179 322d 312e 3235 2e30 362b 6473 ulay2-1.25.06+ds │ │ │ │ +00037cc0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00037cd0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00037ce0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00037cf0: 6c75 7469 6f6e 732e 6d32 3a34 3839 353a lutions.m2:4895: │ │ │ │ +00037d00: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00037d10: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00037d20: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00037d30: 204e 6f64 653a 204c 6966 742c 204e 6578 Node: Lift, Nex │ │ │ │ +00037d40: 743a 206d 616b 6546 696e 6974 6552 6573 t: makeFiniteRes │ │ │ │ +00037d50: 6f6c 7574 696f 6e2c 2050 7265 763a 206c olution, Prev: l │ │ │ │ +00037d60: 6179 6572 6564 5265 736f 6c75 7469 6f6e ayeredResolution │ │ │ │ +00037d70: 2c20 5570 3a20 546f 700a 0a4c 6966 7420 , Up: Top..Lift │ │ │ │ +00037d80: 2d2d 204f 7074 696f 6e20 666f 7220 6e65 -- Option for ne │ │ │ │ +00037d90: 7745 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a wExt.*********** │ │ │ │ +00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00037db0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00037dc0: 2020 2020 6e65 7745 7874 284d 2c4e 2c43 newExt(M,N,C │ │ │ │ +00037dd0: 6865 636b 203d 3e74 7275 6529 0a20 202a heck =>true). * │ │ │ │ +00037de0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00037df0: 2043 6865 636b 2c20 6120 2a6e 6f74 6520 Check, a *note │ │ │ │ +00037e00: 426f 6f6c 6561 6e20 7661 6c75 653a 2028 Boolean value: ( │ │ │ │ +00037e10: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +00037e20: 6c65 616e 2c2c 200a 0a44 6573 6372 6970 lean,, ..Descrip │ │ │ │ +00037e30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00037e40: 0a0a 4d61 6b65 7320 6e65 7745 7874 2070 ..Makes newExt p │ │ │ │ +00037e50: 6572 666f 726d 2076 6172 696f 7573 2063 erform various c │ │ │ │ +00037e60: 6865 636b 7320 6173 2069 7420 636f 6d70 hecks as it comp │ │ │ │ +00037e70: 7574 6573 2e0a 0a53 6565 2061 6c73 6f0a utes...See also. │ │ │ │ +00037e80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00037e90: 6f74 6520 6e65 7745 7874 3a20 6e65 7745 ote newExt: newE │ │ │ │ +00037ea0: 7874 2c20 2d2d 2047 6c6f 6261 6c20 4578 xt, -- Global Ex │ │ │ │ +00037eb0: 7420 666f 7220 6d6f 6475 6c65 7320 6f76 t for modules ov │ │ │ │ +00037ec0: 6572 2061 2063 6f6d 706c 6574 650a 2020 er a complete. │ │ │ │ +00037ed0: 2020 496e 7465 7273 6563 7469 6f6e 0a0a Intersection.. │ │ │ │ +00037ee0: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ +00037ef0: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ +00037f00: 206e 616d 6564 204c 6966 743a 0a3d 3d3d named Lift:.=== │ │ │ │ +00037f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f40: 3d3d 3d3d 0a0a 2020 2a20 226e 6577 4578 ====.. * "newEx │ │ │ │ -00037f50: 7428 2e2e 2e2c 4c69 6674 3d3e 2e2e 2e29 t(...,Lift=>...) │ │ │ │ -00037f60: 2220 2d2d 2073 6565 202a 6e6f 7465 206e " -- see *note n │ │ │ │ -00037f70: 6577 4578 743a 206e 6577 4578 742c 202d ewExt: newExt, - │ │ │ │ -00037f80: 2d20 476c 6f62 616c 2045 7874 2066 6f72 - Global Ext for │ │ │ │ -00037f90: 0a20 2020 206d 6f64 756c 6573 206f 7665 . modules ove │ │ │ │ -00037fa0: 7220 6120 636f 6d70 6c65 7465 2049 6e74 r a complete Int │ │ │ │ -00037fb0: 6572 7365 6374 696f 6e0a 0a46 6f72 2074 ersection..For t │ │ │ │ -00037fc0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00037fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037fe0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00037ff0: 7465 204c 6966 743a 204c 6966 742c 2069 te Lift: Lift, i │ │ │ │ -00038000: 7320 6120 2a6e 6f74 6520 7379 6d62 6f6c s a *note symbol │ │ │ │ -00038010: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00038020: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d Symbol,...------ │ │ │ │ +00037f30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ 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7465 496e 7465 7273 /.CompleteInters │ │ │ │ -000380f0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00038100: 732e 6d32 3a33 3139 383a 302e 0a1f 0a46 s.m2:3198:0....F │ │ │ │ -00038110: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -00038120: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00038130: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -00038140: 206d 616b 6546 696e 6974 6552 6573 6f6c makeFiniteResol │ │ │ │ -00038150: 7574 696f 6e2c 204e 6578 743a 206d 616b ution, Next: mak │ │ │ │ -00038160: 6546 696e 6974 6552 6573 6f6c 7574 696f eFiniteResolutio │ │ │ │ -00038170: 6e43 6f64 696d 322c 2050 7265 763a 204c nCodim2, Prev: L │ │ │ │ -00038180: 6966 742c 2055 703a 2054 6f70 0a0a 6d61 ift, Up: Top..ma │ │ │ │ -00038190: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -000381a0: 6f6e 202d 2d20 6669 6e69 7465 2072 6573 on -- finite res │ │ │ │ -000381b0: 6f6c 7574 696f 6e20 6f66 2061 206d 6174 olution of a mat │ │ │ │ -000381c0: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -000381d0: 6e20 6d6f 6475 6c65 204d 0a2a 2a2a 2a2a n module M.***** │ │ │ │ +00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00038070: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00038080: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00038090: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000380a0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000380b0: 6c61 7932 2d31 2e32 352e 3036 2b64 732f lay2-1.25.06+ds/ │ │ │ │ +000380c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000380d0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +000380e0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000380f0: 7574 696f 6e73 2e6d 323a 3331 3938 3a30 utions.m2:3198:0 │ │ │ │ +00038100: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +00038110: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00038120: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +00038130: 4e6f 6465 3a20 6d61 6b65 4669 6e69 7465 Node: makeFinite │ │ │ │ +00038140: 5265 736f 6c75 7469 6f6e 2c20 4e65 7874 Resolution, Next │ │ │ │ +00038150: 3a20 6d61 6b65 4669 6e69 7465 5265 736f : makeFiniteReso │ │ │ │ +00038160: 6c75 7469 6f6e 436f 6469 6d32 2c20 5072 lutionCodim2, Pr │ │ │ │ +00038170: 6576 3a20 4c69 6674 2c20 5570 3a20 546f ev: Lift, Up: To │ │ │ │ +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 ---------------- │ │ │ │ 00038a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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│ 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 │ │ │ │ -00038b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b10: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038b00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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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 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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} | │ │ │ │ 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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 |.+- │ │ │ │ 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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 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│ │ +0003d200: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +0003d210: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0003d220: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +0003d230: 204e 6f64 653a 206d 616b 6546 696e 6974 Node: makeFinit │ │ │ │ +0003d240: 6552 6573 6f6c 7574 696f 6e43 6f64 696d eResolutionCodim │ │ │ │ +0003d250: 322c 204e 6578 743a 206d 616b 6548 6f6d 2, Next: makeHom │ │ │ │ +0003d260: 6f74 6f70 6965 732c 2050 7265 763a 206d otopies, Prev: m │ │ │ │ +0003d270: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ +0003d280: 696f 6e2c 2055 703a 2054 6f70 0a0a 6d61 ion, Up: Top..ma │ │ │ │ +0003d290: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d2a0: 6f6e 436f 6469 6d32 202d 2d20 4d61 7073 onCodim2 -- Maps │ │ │ │ +0003d2b0: 2061 7373 6f63 6961 7465 6420 746f 2074 associated to t │ │ │ │ +0003d2c0: 6865 2066 696e 6974 6520 7265 736f 6c75 he finite resolu │ │ │ │ +0003d2d0: 7469 6f6e 206f 6620 6120 6869 6768 2073 tion of a high s │ │ │ │ +0003d2e0: 797a 7967 7920 6d6f 6475 6c65 2069 6e20 yzygy module in │ │ │ │ +0003d2f0: 636f 6469 6d20 320a 2a2a 2a2a 2a2a 2a2a codim 2.******** │ │ │ │ +0003d300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 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 │ │ │ │ -0003d390: 6974 6552 6573 6f6c 7574 696f 6e43 6f64 iteResolutionCod │ │ │ │ -0003d3a0: 696d 3228 6666 2c6d 6629 0a20 202a 2049 im2(ff,mf). * I │ │ │ │ -0003d3b0: 6e70 7574 733a 0a20 2020 2020 202a 206d nputs:. * m │ │ │ │ -0003d3c0: 662c 2061 202a 6e6f 7465 206c 6973 743a f, a *note list: │ │ │ │ -0003d3d0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0003d3e0: 6973 742c 2c20 6d61 7472 6978 2066 6163 ist,, matrix fac │ │ │ │ -0003d3f0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020 torization. │ │ │ │ -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 │ │ │ │ -0003d430: 6775 6c61 7220 7365 7175 656e 6365 0a20 gular sequence. │ │ │ │ -0003d440: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ -0003d450: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ -0003d460: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ -0003d470: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -0003d480: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ -0003d490: 2020 2020 202a 2043 6865 636b 203d 3e20 * Check => │ │ │ │ -0003d4a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -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 │ │ │ │ -0003d4f0: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ -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 │ │ │ │ -0003d530: 6469 6d20 3220 6d61 7472 6978 2066 6163 dim 2 matrix fac │ │ │ │ -0003d540: 746f 7269 7a61 7469 6f6e 2c20 6d61 6b65 torization, make │ │ │ │ -0003d550: 7320 616c 6c20 7468 6520 636f 6d70 6f6e s all the compon │ │ │ │ -0003d560: 656e 7473 206f 6620 7468 650a 6469 6666 ents of the.diff │ │ │ │ -0003d570: 6572 656e 7469 616c 2061 6e64 206f 6620 erential and of │ │ │ │ -0003d580: 7468 6520 686f 6d6f 746f 7069 6573 2074 the homotopies t │ │ │ │ -0003d590: 6861 7420 6172 6520 7265 6c65 7661 6e74 hat are relevant │ │ │ │ -0003d5a0: 2074 6f20 7468 6520 6669 6e69 7465 2072 to the finite r │ │ │ │ -0003d5b0: 6573 6f6c 7574 696f 6e2c 0a61 7320 696e esolution,.as in │ │ │ │ -0003d5c0: 2034 2e32 2e33 206f 6620 4569 7365 6e62 4.2.3 of Eisenb │ │ │ │ -0003d5d0: 7564 2d50 6565 7661 2022 4d69 6e69 6d61 ud-Peeva "Minima │ │ │ │ -0003d5e0: 6c20 4672 6565 2052 6573 6f6c 7574 696f l Free Resolutio │ │ │ │ -0003d5f0: 6e73 2061 6e64 2048 6967 6865 7220 4d61 ns and Higher Ma │ │ │ │ -0003d600: 7472 6978 0a46 6163 746f 7269 7a61 7469 trix.Factorizati │ │ │ │ -0003d610: 6f6e 7322 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons"..+--------- │ │ │ │ +0003d360: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0003d370: 2020 2020 2020 206d 6170 7320 3d20 6d61 maps = ma │ │ │ │ +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 │ │ │ │ +0003d3e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ +0003d3f0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0003d400: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0003d410: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0003d420: 2c2c 2072 6567 756c 6172 2073 6571 7565 ,, regular seque │ │ │ │ +0003d430: 6e63 650a 2020 2a20 2a6e 6f74 6520 4f70 nce. * *note Op │ │ │ │ +0003d440: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ +0003d450: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ +0003d460: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ +0003d470: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ +0003d480: 732c 3a0a 2020 2020 2020 2a20 4368 6563 s,:. * Chec │ │ │ │ +0003d490: 6b20 3d3e 202e 2e2e 2c20 6465 6661 756c k => ..., defaul │ │ │ │ +0003d4a0: 7420 7661 6c75 6520 6661 6c73 650a 2020 t value false. │ │ │ │ +0003d4b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0003d4c0: 202a 206d 6170 732c 2061 202a 6e6f 7465 * maps, a *note │ │ │ │ +0003d4d0: 2068 6173 6820 7461 626c 653a 2028 4d61 hash table: (Ma │ │ │ │ +0003d4e0: 6361 756c 6179 3244 6f63 2948 6173 6854 caulay2Doc)HashT │ │ │ │ +0003d4f0: 6162 6c65 2c2c 206d 616e 7920 6d61 7073 able,, many maps │ │ │ │ +0003d500: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0003d510: 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e =========..Given │ │ │ │ +0003d520: 2061 2063 6f64 696d 2032 206d 6174 7269 a codim 2 matri │ │ │ │ +0003d530: 7820 6661 6374 6f72 697a 6174 696f 6e2c x factorization, │ │ │ │ +0003d540: 206d 616b 6573 2061 6c6c 2074 6865 2063 makes all the c │ │ │ │ +0003d550: 6f6d 706f 6e65 6e74 7320 6f66 2074 6865 omponents of the │ │ │ │ +0003d560: 0a64 6966 6665 7265 6e74 6961 6c20 616e .differential an │ │ │ │ +0003d570: 6420 6f66 2074 6865 2068 6f6d 6f74 6f70 d of the homotop │ │ │ │ +0003d580: 6965 7320 7468 6174 2061 7265 2072 656c ies that are rel │ │ │ │ +0003d590: 6576 616e 7420 746f 2074 6865 2066 696e evant to the fin │ │ │ │ +0003d5a0: 6974 6520 7265 736f 6c75 7469 6f6e 2c0a ite resolution,. │ │ │ │ +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 │ │ │ │ +0003d5d0: 696e 696d 616c 2046 7265 6520 5265 736f inimal Free Reso │ │ │ │ +0003d5e0: 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2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003d680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003d690: 0a7c 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 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ -0003d720: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d710: 2020 2020 207c 0a7c 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 207c 0a7c 6f31 203a 2051 756f 7469 |.|o1 : Quoti │ │ │ │ -0003d770: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0003d750: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +0003d760: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +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 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003d790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0003d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0003d7f0: 2053 203d 206b 6b5b 612c 625d 2020 2020 S = kk[a,b] │ │ │ │ +0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +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 |.| │ │ │ │ +0003d820: 207c 0a7c 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 2020 2020 2020 2020 │ │ │ │ -0003d860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003d870: 0a7c 6f32 203d 2053 2020 2020 2020 2020 .|o2 = S │ │ │ │ +0003d860: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 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2020 2020 2a20 622c 2061 6e20 2a6e * b, an *n │ │ │ │ -0003f780: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -0003f790: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -0003f7a0: 686f 7720 6661 7220 6261 636b 2074 6f20 how far back to │ │ │ │ -0003f7b0: 636f 6d70 7574 6520 7468 650a 2020 2020 compute the. │ │ │ │ -0003f7c0: 2020 2020 686f 6d6f 746f 7069 6573 2028 homotopies ( │ │ │ │ -0003f7d0: 6465 6661 756c 7473 2074 6f20 6c65 6e67 defaults to leng │ │ │ │ -0003f7e0: 7468 206f 6620 4629 0a20 202a 204f 7574 th of F). * Out │ │ │ │ -0003f7f0: 7075 7473 3a0a 2020 2020 2020 2a20 482c puts:. * H, │ │ │ │ -0003f800: 2061 202a 6e6f 7465 2068 6173 6820 7461 a *note hash ta │ │ │ │ -0003f810: 626c 653a 2028 4d61 6361 756c 6179 3244 ble: (Macaulay2D │ │ │ │ -0003f820: 6f63 2948 6173 6854 6162 6c65 2c2c 2067 oc)HashTable,, g │ │ │ │ -0003f830: 6976 6573 2074 6865 2068 6967 6865 720a ives the higher. │ │ │ │ -0003f840: 2020 2020 2020 2020 686f 6d6f 746f 7079 homotopy │ │ │ │ -0003f850: 2066 726f 6d20 465f 6920 636f 7272 6573 from F_i corres │ │ │ │ -0003f860: 706f 6e64 696e 6720 746f 2061 206d 6f6e ponding to a mon │ │ │ │ -0003f870: 6f6d 6961 6c20 7769 7468 2065 7870 6f6e omial with expon │ │ │ │ -0003f880: 656e 7420 7665 6374 6f72 204c 2061 730a ent vector L as. │ │ │ │ -0003f890: 2020 2020 2020 2020 7468 6520 7661 6c75 the valu │ │ │ │ -0003f8a0: 6520 2448 235c 7b4c 2c69 5c7d 240a 0a44 e $H#\{L,i\}$..D │ │ │ │ -0003f8b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0003f8c0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120 ======..Given a │ │ │ │ -0003f8d0: 2431 5c74 696d 6573 206e 2420 6d61 7472 $1\times n$ matr │ │ │ │ -0003f8e0: 6978 2066 2c20 616e 6420 6120 6368 6169 ix f, and a chai │ │ │ │ -0003f8f0: 6e20 636f 6d70 6c65 7820 462c 2074 6865 n complex F, the │ │ │ │ -0003f900: 2073 6372 6970 7420 6174 7465 6d70 7473 script attempts │ │ │ │ -0003f910: 2074 6f0a 6d61 6b65 2061 2066 616d 696c to.make a famil │ │ │ │ -0003f920: 7920 6f66 2068 6967 6865 7220 686f 6d6f y of higher homo │ │ │ │ -0003f930: 746f 7069 6573 206f 6e20 4620 666f 7220 topies on F for │ │ │ │ -0003f940: 7468 6520 656c 656d 656e 7473 206f 6620 the elements of │ │ │ │ -0003f950: 662c 2069 6e20 7468 6520 7365 6e73 650a f, in the sense. │ │ │ │ -0003f960: 6465 7363 7269 6265 642c 2066 6f72 2065 described, for e │ │ │ │ -0003f970: 7861 6d70 6c65 2c20 696e 2045 6973 656e xample, in Eisen │ │ │ │ -0003f980: 6275 6420 2245 6e72 6963 6865 6420 4672 bud "Enriched Fr │ │ │ │ -0003f990: 6565 2052 6573 6f6c 7574 696f 6e73 2061 ee Resolutions a │ │ │ │ -0003f9a0: 6e64 2043 6861 6e67 6520 6f66 0a52 696e nd Change of.Rin │ │ │ │ -0003f9b0: 6773 222e 0a0a 5468 6520 6f75 7470 7574 gs"...The output │ │ │ │ -0003f9c0: 2069 7320 6120 6861 7368 2074 6162 6c65 is a hash table │ │ │ │ -0003f9d0: 2077 6974 6820 656e 7472 6965 7320 6f66 with entries of │ │ │ │ -0003f9e0: 2074 6865 2066 6f72 6d20 245c 7b4a 2c69 the form $\{J,i │ │ │ │ -0003f9f0: 5c7d 3d3e 7324 2c20 7768 6572 6520 4a20 \}=>s$, where J │ │ │ │ -0003fa00: 6973 2061 0a6c 6973 7420 6f66 206e 6f6e is a.list of non │ │ │ │ -0003fa10: 2d6e 6567 6174 6976 6520 696e 7465 6765 -negative intege │ │ │ │ -0003fa20: 7273 2c20 6f66 206c 656e 6774 6820 6e20 rs, of length n │ │ │ │ -0003fa30: 616e 6420 2448 5c23 5c7b 4a2c 695c 7d3a and $H\#\{J,i\}: │ │ │ │ -0003fa40: 2046 5f69 2d3e 465f 7b69 2b32 7c4a 7c2d F_i->F_{i+2|J|- │ │ │ │ -0003fa50: 317d 240a 6172 6520 6d61 7073 2073 6174 1}$.are maps sat │ │ │ │ -0003fa60: 6973 6679 696e 6720 7468 6520 636f 6e64 isfying the cond │ │ │ │ -0003fa70: 6974 696f 6e73 2024 2420 485c 235c 7b65 itions $$ H\#\{e │ │ │ │ -0003fa80: 302c 695c 7d20 3d20 643b 2024 2420 616e 0,i\} = d; $$ an │ │ │ │ -0003fa90: 6420 2424 0a48 235c 7b65 302c 692b 315c d $$.H#\{e0,i+1\ │ │ │ │ -0003faa0: 7d2a 4823 5c7b 652c 695c 7d2b 4823 5c7b }*H#\{e,i\}+H#\{ │ │ │ │ -0003fab0: 652c 692d 315c 7d48 235c 7b65 302c 695c e,i-1\}H#\{e0,i\ │ │ │ │ -0003fac0: 7d20 3d20 665f 692c 2024 2420 7768 6572 } = f_i, $$ wher │ │ │ │ -0003fad0: 6520 2465 3020 3d0a 5c7b 302c 5c64 6f74 e $e0 =.\{0,\dot │ │ │ │ -0003fae0: 732c 305c 7d24 2061 6e64 2024 6524 2069 s,0\}$ and $e$ i │ │ │ │ -0003faf0: 7320 7468 6520 696e 6465 7820 6f66 2064 s the index of d │ │ │ │ -0003fb00: 6567 7265 6520 3120 7769 7468 2061 2031 egree 1 with a 1 │ │ │ │ -0003fb10: 2069 6e20 7468 6520 2469 242d 7468 2070 in the $i$-th p │ │ │ │ -0003fb20: 6c61 6365 3b0a 616e 642c 2066 6f72 2065 lace;.and, for e │ │ │ │ -0003fb30: 6163 6820 696e 6465 7820 6c69 7374 2049 ach index list I │ │ │ │ -0003fb40: 2077 6974 6820 7c49 7c3c 3d64 2c20 2424 with |I|<=d, $$ │ │ │ │ -0003fb50: 2073 756d 5f7b 4a3c 497d 2048 235c 7b49 sum_{Js$, whe │ │ │ │ +0003f9f0: 7265 204a 2069 7320 610a 6c69 7374 206f re J is a.list o │ │ │ │ +0003fa00: 6620 6e6f 6e2d 6e65 6761 7469 7665 2069 f non-negative i │ │ │ │ +0003fa10: 6e74 6567 6572 732c 206f 6620 6c65 6e67 ntegers, of leng │ │ │ │ +0003fa20: 7468 206e 2061 6e64 2024 485c 235c 7b4a th n and $H\#\{J │ │ │ │ +0003fa30: 2c69 5c7d 3a20 465f 692d 3e46 5f7b 692b ,i\}: F_i->F_{i+ │ │ │ │ +0003fa40: 327c 4a7c 2d31 7d24 0a61 7265 206d 6170 2|J|-1}$.are map │ │ │ │ +0003fa50: 7320 7361 7469 7366 7969 6e67 2074 6865 s satisfying the │ │ │ │ +0003fa60: 2063 6f6e 6469 7469 6f6e 7320 2424 2048 conditions $$ H │ │ │ │ +0003fa70: 5c23 5c7b 6530 2c69 5c7d 203d 2064 3b20 \#\{e0,i\} = d; │ │ │ │ +0003fa80: 2424 2061 6e64 2024 240a 4823 5c7b 6530 $$ and $$.H#\{e0 │ │ │ │ +0003fa90: 2c69 2b31 5c7d 2a48 235c 7b65 2c69 5c7d ,i+1\}*H#\{e,i\} │ │ │ │ +0003faa0: 2b48 235c 7b65 2c69 2d31 5c7d 4823 5c7b +H#\{e,i-1\}H#\{ │ │ │ │ +0003fab0: 6530 2c69 5c7d 203d 2066 5f69 2c20 2424 e0,i\} = f_i, $$ │ │ │ │ +0003fac0: 2077 6865 7265 2024 6530 203d 0a5c 7b30 where $e0 =.\{0 │ │ │ │ +0003fad0: 2c5c 646f 7473 2c30 5c7d 2420 616e 6420 ,\dots,0\}$ and │ │ │ │ +0003fae0: 2465 2420 6973 2074 6865 2069 6e64 6578 $e$ is the index │ │ │ │ +0003faf0: 206f 6620 6465 6772 6565 2031 2077 6974 of degree 1 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│ -000411c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000411d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000411e0: 2020 2020 2020 2020 2020 7b33 7d20 7c20 {3} | │ │ │ │ -000411f0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ -00041200: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00041210: 2020 2020 2020 7b7b 312c 2030 2c20 317d {{1, 0, 1} │ │ │ │ -00041220: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020 , -1} => 0 │ │ │ │ -00041230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041240: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041250: 2020 2020 7b7b 312c 2031 2c20 307d 2c20 {{1, 1, 0}, │ │ │ │ -00041260: 2d31 7d20 3d3e 2030 2020 2020 2020 2020 -1} => 0 │ │ │ │ -00041270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041280: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041290: 2020 7b7b 322c 2030 2c20 307d 2c20 2d31 {{2, 0, 0}, -1 │ │ │ │ -000412a0: 7d20 3d3e 2030 2020 2020 2020 2020 2020 } => 0 │ │ │ │ -000412b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000412c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000411a0: 207b 337d 207c 2030 2030 2030 2030 2030 {3} | 0 0 0 0 0 │ │ │ │ +000411b0: 2031 207c 2020 2020 2020 207c 0a7c 2020 1 | |.| │ │ │ │ +000411c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000411d0: 2020 2020 2020 2020 2020 2020 2020 207b { │ │ │ │ +000411e0: 337d 207c 2030 2030 2030 2030 2030 2030 3} | 0 0 0 0 0 0 │ │ │ │ +000411f0: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ +00041200: 2020 2020 2020 2020 2020 207b 7b31 2c20 {{1, │ │ │ │ +00041210: 302c 2031 7d2c 202d 317d 203d 3e20 3020 0, 1}, -1} => 0 │ │ │ │ +00041220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041230: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00041240: 2020 2020 2020 2020 207b 7b31 2c20 312c {{1, 1, │ │ │ │ +00041250: 2030 7d2c 202d 317d 203d 3e20 3020 2020 0}, -1} => 0 │ │ │ │ +00041260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041270: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00041280: 2020 2020 2020 207b 7b32 2c20 302c 2030 {{2, 0, 0 │ │ │ │ +00041290: 7d2c 202d 317d 203d 3e20 3020 2020 2020 }, -1} => 0 │ │ │ │ +000412a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000412b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000412c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000412d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ │ │ -00041380: 6973 2063 6173 6520 7468 6520 6869 6768 is case the high │ │ │ │ -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 │ │ │ │ +00041380: 2068 6967 6865 7220 686f 6d6f 746f 7069 higher homotopi │ │ │ │ +00041390: 6573 2061 7265 2030 3a0a 0a2b 2d2d 2d2d es are 0:..+---- │ │ │ │ +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 │ │ │ │ -000413f0: 203d 2073 6f72 7420 7365 6c65 6374 286b = sort select(k │ │ │ │ -00041400: 6579 7320 686f 6d6f 742c 206b 2d3e 2868 eys homot, k->(h │ │ │ │ -00041410: 6f6d 6f74 236b 213d 3020 616e 6420 7375 omot#k!=0 and su │ │ │ │ -00041420: 6d28 6b5f 3029 3e31 2929 7c0a 7c20 2020 m(k_0)>1))|.| │ │ │ │ +000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000413e0: 3620 3a20 4c20 3d20 736f 7274 2073 656c 6 : L = sort sel │ │ │ │ +000413f0: 6563 7428 6b65 7973 2068 6f6d 6f74 2c20 ect(keys homot, │ │ │ │ +00041400: 6b2d 3e28 686f 6d6f 7423 6b21 3d30 2061 k->(homot#k!=0 a │ │ │ │ +00041410: 6e64 2073 756d 286b 5f30 293e 3129 297c nd sum(k_0)>1))| │ │ │ │ +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 │ │ │ │ -00041580: 6865 206f 7468 6572 2068 616e 642c 2069 he other hand, i │ │ │ │ -00041590: 6620 7765 2074 616b 6520 6120 636f 6d70 f we take a comp │ │ │ │ -000415a0: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -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 │ │ │ │ -000415e0: 6361 7465 6420 7369 7475 6174 696f 6e2c cated situation, │ │ │ │ -000415f0: 2074 6865 2070 726f 6772 616d 2067 6976 the program giv │ │ │ │ -00041600: 6573 206e 6f6e 7a65 726f 2068 6967 6865 es nonzero highe │ │ │ │ -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 │ │ │ │ +000415a0: 6563 7469 6f6e 2061 6e64 2073 6f6d 6574 ection and somet │ │ │ │ +000415b0: 6869 6e67 2063 6f6e 7461 696e 6564 0a69 hing contained.i │ │ │ │ +000415c0: 6e20 6974 2069 6e20 6120 6d6f 7265 2063 n it in a more c │ │ │ │ +000415d0: 6f6d 706c 6963 6174 6564 2073 6974 7561 omplicated situa │ │ │ │ +000415e0: 7469 6f6e 2c20 7468 6520 7072 6f67 7261 tion, the progra │ │ │ │ +000415f0: 6d20 6769 7665 7320 6e6f 6e7a 6572 6f20 m gives nonzero │ │ │ │ +00041600: 6869 6768 6572 0a68 6f6d 6f74 6f70 6965 higher.homotopie │ │ │ │ +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 │ │ │ │ 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│ │ │ │ -000537b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000537c0: 207c 0a7c 6f31 3420 3d20 7b7b 7b30 2c20 |.|o14 = {{{0, │ │ │ │ -000537d0: 327d 2c20 307d 2c20 7b7b 302c 2032 7d2c 2}, 0}, {{0, 2}, │ │ │ │ -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 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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 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│ │ │ │ -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 } │ │ │ │ -00054950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054960: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00054920: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00054930: 3920 3d20 7b7d 2020 2020 2020 2020 2020 9 = {} │ │ │ │ +00054940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00054950: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00054960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054990: 2020 2020 7c0a 7c6f 3139 203a 204c 6973 |.|o19 : Lis │ │ │ │ -000549a0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -000549b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000549c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00054980: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +00054990: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +000549a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000549b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000549c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000549e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000549f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ -00054a00: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00054a10: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ -00054a20: 313a 206d 616b 6548 6f6d 6f74 6f70 6965 1: makeHomotopie │ │ │ │ -00054a30: 7331 2c20 2d2d 2072 6574 7572 6e73 2061 s1, -- returns a │ │ │ │ -00054a40: 2073 7973 7465 6d20 6f66 2066 6972 7374 system of first │ │ │ │ -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 │ │ │ │ -00054ab0: 7472 6978 2c43 6f6d 706c 6578 2922 0a20 trix,Complex)". │ │ │ │ -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 │ │ │ │ -00054af0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00054b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00054b10: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00054b20: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -00054b30: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -00054b40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00054b50: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00054b60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00054b70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000549e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +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 │ │ │ │ -00054bd0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00054be0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00054bf0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00054c00: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00054c10: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -00054c20: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00054c30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -00054c40: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00054c50: 6f6e 732e 6d32 3a33 3737 313a 302e 0a1f ons.m2:3771:0... │ │ │ │ -00054c60: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00054c70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00054c80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -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 │ │ │ │ -00054d00: 7379 7374 656d 206f 6620 6669 7273 7420 system of first │ │ │ │ -00054d10: 686f 6d6f 746f 7069 6573 0a2a 2a2a 2a2a homotopies.***** │ │ │ │ +00054bc0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00054bd0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00054be0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00054bf0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00054c00: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+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 │ │ │ │ -00054da0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ -00054db0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ -00054dc0: 2031 786e 206d 6174 7269 7820 6f66 2065 1xn matrix of e │ │ │ │ -00054dd0: 6c65 6d65 6e74 7320 6f66 2053 0a20 2020 lements of S. │ │ │ │ -00054de0: 2020 202a 2046 2c20 6120 2a6e 6f74 6520 * F, a *note │ │ │ │ -00054df0: 636f 6d70 6c65 783a 2028 436f 6d70 6c65 complex: (Comple │ │ │ │ -00054e00: 7865 7329 436f 6d70 6c65 782c 2c20 6164 xes)Complex,, ad │ │ │ │ -00054e10: 6d69 7474 696e 6720 686f 6d6f 746f 7069 mitting homotopi │ │ │ │ -00054e20: 6573 2066 6f72 2074 6865 0a20 2020 2020 es for the. │ │ │ │ -00054e30: 2020 2065 6e74 7269 6573 206f 6620 660a entries of f. │ │ │ │ -00054e40: 2020 2020 2020 2a20 642c 2061 6e20 2a6e * d, an *n │ │ │ │ -00054e50: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -00054e60: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -00054e70: 686f 7720 6661 7220 6261 636b 2074 6f20 how far back to │ │ │ │ -00054e80: 636f 6d70 7574 6520 7468 650a 2020 2020 compute the. │ │ │ │ -00054e90: 2020 2020 686f 6d6f 746f 7069 6573 2028 homotopies ( │ │ │ │ -00054ea0: 6465 6661 756c 7473 2074 6f20 6c65 6e67 defaults to leng │ │ │ │ -00054eb0: 7468 206f 6620 4629 0a20 202a 204f 7574 th of F). * Out │ │ │ │ -00054ec0: 7075 7473 3a0a 2020 2020 2020 2a20 482c puts:. * H, │ │ │ │ -00054ed0: 2061 202a 6e6f 7465 2068 6173 6820 7461 a *note hash ta │ │ │ │ -00054ee0: 626c 653a 2028 4d61 6361 756c 6179 3244 ble: (Macaulay2D │ │ │ │ -00054ef0: 6f63 2948 6173 6854 6162 6c65 2c2c 2067 oc)HashTable,, g │ │ │ │ -00054f00: 6976 6573 2074 6865 2068 6f6d 6f74 6f70 ives the homotop │ │ │ │ -00054f10: 790a 2020 2020 2020 2020 6672 6f6d 2046 y. from F │ │ │ │ -00054f20: 5f69 2063 6f72 7265 7370 6f6e 6469 6e67 _i corresponding │ │ │ │ -00054f30: 2074 6f20 665f 6a20 6173 2074 6865 2076 to f_j as the v │ │ │ │ -00054f40: 616c 7565 2024 4823 5c7b 6a2c 695c 7d24 alue $H#\{j,i\}$ │ │ │ │ -00054f50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00054f60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 616d 6520 =========..Same │ │ │ │ -00054f70: 6173 206d 616b 6548 6f6d 6f74 6f70 6965 as makeHomotopie │ │ │ │ -00054f80: 732c 2062 7574 206f 6e6c 7920 636f 6d70 s, but only comp │ │ │ │ -00054f90: 7574 6573 2074 6865 206f 7264 696e 6172 utes the ordinar │ │ │ │ -00054fa0: 7920 686f 6d6f 746f 7069 6573 2c20 6e6f y homotopies, no │ │ │ │ -00054fb0: 7420 7468 650a 6869 6768 6572 206f 6e65 t the.higher one │ │ │ │ -00054fc0: 732e 2055 7365 6420 696e 2065 7874 6572 s. Used in exter │ │ │ │ -00054fd0: 696f 7254 6f72 4d6f 6475 6c65 0a0a 5365 iorTorModule..Se │ │ │ │ -00054fe0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -00054ff0: 0a20 202a 202a 6e6f 7465 206d 616b 6548 . * *note makeH │ │ │ │ -00055000: 6f6d 6f74 6f70 6965 733a 206d 616b 6548 omotopies: makeH │ │ │ │ -00055010: 6f6d 6f74 6f70 6965 732c 202d 2d20 7265 omotopies, -- re │ │ │ │ -00055020: 7475 726e 7320 6120 7379 7374 656d 206f turns a system o │ │ │ │ -00055030: 6620 6869 6768 6572 0a20 2020 2068 6f6d f higher. hom │ │ │ │ -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 │ │ │ │ -00055080: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ -00055090: 616e 0a20 2020 2065 7874 6572 696f 7220 an. exterior │ │ │ │ -000550a0: 616c 6765 6272 6120 6f72 2062 6967 7261 algebra or bigra │ │ │ │ -000550b0: 6465 6420 616c 6765 6272 610a 0a57 6179 ded algebra..Way │ │ │ │ -000550c0: 7320 746f 2075 7365 206d 616b 6548 6f6d s to use makeHom │ │ │ │ -000550d0: 6f74 6f70 6965 7331 3a0a 3d3d 3d3d 3d3d otopies1:.====== │ │ │ │ -000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000550f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ -00055100: 6548 6f6d 6f74 6f70 6965 7331 284d 6174 eHomotopies1(Mat │ │ │ │ -00055110: 7269 782c 436f 6d70 6c65 7829 220a 2020 rix,Complex)". │ │ │ │ -00055120: 2a20 226d 616b 6548 6f6d 6f74 6f70 6965 * "makeHomotopie │ │ │ │ -00055130: 7331 284d 6174 7269 782c 436f 6d70 6c65 s1(Matrix,Comple │ │ │ │ -00055140: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -00055150: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00055160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00055170: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00055180: 6d61 6b65 486f 6d6f 746f 7069 6573 313a makeHomotopies1: │ │ │ │ -00055190: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ -000551a0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -000551b0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -000551c0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -000551d0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +00054d40: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +00054d50: 6765 3a20 0a20 2020 2020 2020 2048 203d ge: . H = │ │ │ │ +00054d60: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ +00054d70: 2866 2c46 2c64 290a 2020 2a20 496e 7075 (f,F,d). * Inpu │ │ │ │ +00054d80: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ +00054d90: 202a 6e6f 7465 206d 6174 7269 783a 2028 *note matrix: ( │ │ │ │ +00054da0: 4d61 6361 756c 6179 3244 6f63 294d 6174 Macaulay2Doc)Mat │ │ │ │ +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 2e30 362b 6473 2f4d ay2-1.25.06+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 3036 acaulay2-1.25.06 │ │ │ │ +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 │ │ │ │ +00055580: 6779 206f 6620 430a 0a44 6573 6372 6970 gy of C..Descrip │ │ │ │ +00055590: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +000555a0: 0a0a 5468 6520 7363 7269 7074 2063 616c ..The script cal │ │ │ │ +000555b0: 6c73 206d 616b 6548 6f6d 6f74 6f70 6965 ls makeHomotopie │ │ │ │ +000555c0: 7331 2074 6f20 7072 6f64 7563 6520 686f s1 to produce ho │ │ │ │ +000555d0: 6d6f 746f 7069 6573 2066 6f72 2074 6865 motopies for the │ │ │ │ +000555e0: 2066 665f 6920 6f6e 2043 2c20 616e 640a ff_i on C, and. │ │ │ │ +000555f0: 7468 656e 2063 6f6d 7075 7465 7320 7468 then computes th │ │ │ │ +00055600: 6569 7220 6163 7469 6f6e 206f 6e20 7468 eir action on th │ │ │ │ +00055610: 6520 486f 6d6f 6c6f 6779 206f 6620 432e e Homology of C. │ │ │ │ +00055620: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00055630: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065 ===.. * *note e │ │ │ │ +00055640: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ +00055650: 3a20 6578 7465 7269 6f72 546f 724d 6f64 : exteriorTorMod │ │ │ │ +00055660: 756c 652c 202d 2d20 546f 7220 6173 2061 ule, -- Tor as a │ │ │ │ +00055670: 206d 6f64 756c 6520 6f76 6572 2061 6e0a module over an. │ │ │ │ +00055680: 2020 2020 6578 7465 7269 6f72 2061 6c67 exterior alg │ │ │ │ +00055690: 6562 7261 206f 7220 6269 6772 6164 6564 ebra or bigraded │ │ │ │ +000556a0: 2061 6c67 6562 7261 0a20 202a 202a 6e6f algebra. * *no │ │ │ │ +000556b0: 7465 2065 7874 6572 696f 7245 7874 4d6f te exteriorExtMo │ │ │ │ +000556c0: 6475 6c65 3a20 6578 7465 7269 6f72 4578 dule: exteriorEx │ │ │ │ +000556d0: 744d 6f64 756c 652c 202d 2d20 4578 7428 tModule, -- Ext( │ │ │ │ +000556e0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29 M,k) or Ext(M,N) │ │ │ │ +000556f0: 2061 7320 610a 2020 2020 6d6f 6475 6c65 as a. module │ │ │ │ +00055700: 206f 7665 7220 616e 2065 7874 6572 696f over an exterio │ │ │ │ +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 .=============== │ │ │ │ 00055750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -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 │ │ │ │ -00055790: 284d 6174 7269 782c 436f 6d70 6c65 7829 (Matrix,Complex) │ │ │ │ -000557a0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -000557b0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -000557c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -000557d0: 6a65 6374 202a 6e6f 7465 206d 616b 6548 ject *note makeH │ │ │ │ -000557e0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c omotopiesOnHomol │ │ │ │ -000557f0: 6f67 793a 206d 616b 6548 6f6d 6f74 6f70 ogy: makeHomotop │ │ │ │ -00055800: 6965 734f 6e48 6f6d 6f6c 6f67 792c 2069 iesOnHomology, i │ │ │ │ -00055810: 7320 6120 2a6e 6f74 650a 6d65 7468 6f64 s a *note.method │ │ │ │ -00055820: 2066 756e 6374 696f 6e3a 2028 4d61 6361 function: (Maca │ │ │ │ -00055830: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00055840: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +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 │ │ │ │ +00055830: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00055840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ -000558a0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -000558b0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -000558c0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -000558d0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -000558e0: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ -000558f0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00055900: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00055910: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00055920: 6e73 2e6d 323a 3236 3934 3a30 2e0a 1f0a ns.m2:2694:0.... │ │ │ │ -00055930: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00055940: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00055950: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00055960: 3a20 6d61 6b65 4d6f 6475 6c65 2c20 4e65 : makeModule, Ne │ │ │ │ -00055970: 7874 3a20 6d61 6b65 542c 2050 7265 763a xt: makeT, Prev: │ │ │ │ -00055980: 206d 616b 6548 6f6d 6f74 6f70 6965 734f makeHomotopiesO │ │ │ │ -00055990: 6e48 6f6d 6f6c 6f67 792c 2055 703a 2054 nHomology, Up: T │ │ │ │ -000559a0: 6f70 0a0a 6d61 6b65 4d6f 6475 6c65 202d op..makeModule - │ │ │ │ -000559b0: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65 - makes a Module │ │ │ │ -000559c0: 206f 7574 206f 6620 6120 636f 6c6c 6563 out of a collec │ │ │ │ -000559d0: 7469 6f6e 206f 6620 6d6f 6475 6c65 7320 tion of modules │ │ │ │ -000559e0: 616e 6420 6d61 7073 0a2a 2a2a 2a2a 2a2a and maps.******* │ │ │ │ +00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00055890: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +000558a0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +000558b0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +000558c0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +000558d0: 756c 6179 322d 312e 3235 2e30 362b 6473 ulay2-1.25.06+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 │ │ │ │ +00055910: 6c75 7469 6f6e 732e 6d32 3a32 3639 343a lutions.m2:2694: │ │ │ │ +00055920: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00055930: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00055940: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00055950: 204e 6f64 653a 206d 616b 654d 6f64 756c Node: makeModul │ │ │ │ +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 │ │ │ │ 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│ │ │ -00055b60: 202a 6e6f 7465 2068 6173 6820 7461 626c *note hash tabl │ │ │ │ -00055b70: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00055b80: 2948 6173 6854 6162 6c65 2c2c 206d 6170 )HashTable,, map │ │ │ │ -00055b90: 7320 6265 7477 6565 6e20 7468 650a 2020 s between the. │ │ │ │ -00055ba0: 2020 2020 2020 6772 6164 6564 2063 6f6d graded com │ │ │ │ -00055bb0: 706f 6e65 6e74 7320 7468 6174 2077 696c ponents that wil │ │ │ │ -00055bc0: 6c20 6265 2074 6865 2061 6374 696f 6e20 l be the action │ │ │ │ -00055bd0: 6f66 2074 6865 2076 6172 6961 626c 6573 of the variables │ │ │ │ -00055be0: 2069 6e20 450a 2020 2a20 4f75 7470 7574 in E. * Output │ │ │ │ -00055bf0: 733a 0a20 2020 2020 202a 204d 2c20 6120 s:. * M, a │ │ │ │ -00055c00: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -00055c10: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -00055c20: 6c65 2c2c 2067 7261 6465 6420 6d6f 6475 le,, graded modu │ │ │ │ -00055c30: 6c65 7320 7768 6f73 650a 2020 2020 2020 les whose. │ │ │ │ -00055c40: 2020 636f 6d70 6f6e 656e 7473 2061 7265 components are │ │ │ │ -00055c50: 2067 6976 656e 2062 7920 480a 0a44 6573 given by H..Des │ │ │ │ -00055c60: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00055c70: 3d3d 3d3d 0a0a 5468 6520 4861 7368 7461 ====..The Hashta │ │ │ │ -00055c80: 626c 6520 4820 7368 6f75 6c64 2068 6176 ble H should hav │ │ │ │ -00055c90: 6520 636f 6e73 6563 7574 6976 6520 696e e consecutive in │ │ │ │ -00055ca0: 7465 6765 7220 6b65 7973 2069 5f30 2e2e teger keys i_0.. │ │ │ │ -00055cb0: 695f 302c 2073 6179 2c20 7769 7468 2076 i_0, say, with v │ │ │ │ -00055cc0: 616c 7565 730a 4823 6920 7468 6174 2061 alues.H#i that a │ │ │ │ -00055cd0: 7265 206d 6f64 756c 6573 206f 7665 7220 re modules over │ │ │ │ -00055ce0: 6120 7269 6e67 2053 4520 7768 6f73 6520 a ring SE whose │ │ │ │ -00055cf0: 7661 7269 6162 6c65 7320 696e 636c 7564 variables includ │ │ │ │ -00055d00: 6520 7468 6520 656c 656d 656e 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6120 6f6e 2078 2c79 r algebra on x,y │ │ │ │ -00056110: 2c20 7374 6172 7469 6e67 2077 6974 6820 , starting with │ │ │ │ -00056120: 7468 6520 636f 6e73 7472 7563 7469 6f6e the construction │ │ │ │ -00056130: 206f 6620 6120 6d6f 6475 6c65 0a6f 7665 of a module.ove │ │ │ │ -00056140: 7220 6120 6269 686f 6d6f 6765 6e65 6f75 r a bihomogeneou │ │ │ │ -00056150: 7320 7269 6e67 2e0a 0a2b 2d2d 2d2d 2d2d s ring...+------ │ │ │ │ +00055a20: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00055a30: 2020 2020 2020 2020 4d20 3d20 6d61 6b65 M = make │ │ │ │ +00055a40: 4d6f 6475 6c65 2848 2c45 2c70 6869 290a Module(H,E,phi). │ │ │ │ +00055a50: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00055a60: 2020 2a20 482c 2061 202a 6e6f 7465 2068 * H, a *note h │ │ │ │ +00055a70: 6173 6820 7461 626c 653a 2028 4d61 6361 ash table: (Maca │ │ │ │ +00055a80: 756c 6179 3244 6f63 2948 6173 6854 6162 ulay2Doc)HashTab │ │ │ │ +00055a90: 6c65 2c2c 2067 7261 6465 6420 636f 6d70 le,, graded comp │ │ │ │ +00055aa0: 6f6e 656e 7473 2074 6861 740a 2020 2020 onents that. │ │ │ │ +00055ab0: 2020 2020 6172 6520 6d6f 6475 6c65 732c are modules, │ │ │ │ +00055ac0: 2074 6f20 6d61 6b65 2069 6e74 6f20 6173 to make into as │ │ │ │ +00055ad0: 2073 696e 676c 6520 6d6f 6475 6c65 0a20 single module. │ │ │ │ +00055ae0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ +00055af0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +00055b00: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +00055b10: 204d 6174 7269 7820 6f66 2076 6172 6961 Matrix of varia │ │ │ │ +00055b20: 626c 6573 2077 686f 7365 0a20 2020 2020 bles whose. │ │ │ │ +00055b30: 2020 2061 6374 696f 6e20 7769 6c6c 2064 action will d │ │ │ │ +00055b40: 6566 696e 6564 0a20 2020 2020 202a 2070 efined. * p │ │ │ │ +00055b50: 6869 2c20 6120 2a6e 6f74 6520 6861 7368 hi, a *note hash │ │ │ │ +00055b60: 2074 6162 6c65 3a20 284d 6163 6175 6c61 table: (Macaula │ │ │ │ +00055b70: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ +00055b80: 2c20 6d61 7073 2062 6574 7765 656e 2074 , maps between t │ │ │ │ +00055b90: 6865 0a20 2020 2020 2020 2067 7261 6465 he. grade │ │ │ │ +00055ba0: 6420 636f 6d70 6f6e 656e 7473 2074 6861 d components tha │ │ │ │ +00055bb0: 7420 7769 6c6c 2062 6520 7468 6520 6163 t will be the ac │ │ │ │ +00055bc0: 7469 6f6e 206f 6620 7468 6520 7661 7269 tion of the vari │ │ │ │ +00055bd0: 6162 6c65 7320 696e 2045 0a20 202a 204f ables in E. * O │ │ │ │ +00055be0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00055bf0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +00055c00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +00055c10: 294d 6f64 756c 652c 2c20 6772 6164 6564 )Module,, graded │ │ │ │ +00055c20: 206d 6f64 756c 6573 2077 686f 7365 0a20 modules whose. │ │ │ │ +00055c30: 2020 2020 2020 2063 6f6d 706f 6e65 6e74 component │ │ │ │ +00055c40: 7320 6172 6520 6769 7665 6e20 6279 2048 s are given by H │ │ │ │ +00055c50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00055c60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2048 =========..The H │ │ │ │ +00055c70: 6173 6874 6162 6c65 2048 2073 686f 756c ashtable H shoul │ │ │ │ +00055c80: 6420 6861 7665 2063 6f6e 7365 6375 7469 d have consecuti │ │ │ │ +00055c90: 7665 2069 6e74 6567 6572 206b 6579 7320 ve integer keys │ │ │ │ +00055ca0: 695f 302e 2e69 5f30 2c20 7361 792c 2077 i_0..i_0, say, w │ │ │ │ +00055cb0: 6974 6820 7661 6c75 6573 0a48 2369 2074 ith values.H#i t │ │ │ │ +00055cc0: 6861 7420 6172 6520 6d6f 6475 6c65 7320 hat are modules │ │ │ │ +00055cd0: 6f76 6572 2061 2072 696e 6720 5345 2077 over a ring SE w │ │ │ │ +00055ce0: 686f 7365 2076 6172 6961 626c 6573 2069 hose variables i │ │ │ │ +00055cf0: 6e63 6c75 6465 2074 6865 2065 6c65 6d65 nclude the eleme │ │ │ │ +00055d00: 6e74 7320 6f66 2045 2e0a 453a 205c 6f70 nts of E..E: \op │ │ │ │ +00055d10: 6c75 7320 5345 5e7b 645f 697d 205c 746f lus SE^{d_i} \to │ │ │ │ +00055d20: 2053 455e 3120 6973 2061 206d 6174 7269 SE^1 is a matri │ │ │ │ +00055d30: 7820 6f66 2063 2076 6172 6961 626c 6573 x of c variables │ │ │ │ +00055d40: 2066 726f 6d20 5345 2048 2069 7320 6120 from SE H is a │ │ │ │ +00055d50: 6861 7368 5461 626c 650a 6f66 206d 2070 hashTable.of m p │ │ │ │ +00055d60: 6169 7273 207b 692c 2074 5f69 7d2c 2077 airs {i, t_i}, w │ │ │ │ +00055d70: 6865 7265 2074 6865 2074 5f69 2061 7265 here the t_i are │ │ │ │ +00055d80: 2052 452d 6d6f 6475 6c65 732c 2061 6e64 RE-modules, and │ │ │ │ +00055d90: 2074 6865 2069 2061 7265 2063 6f6e 7365 the i are conse │ │ │ │ +00055da0: 6375 7469 7665 0a69 6e74 6567 6572 2e20 cutive.integer. │ │ │ │ +00055db0: 7068 6920 6973 2061 2068 6173 682d 7461 phi is a hash-ta │ │ │ │ +00055dc0: 626c 6520 6f66 2068 6f6d 6f67 656e 656f ble of homogeneo │ │ │ │ +00055dd0: 7573 206d 6170 7320 7068 6923 7b6a 2c69 us maps phi#{j,i │ │ │ │ +00055de0: 7d3a 2048 2369 2a2a 465f 6a5c 746f 2048 }: H#i**F_j\to H │ │ │ │ +00055df0: 2328 692b 3129 0a77 6865 7265 2046 5f6a #(i+1).where F_j │ │ │ │ +00055e00: 203d 2073 6f75 7263 6520 2845 5f7b 6a7d = source (E_{j} │ │ │ │ +00055e10: 203d 206d 6174 7269 7820 7b7b 655f 6a7d = matrix {{e_j} │ │ │ │ +00055e20: 7d29 2e20 5468 7573 2074 6865 206d 6170 }). Thus the map │ │ │ │ +00055e30: 7320 7023 7b6a 2c69 7d20 3d20 2845 5f6a s p#{j,i} = (E_j │ │ │ │ +00055e40: 207c 7c0a 2d70 6869 237b 6a2c 697d 293a ||.-phi#{j,i}): │ │ │ │ +00055e50: 2074 5f69 2a2a 465f 6a20 5c74 6f20 745f t_i**F_j \to t_ │ │ │ │ +00055e60: 692b 2b74 5f7b 2869 2b31 297d 2c20 6172 i++t_{(i+1)}, ar │ │ │ │ +00055e70: 6520 686f 6d6f 6765 6e65 6f75 732e 2054 e homogeneous. T │ │ │ │ +00055e80: 6865 2073 6372 6970 7420 7265 7475 726e he script return │ │ │ │ +00055e90: 7320 4d0a 3d20 5c6f 706c 7573 5f69 2054 s M.= \oplus_i T │ │ │ │ +00055ea0: 5f20 6173 2061 6e20 5345 2d6d 6f64 756c _ as an SE-modul │ │ │ │ +00055eb0: 652c 2063 6f6d 7075 7465 6420 6173 2074 e, computed as t │ │ │ │ +00055ec0: 6865 2071 756f 7469 656e 7420 6f66 2050 he quotient of P │ │ │ │ +00055ed0: 203a 3d20 5c6f 706c 7573 2054 5f69 0a6f := \oplus T_i.o │ │ │ │ +00055ee0: 6274 6169 6e65 6420 6279 2066 6163 746f btained by facto │ │ │ │ +00055ef0: 7269 6e67 206f 7574 2074 6865 2073 756d ring out the sum │ │ │ │ +00055f00: 206f 6620 7468 6520 696d 6167 6573 206f of the images o │ │ │ │ +00055f10: 6620 7468 6520 6d61 7073 2070 237b 6a2c f the maps p#{j, │ │ │ │ +00055f20: 697d 0a0a 5468 6520 4861 7368 7461 626c i}..The Hashtabl │ │ │ │ +00055f30: 6520 7068 6920 6861 7320 6b65 7973 206f e phi has keys o │ │ │ │ +00055f40: 6620 7468 6520 666f 726d 207b 6a2c 697d f the form {j,i} │ │ │ │ +00055f50: 2077 6865 7265 206a 2072 756e 7320 6672 where j runs fr │ │ │ │ +00055f60: 6f6d 2030 2074 6f20 632d 312c 2069 2061 om 0 to c-1, i a │ │ │ │ +00055f70: 6e64 0a69 2b31 2061 7265 206b 6579 7320 nd.i+1 are keys │ │ │ │ +00055f80: 6f66 2048 2c20 616e 6420 7068 6923 7b6a of H, and phi#{j │ │ │ │ +00055f90: 2c69 7d20 6973 2074 6865 206d 6170 2066 ,i} is the map f │ │ │ │ +00055fa0: 726f 6d20 2873 6f75 7263 6520 455f 7b69 rom (source E_{i │ │ │ │ +00055fb0: 7d29 2a2a 4823 6920 746f 2048 2328 692b })**H#i to H#(i+ │ │ │ │ +00055fc0: 3129 0a74 6861 7420 7769 6c6c 2062 6520 1).that will be │ │ │ │ +00055fd0: 6964 656e 7469 6669 6564 2077 6974 6820 identified with │ │ │ │ +00055fe0: 7468 6520 6163 7469 6f6e 206f 6620 455f the action of E_ │ │ │ │ +00055ff0: 7b6a 7d2e 0a0a 5468 6520 7363 7269 7074 {j}...The script │ │ │ │ +00056000: 2069 7320 7573 6564 2069 6e20 626f 7468 is used in both │ │ │ │ +00056010: 2074 6865 2073 696e 676c 7920 6772 6164 the singly grad │ │ │ │ +00056020: 6564 2063 6173 652c 2066 6f72 2065 7861 ed case, for exa │ │ │ │ +00056030: 6d70 6c65 2069 6e0a 6578 7465 7269 6f72 mple in.exterior │ │ │ │ +00056040: 546f 724d 6f64 756c 6528 6666 2c4d 2920 TorModule(ff,M) │ │ │ │ +00056050: 616e 6420 696e 2074 6865 2062 6967 7261 and in the bigra │ │ │ │ +00056060: 6465 6420 6361 7365 2c20 666f 7220 6578 ded case, for ex │ │ │ │ +00056070: 616d 706c 6520 696e 0a65 7874 6572 696f ample in.exterio │ │ │ │ +00056080: 7254 6f72 4d6f 6475 6c65 2866 662c 4d2c rTorModule(ff,M, │ │ │ │ +00056090: 4e29 2e0a 0a49 6e20 7468 6520 666f 6c6c N)...In the foll │ │ │ │ +000560a0: 6f77 696e 6720 7765 2075 7365 206d 616b owing we use mak │ │ │ │ +000560b0: 654d 6f64 756c 6520 746f 2063 6f6e 7374 eModule to const │ │ │ │ +000560c0: 7275 6374 2062 7920 6861 6e64 2061 2066 ruct by hand a f │ │ │ │ +000560d0: 7265 6520 6d6f 6475 6c65 206f 6620 7261 ree module of ra │ │ │ │ +000560e0: 6e6b 2031 0a6f 7665 7220 7468 6520 6578 nk 1.over the ex │ │ │ │ +000560f0: 7465 7269 6f72 2061 6c67 6562 7261 206f terior algebra o │ │ │ │ +00056100: 6e20 782c 792c 2073 7461 7274 696e 6720 n x,y, starting │ │ │ │ +00056110: 7769 7468 2074 6865 2063 6f6e 7374 7275 with the constru │ │ │ │ +00056120: 6374 696f 6e20 6f66 2061 206d 6f64 756c ction of a modul │ │ │ │ +00056130: 650a 6f76 6572 2061 2062 6968 6f6d 6f67 e.over a bihomog │ │ │ │ +00056140: 656e 656f 7573 2072 696e 672e 0a0a 2b2d eneous ring...+- │ │ │ │ +00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000561a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ -000561b0: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63 E = ZZ/101[a,b,c │ │ │ │ -000561c0: 2c78 2c79 2c44 6567 7265 6573 3d3e 746f ,x,y,Degrees=>to │ │ │ │ -000561d0: 4c69 7374 2833 3a7b 312c 307d 297c 746f List(3:{1,0})|to │ │ │ │ -000561e0: 4c69 7374 2832 3a7b 312c 317d 292c 2020 List(2:{1,1}), │ │ │ │ -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[ │ │ │ │ +000561b0: 612c 622c 632c 782c 792c 4465 6772 6565 a,b,c,x,y,Degree │ │ │ │ +000561c0: 733d 3e74 6f4c 6973 7428 333a 7b31 2c30 s=>toList(3:{1,0 │ │ │ │ +000561d0: 7d29 7c74 6f4c 6973 7428 323a 7b31 2c31 })|toList(2:{1,1 │ │ │ │ +000561e0: 7d29 2c20 2020 2020 2020 2020 7c0a 7c20 }), |.| │ │ │ │ +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 │ │ │ │ -00056310: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ -00056320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +000562f0: 6e67 2c20 3220 736b 6577 2063 6f6d 6d75 ng, 2 skew commu │ │ │ │ +00056300: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +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 ---------------- │ │ │ │ 00056340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056380: 2d2d 2d2d 2d2d 2d7c 0a7c 536b 6577 436f -------|.|SkewCo │ │ │ │ -00056390: 6d6d 7574 6174 6976 653d 3e7b 782c 797d mmutative=>{x,y} │ │ │ │ -000563a0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c53 ------------|.|S │ │ │ │ +00056380: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e kewCommutative=> │ │ │ │ +00056390: 7b78 2c79 7d5d 2020 2020 2020 2020 2020 {x,y}] │ │ │ │ +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 |.+- │ │ │ │ +000563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000563e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -00056420: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -00056430: 4520 3d20 5345 2f69 6465 616c 2261 322c E = SE/ideal"a2, │ │ │ │ -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 │ │ │ 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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 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│ │ │ +000578c0: 3130 203a 2071 203d 206d 6170 285a 5a2f 10 : q = map(ZZ/ │ │ │ │ +000578d0: 3130 315b 782c 792c 2053 6b65 7743 6f6d 101[x,y, SkewCom │ │ │ │ +000578e0: 6d75 7461 7469 7665 203d 3e20 7472 7565 mutative => true │ │ │ │ +000578f0: 2c20 4465 6772 6565 4d61 7020 3d3e 2064 , DegreeMap => d │ │ │ │ +00057900: 2d3e 7b64 5f31 7d5d 2c20 2020 7c0a 7c20 ->{d_1}], |.| │ │ │ │ +00057910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00057970: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ +00057950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00057960: 2020 2020 2020 2020 2020 205a 5a20 2020 ZZ │ │ │ │ +00057970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057980: 2020 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2022 =========.. * " │ │ │ │ -00058160: 6d61 6b65 4d6f 6475 6c65 2848 6173 6854 makeModule(HashT │ │ │ │ -00058170: 6162 6c65 2c4d 6174 7269 782c 4861 7368 able,Matrix,Hash │ │ │ │ -00058180: 5461 626c 6529 220a 0a46 6f72 2074 6865 Table)"..For the │ │ │ │ -00058190: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -000581a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -000581b0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -000581c0: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ -000581d0: 654d 6f64 756c 652c 2069 7320 6120 2a6e eModule, is a *n │ │ │ │ -000581e0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -000581f0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00058200: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00058210: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00057f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 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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: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 ---------------- │ │ │ │ -00059090: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d --+..Caveat.==== │ │ │ │ -000590a0: 3d3d 0a0a 5363 7269 7074 2061 7373 756d ==..Script assum │ │ │ │ -000590b0: 6573 2074 6861 7420 7269 6e67 2046 203d es that ring F = │ │ │ │ -000590c0: 3d20 2872 696e 6720 6666 292f 2869 6465 = (ring ff)/(ide │ │ │ │ -000590d0: 616c 2066 6629 2e20 4974 206d 6967 6874 al ff). It might │ │ │ │ -000590e0: 2062 6520 6d6f 7265 2075 7365 6675 6c20 be more useful │ │ │ │ -000590f0: 746f 0a72 6574 7572 6e20 7468 6520 6f70 to.return the op │ │ │ │ -00059100: 6572 6174 6f72 7320 6173 206d 6174 7269 erators as matri │ │ │ │ -00059110: 6365 7320 6f76 6572 2053 2072 6174 6865 ces over S rathe │ │ │ │ -00059120: 7220 7468 616e 206f 7665 7220 522c 2073 r than over R, s │ │ │ │ -00059130: 696e 6365 2074 6869 7320 6973 2077 6861 ince this is wha │ │ │ │ -00059140: 740a 7765 2764 206e 6565 6420 666f 7220 t.we'd need for │ │ │ │ -00059150: 7468 696e 6773 206c 696b 6520 6d61 7472 things like matr │ │ │ │ -00059160: 6978 4661 6374 6f72 697a 6174 696f 6e20 ixFactorization │ │ │ │ -00059170: 2877 6865 7265 2074 6869 7320 7072 6f63 (where this proc │ │ │ │ -00059180: 6573 7320 6375 7272 656e 746c 790a 646f ess currently.do │ │ │ │ -00059190: 6e65 206f 6e20 7468 6520 666c 792c 206e ne on the fly, n │ │ │ │ -000591a0: 6f74 2063 616c 6c69 6e67 206d 616b 6554 ot calling makeT │ │ │ │ -000591b0: 290a 0a57 6179 7320 746f 2075 7365 206d )..Ways to use m │ │ │ │ -000591c0: 616b 6554 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d akeT:.========== │ │ │ │ -000591d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ -000591e0: 616b 6554 284d 6174 7269 782c 436f 6d70 akeT(Matrix,Comp │ │ │ │ -000591f0: 6c65 782c 5a5a 2922 0a0a 466f 7220 7468 lex,ZZ)"..For th │ │ │ │ -00059200: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00059210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00059220: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00059230: 6520 6d61 6b65 543a 206d 616b 6554 2c20 e makeT: makeT, │ │ │ │ -00059240: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00059250: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00059260: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00059270: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00059080: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 -------+..Caveat │ │ │ │ +00059090: 0a3d 3d3d 3d3d 3d0a 0a53 6372 6970 7420 .======..Script │ │ │ │ +000590a0: 6173 7375 6d65 7320 7468 6174 2072 696e assumes that rin │ │ │ │ +000590b0: 6720 4620 3d3d 2028 7269 6e67 2066 6629 g F == (ring ff) │ │ │ │ +000590c0: 2f28 6964 6561 6c20 6666 292e 2049 7420 /(ideal ff). It │ │ │ │ +000590d0: 6d69 6768 7420 6265 206d 6f72 6520 7573 might be more us │ │ │ │ +000590e0: 6566 756c 2074 6f0a 7265 7475 726e 2074 eful to.return t │ │ │ │ +000590f0: 6865 206f 7065 7261 746f 7273 2061 7320 he operators as │ │ │ │ +00059100: 6d61 7472 6963 6573 206f 7665 7220 5320 matrices over S │ │ │ │ +00059110: 7261 7468 6572 2074 6861 6e20 6f76 6572 rather than over │ │ │ │ +00059120: 2052 2c20 7369 6e63 6520 7468 6973 2069 R, since this i │ │ │ │ +00059130: 7320 7768 6174 0a77 6527 6420 6e65 6564 s what.we'd need │ │ │ │ +00059140: 2066 6f72 2074 6869 6e67 7320 6c69 6b65 for things like │ │ │ │ +00059150: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +00059160: 7469 6f6e 2028 7768 6572 6520 7468 6973 tion (where this │ │ │ │ +00059170: 2070 726f 6365 7373 2063 7572 7265 6e74 process current │ │ │ │ +00059180: 6c79 0a64 6f6e 6520 6f6e 2074 6865 2066 ly.done on the f │ │ │ │ +00059190: 6c79 2c20 6e6f 7420 6361 6c6c 696e 6720 ly, not calling │ │ │ │ +000591a0: 6d61 6b65 5429 0a0a 5761 7973 2074 6f20 makeT)..Ways to │ │ │ │ +000591b0: 7573 6520 6d61 6b65 543a 0a3d 3d3d 3d3d use makeT:.===== │ │ │ │ +000591c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000591d0: 202a 2022 6d61 6b65 5428 4d61 7472 6978 * "makeT(Matrix │ │ │ │ +000591e0: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46 ,Complex,ZZ)"..F │ │ │ │ +000591f0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00059200: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00059210: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00059220: 202a 6e6f 7465 206d 616b 6554 3a20 6d61 *note makeT: ma │ │ │ │ +00059230: 6b65 542c 2069 7320 6120 2a6e 6f74 6520 keT, is a *note │ │ │ │ +00059240: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00059250: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00059260: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00059270: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00059280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000592a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000592b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000592c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -000592d0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -000592e0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -000592f0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00059300: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00059310: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -00059320: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00059330: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -00059340: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00059350: 6f6e 732e 6d32 3a33 3534 383a 302e 0a1f ons.m2:3548:0... │ │ │ │ -00059360: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00059370: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00059380: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -00059390: 653a 206d 6174 7269 7846 6163 746f 7269 e: matrixFactori │ │ │ │ -000593a0: 7a61 7469 6f6e 2c20 4e65 7874 3a20 6d66 zation, Next: mf │ │ │ │ -000593b0: 426f 756e 642c 2050 7265 763a 206d 616b Bound, Prev: mak │ │ │ │ -000593c0: 6554 2c20 5570 3a20 546f 700a 0a6d 6174 eT, Up: Top..mat │ │ │ │ -000593d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -000593e0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ -000593f0: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ -00059400: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ -00059410: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ation.********** │ │ │ │ +000592c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000592d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000592e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000592f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00059300: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +00059310: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00059320: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +00059330: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00059340: 6f6c 7574 696f 6e73 2e6d 323a 3335 3438 olutions.m2:3548 │ │ │ │ +00059350: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00059360: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00059370: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +00059380: 2c20 4e6f 6465 3a20 6d61 7472 6978 4661 , Node: matrixFa │ │ │ │ +00059390: 6374 6f72 697a 6174 696f 6e2c 204e 6578 ctorization, Nex │ │ │ │ +000593a0: 743a 206d 6642 6f75 6e64 2c20 5072 6576 t: mfBound, Prev │ │ │ │ +000593b0: 3a20 6d61 6b65 542c 2055 703a 2054 6f70 : makeT, Up: Top │ │ │ │ +000593c0: 0a0a 6d61 7472 6978 4661 6374 6f72 697a ..matrixFactoriz │ │ │ │ +000593d0: 6174 696f 6e20 2d2d 204d 6170 7320 696e ation -- Maps in │ │ │ │ +000593e0: 2061 2068 6967 6865 7220 636f 6469 6d65 a higher codime │ │ │ │ +000593f0: 6e73 696f 6e20 6d61 7472 6978 2066 6163 nsion matrix fac │ │ │ │ +00059400: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a torization.***** │ │ │ │ +00059410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00059450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00059460: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00059470: 2020 2020 4d46 203d 206d 6174 7269 7846 MF = matrixF │ │ │ │ -00059480: 6163 746f 7269 7a61 7469 6f6e 2866 662c actorization(ff, │ │ │ │ -00059490: 4d29 0a20 202a 2049 6e70 7574 733a 0a20 M). * Inputs:. │ │ │ │ -000594a0: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -000594b0: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -000594c0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -000594d0: 2c20 6120 7375 6666 6963 6965 6e74 6c79 , a sufficiently │ │ │ │ -000594e0: 2067 656e 6572 616c 0a20 2020 2020 2020 general. │ │ │ │ -000594f0: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ -00059500: 6520 696e 2061 2072 696e 6720 530a 2020 e in a ring S. │ │ │ │ -00059510: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -00059520: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -00059530: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00059540: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d a maximal Cohen- │ │ │ │ -00059550: 4d61 6361 756c 6179 0a20 2020 2020 2020 Macaulay. │ │ │ │ -00059560: 206d 6f64 756c 6520 6f76 6572 2053 2f69 module over S/i │ │ │ │ -00059570: 6465 616c 2066 660a 2020 2a20 2a6e 6f74 deal ff. * *not │ │ │ │ -00059580: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ -00059590: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ -000595a0: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ -000595b0: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -000595c0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -000595d0: 4175 676d 656e 7461 7469 6f6e 203d 3e20 Augmentation => │ │ │ │ -000595e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -000595f0: 7565 2074 7275 650a 2020 2020 2020 2a20 ue true. * │ │ │ │ -00059600: 4368 6563 6b20 3d3e 202e 2e2e 2c20 6465 Check => ..., de │ │ │ │ -00059610: 6661 756c 7420 7661 6c75 6520 6661 6c73 fault value fals │ │ │ │ -00059620: 650a 2020 2020 2020 2a20 4c61 7965 7265 e. * Layere │ │ │ │ -00059630: 6420 3d3e 202e 2e2e 2c20 6465 6661 756c d => ..., defaul │ │ │ │ -00059640: 7420 7661 6c75 6520 7472 7565 0a20 2020 t value true. │ │ │ │ -00059650: 2020 202a 2056 6572 626f 7365 203d 3e20 * Verbose => │ │ │ │ -00059660: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00059670: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -00059680: 7075 7473 3a0a 2020 2020 2020 2a20 4d46 puts:. * MF │ │ │ │ -00059690: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -000596a0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -000596b0: 7374 2c2c 205c 7b64 2c68 2c67 616d 6d61 st,, \{d,h,gamma │ │ │ │ -000596c0: 5c7d 2c20 7768 6572 6520 643a 415f 3120 \}, where d:A_1 │ │ │ │ -000596d0: 5c74 6f0a 2020 2020 2020 2020 415f 3020 \to. A_0 │ │ │ │ -000596e0: 616e 6420 683a 205c 6f70 6c75 7320 415f and h: \oplus A_ │ │ │ │ -000596f0: 3028 7029 205c 746f 2041 5f31 2069 7320 0(p) \to A_1 is │ │ │ │ -00059700: 7468 6520 6469 7265 6374 2073 756d 206f the direct sum o │ │ │ │ -00059710: 6620 7061 7274 6961 6c0a 2020 2020 2020 f partial. │ │ │ │ -00059720: 2020 686f 6d6f 746f 7069 6573 2c20 616e homotopies, an │ │ │ │ -00059730: 6420 6761 6d6d 613a 2041 5f30 202d 3e4d d gamma: A_0 ->M │ │ │ │ -00059740: 2069 7320 7468 6520 6175 676d 656e 7461 is the augmenta │ │ │ │ -00059750: 7469 6f6e 2028 7265 7475 726e 6564 206f tion (returned o │ │ │ │ -00059760: 6e6c 7920 6966 0a20 2020 2020 2020 2041 nly if. A │ │ │ │ -00059770: 7567 6d65 6e74 6174 696f 6e20 3d3e 7472 ugmentation =>tr │ │ │ │ -00059780: 7565 290a 0a44 6573 6372 6970 7469 6f6e ue)..Description │ │ │ │ -00059790: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -000597a0: 6520 696e 7075 7420 6d6f 6475 6c65 204d e input module M │ │ │ │ -000597b0: 2073 686f 756c 6420 6265 2061 206d 6178 should be a max │ │ │ │ -000597c0: 696d 616c 2043 6f68 656e 2d4d 6163 6175 imal Cohen-Macau │ │ │ │ -000597d0: 6c61 7920 6d6f 6475 6c65 206f 7665 7220 lay module over │ │ │ │ -000597e0: 5220 3d20 532f 6964 6561 6c0a 6666 2e20 R = S/ideal.ff. │ │ │ │ -000597f0: 2049 6620 4d20 6973 2069 6e20 6661 6374 If M is in fact │ │ │ │ -00059800: 2061 2022 6869 6768 2073 797a 7967 7922 a "high syzygy" │ │ │ │ -00059810: 2c20 7468 656e 2074 6865 2066 756e 6374 , then the funct │ │ │ │ -00059820: 696f 6e0a 6d61 7472 6978 4661 6374 6f72 ion.matrixFactor │ │ │ │ -00059830: 697a 6174 696f 6e28 6666 2c4d 2c4c 6179 ization(ff,M,Lay │ │ │ │ -00059840: 6572 6564 3d3e 6661 6c73 6529 2075 7365 ered=>false) use │ │ │ │ -00059850: 7320 6120 6469 6666 6572 656e 742c 2066 s a different, f │ │ │ │ -00059860: 6173 7465 7220 616c 676f 7269 7468 6d0a aster algorithm. │ │ │ │ -00059870: 7768 6963 6820 6f6e 6c79 2077 6f72 6b73 which only works │ │ │ │ -00059880: 2069 6e20 7468 6520 6869 6768 2073 797a in the high syz │ │ │ │ -00059890: 7967 7920 6361 7365 2e0a 0a49 6e20 616c ygy case...In al │ │ │ │ -000598a0: 6c20 6578 616d 706c 6573 2077 6520 6b6e l examples we kn │ │ │ │ -000598b0: 6f77 2c20 4d20 6361 6e20 6265 2063 6f6e ow, M can be con │ │ │ │ -000598c0: 7369 6465 7265 6420 6120 2268 6967 6820 sidered a "high │ │ │ │ -000598d0: 7379 7a79 6779 2220 6173 206c 6f6e 6720 syzygy" as long │ │ │ │ -000598e0: 6173 0a45 7874 5e7b 6576 656e 7d5f 5228 as.Ext^{even}_R( │ │ │ │ -000598f0: 4d2c 6b29 2061 6e64 2045 7874 5e7b 6f64 M,k) and Ext^{od │ │ │ │ -00059900: 647d 5f52 284d 2c6b 2920 6861 7665 206e d}_R(M,k) have n │ │ │ │ -00059910: 6567 6174 6976 6520 7265 6775 6c61 7269 egative regulari │ │ │ │ -00059920: 7479 206f 7665 7220 7468 6520 7269 6e67 ty over the ring │ │ │ │ -00059930: 0a6f 6620 4349 206f 7065 7261 746f 7273 .of CI operators │ │ │ │ -00059940: 2028 7265 6772 6164 6564 2077 6974 6820 (regraded with │ │ │ │ -00059950: 7661 7269 6162 6c65 7320 6f66 2064 6567 variables of deg │ │ │ │ -00059960: 7265 6520 312e 2048 6f77 6576 6572 2c20 ree 1. However, │ │ │ │ -00059970: 7468 6520 6265 7374 2072 6573 756c 740a the best result. │ │ │ │ -00059980: 7765 2063 616e 2070 726f 7665 2069 7320 we can prove is │ │ │ │ -00059990: 7468 6174 2069 7420 7375 6666 6963 6573 that it suffices │ │ │ │ -000599a0: 2074 6f20 6861 7665 2072 6567 756c 6172 to have regular │ │ │ │ -000599b0: 6974 7920 3c20 2d28 322a 6469 6d20 522b ity < -(2*dim R+ │ │ │ │ -000599c0: 3129 2e0a 0a57 6865 6e20 7468 6520 6f70 1)...When the op │ │ │ │ -000599d0: 7469 6f6e 616c 2069 6e70 7574 2043 6865 tional input Che │ │ │ │ -000599e0: 636b 3d3d 7472 7565 2028 7468 6520 6465 ck==true (the de │ │ │ │ -000599f0: 6661 756c 7420 6973 2043 6865 636b 3d3d fault is Check== │ │ │ │ -00059a00: 6661 6c73 6529 2c20 7468 650a 7072 6f70 false), the.prop │ │ │ │ -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 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│ │ │ -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 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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: 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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 2e30 362b 6473 ulay2-1.25.06+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: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+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 2e30 macaulay2-1.25.0 │ │ │ │ -0005bd10: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+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 3036 2b64 732f 4d32 2f4d 6163 .25.06+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 │ │ │ │ -0005d2f0: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ -0005d300: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ -0005d310: 7465 0a20 2020 2069 6e74 6572 7365 6374 te. intersect │ │ │ │ -0005d320: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -0005d330: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -0005d340: 696e 670a 2020 2a20 2a6e 6f74 6520 4578 ing. * *note Ex │ │ │ │ -0005d350: 744d 6f64 756c 6544 6174 613a 2045 7874 tModuleData: Ext │ │ │ │ -0005d360: 4d6f 6475 6c65 4461 7461 2c20 2d2d 2045 ModuleData, -- E │ │ │ │ -0005d370: 7665 6e20 616e 6420 6f64 6420 4578 7420 ven and odd Ext │ │ │ │ -0005d380: 6d6f 6475 6c65 7320 616e 6420 7468 6569 modules and thei │ │ │ │ -0005d390: 720a 2020 2020 7265 6775 6c61 7269 7479 r. regularity │ │ │ │ -0005d3a0: 0a20 202a 202a 6e6f 7465 2068 664d 6f64 . * *note hfMod │ │ │ │ -0005d3b0: 756c 6541 7345 7874 3a20 6866 4d6f 6475 uleAsExt: hfModu │ │ │ │ -0005d3c0: 6c65 4173 4578 742c 202d 2d20 7072 6564 leAsExt, -- pred │ │ │ │ -0005d3d0: 6963 7420 6265 7474 6920 6e75 6d62 6572 ict betti number │ │ │ │ -0005d3e0: 7320 6f66 0a20 2020 206d 6f64 756c 6541 s of. moduleA │ │ │ │ -0005d3f0: 7345 7874 284d 2c52 290a 0a57 6179 7320 sExt(M,R)..Ways │ │ │ │ -0005d400: 746f 2075 7365 206d 6f64 756c 6541 7345 to use moduleAsE │ │ │ │ -0005d410: 7874 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d xt:.============ │ │ │ │ -0005d420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0005d430: 2a20 226d 6f64 756c 6541 7345 7874 284d * "moduleAsExt(M │ │ │ │ -0005d440: 6f64 756c 652c 5269 6e67 2922 0a0a 466f odule,Ring)"..Fo │ │ │ │ -0005d450: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0005d460: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005d470: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0005d480: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ -0005d490: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ -0005d4a0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0005d4b0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -0005d4c0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0005d4d0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ +0005d100: 6174 0a3d 3d3d 3d3d 3d0a 0a54 6865 2065 at.======..The e │ │ │ │ +0005d110: 6c65 6d65 6e74 7320 665f 312e 2e66 5f63 lements f_1..f_c │ │ │ │ +0005d120: 206d 7573 7420 6265 2068 6f6d 6f67 656e must be homogen │ │ │ │ +0005d130: 656f 7573 206f 6620 7468 6520 7361 6d65 eous of the same │ │ │ │ +0005d140: 2064 6567 7265 652e 2054 6865 2073 6372 degree. The scr │ │ │ │ +0005d150: 6970 7420 636f 756c 640a 6265 2072 6577 ipt could.be rew │ │ │ │ +0005d160: 7269 7474 656e 2074 6f20 6163 636f 6d6d ritten to accomm │ │ │ │ +0005d170: 6f64 6174 6520 6469 6666 6572 656e 7420 odate different │ │ │ │ +0005d180: 6465 6772 6565 732c 2062 7574 206f 6e6c degrees, but onl │ │ │ │ +0005d190: 7920 6279 2067 6f69 6e67 2074 6f20 7468 y by going to th │ │ │ │ +0005d1a0: 6520 6c6f 6361 6c0a 6361 7465 676f 7279 e local.category │ │ │ │ +0005d1b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0005d1c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045 ===.. * *note E │ │ │ │ +0005d1d0: 7874 4d6f 6475 6c65 3a20 4578 744d 6f64 xtModule: ExtMod │ │ │ │ +0005d1e0: 756c 652c 202d 2d20 4578 745e 2a28 4d2c ule, -- Ext^*(M, │ │ │ │ +0005d1f0: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ +0005d200: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ +0005d210: 6173 0a20 2020 206d 6f64 756c 6520 6f76 as. module ov │ │ │ │ +0005d220: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ +0005d230: 696e 670a 2020 2a20 2a6e 6f74 6520 6576 ing. * *note ev │ │ │ │ +0005d240: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +0005d250: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065 nExtModule, -- e │ │ │ │ +0005d260: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ +0005d270: 2a28 4d2c 6b29 206f 7665 7220 610a 2020 *(M,k) over a. │ │ │ │ +0005d280: 2020 636f 6d70 6c65 7465 2069 6e74 6572 complete inter │ │ │ │ +0005d290: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +0005d2a0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +0005d2b0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74 or ring. * *not │ │ │ │ +0005d2c0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20 e oddExtModule: │ │ │ │ +0005d2d0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d oddExtModule, -- │ │ │ │ +0005d2e0: 206f 6464 2070 6172 7420 6f66 2045 7874 odd part of Ext │ │ │ │ +0005d2f0: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063 ^*(M,k) over a c │ │ │ │ +0005d300: 6f6d 706c 6574 650a 2020 2020 696e 7465 omplete. inte │ │ │ │ +0005d310: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +0005d320: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +0005d330: 746f 7220 7269 6e67 0a20 202a 202a 6e6f tor ring. * *no │ │ │ │ +0005d340: 7465 2045 7874 4d6f 6475 6c65 4461 7461 te ExtModuleData │ │ │ │ +0005d350: 3a20 4578 744d 6f64 756c 6544 6174 612c : ExtModuleData, │ │ │ │ +0005d360: 202d 2d20 4576 656e 2061 6e64 206f 6464 -- Even and odd │ │ │ │ +0005d370: 2045 7874 206d 6f64 756c 6573 2061 6e64 Ext modules and │ │ │ │ +0005d380: 2074 6865 6972 0a20 2020 2072 6567 756c their. regul │ │ │ │ +0005d390: 6172 6974 790a 2020 2a20 2a6e 6f74 6520 arity. * *note │ │ │ │ +0005d3a0: 6866 4d6f 6475 6c65 4173 4578 743a 2068 hfModuleAsExt: h │ │ │ │ +0005d3b0: 664d 6f64 756c 6541 7345 7874 2c20 2d2d fModuleAsExt, -- │ │ │ │ +0005d3c0: 2070 7265 6469 6374 2062 6574 7469 206e predict betti n │ │ │ │ +0005d3d0: 756d 6265 7273 206f 660a 2020 2020 6d6f umbers of. mo │ │ │ │ +0005d3e0: 6475 6c65 4173 4578 7428 4d2c 5229 0a0a duleAsExt(M,R).. │ │ │ │ +0005d3f0: 5761 7973 2074 6f20 7573 6520 6d6f 6475 Ways to use modu │ │ │ │ +0005d400: 6c65 4173 4578 743a 0a3d 3d3d 3d3d 3d3d leAsExt:.======= │ │ │ │ +0005d410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0005d420: 3d0a 0a20 202a 2022 6d6f 6475 6c65 4173 =.. * "moduleAs │ │ │ │ +0005d430: 4578 7428 4d6f 6475 6c65 2c52 696e 6729 Ext(Module,Ring) │ │ │ │ +0005d440: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0005d450: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0005d460: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0005d470: 6a65 6374 202a 6e6f 7465 206d 6f64 756c ject *note modul │ │ │ │ +0005d480: 6541 7345 7874 3a20 6d6f 6475 6c65 4173 eAsExt: moduleAs │ │ │ │ +0005d490: 4578 742c 2069 7320 6120 2a6e 6f74 6520 Ext, is a *note │ │ │ │ +0005d4a0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +0005d4b0: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +0005d4c0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0005d4d0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0005d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0005d530: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0005d540: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0005d550: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0005d560: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0005d570: 322d 312e 3235 2e30 362b 6473 2f4d 322f 2-1.25.06+ds/M2/ │ │ │ │ -0005d580: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0005d590: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0005d5a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0005d5b0: 6f6e 732e 6d32 3a33 3039 363a 302e 0a1f ons.m2:3096:0... │ │ │ │ -0005d5c0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0005d5d0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0005d5e0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -0005d5f0: 653a 206e 6577 4578 742c 204e 6578 743a e: newExt, Next: │ │ │ │ -0005d600: 206f 6464 4578 744d 6f64 756c 652c 2050 oddExtModule, P │ │ │ │ -0005d610: 7265 763a 206d 6f64 756c 6541 7345 7874 rev: moduleAsExt │ │ │ │ -0005d620: 2c20 5570 3a20 546f 700a 0a6e 6577 4578 , Up: Top..newEx │ │ │ │ -0005d630: 7420 2d2d 2047 6c6f 6261 6c20 4578 7420 t -- Global Ext │ │ │ │ -0005d640: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ -0005d650: 2061 2063 6f6d 706c 6574 6520 496e 7465 a complete Inte │ │ │ │ -0005d660: 7273 6563 7469 6f6e 0a2a 2a2a 2a2a 2a2a rsection.******* │ │ │ │ +0005d520: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0005d530: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0005d540: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0005d550: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0005d560: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +0005d570: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0005d580: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0005d590: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0005d5a0: 6f6c 7574 696f 6e73 2e6d 323a 3330 3936 olutions.m2:3096 │ │ │ │ +0005d5b0: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0005d5c0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0005d5d0: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0005d5e0: 2c20 4e6f 6465 3a20 6e65 7745 7874 2c20 , Node: newExt, │ │ │ │ +0005d5f0: 4e65 7874 3a20 6f64 6445 7874 4d6f 6475 Next: oddExtModu │ │ │ │ +0005d600: 6c65 2c20 5072 6576 3a20 6d6f 6475 6c65 le, Prev: module │ │ │ │ +0005d610: 4173 4578 742c 2055 703a 2054 6f70 0a0a AsExt, Up: Top.. │ │ │ │ +0005d620: 6e65 7745 7874 202d 2d20 476c 6f62 616c newExt -- Global │ │ │ │ +0005d630: 2045 7874 2066 6f72 206d 6f64 756c 6573 Ext for modules │ │ │ │ +0005d640: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0005d650: 2049 6e74 6572 7365 6374 696f 6e0a 2a2a Intersection.** │ │ │ │ +0005d660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d6a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0005d6b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -0005d6c0: 6e65 7745 7874 284d 2c4e 290a 2020 2a20 newExt(M,N). * │ │ │ │ -0005d6d0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0005d6e0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -0005d6f0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0005d700: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -0005d710: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0005d720: 6563 7469 6f6e 0a20 2020 2020 2020 2052 ection. R │ │ │ │ -0005d730: 6261 720a 2020 2020 2020 2a20 4e2c 2061 bar. * N, a │ │ │ │ -0005d740: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ -0005d750: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ -0005d760: 756c 652c 2c20 6f76 6572 2052 6261 720a ule,, over Rbar. │ │ │ │ -0005d770: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ -0005d780: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ -0005d790: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ -0005d7a0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ -0005d7b0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ -0005d7c0: 2020 2020 2020 2a20 4368 6563 6b20 3d3e * Check => │ │ │ │ -0005d7d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0005d7e0: 6c75 6520 6661 6c73 650a 2020 2020 2020 lue false. │ │ │ │ -0005d7f0: 2a20 4772 6164 696e 6720 3d3e 202e 2e2e * Grading => ... │ │ │ │ -0005d800: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -0005d810: 320a 2020 2020 2020 2a20 4c69 6674 203d 2. * Lift = │ │ │ │ -0005d820: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -0005d830: 616c 7565 2066 616c 7365 0a20 2020 2020 alue false. │ │ │ │ -0005d840: 202a 2056 6172 6961 626c 6573 203d 3e20 * Variables => │ │ │ │ -0005d850: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0005d860: 7565 2073 0a20 202a 204f 7574 7075 7473 ue s. * Outputs │ │ │ │ -0005d870: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -0005d880: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0005d890: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0005d8a0: 652c 2c20 6f76 6572 2061 2072 696e 6720 e,, over a ring │ │ │ │ -0005d8b0: 5320 6d61 6465 2066 726f 6d20 7269 6e67 S made from ring │ │ │ │ -0005d8c0: 0a20 2020 2020 2020 2070 7265 7365 6e74 . present │ │ │ │ -0005d8d0: 6174 696f 6e20 5262 6172 2077 6974 6820 ation Rbar with │ │ │ │ -0005d8e0: 636f 6469 6d20 5262 6172 206e 6577 2076 codim Rbar new v │ │ │ │ -0005d8f0: 6172 6961 626c 6573 0a0a 4465 7363 7269 ariables..Descri │ │ │ │ -0005d900: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0005d910: 3d0a 0a4c 6574 2052 6261 7220 3d20 522f =..Let Rbar = R/ │ │ │ │ -0005d920: 2866 312e 2e66 6329 2c20 6120 636f 6d70 (f1..fc), a comp │ │ │ │ -0005d930: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -0005d940: 6e20 6f66 2063 6f64 696d 656e 7369 6f6e n of codimension │ │ │ │ -0005d950: 2063 2c20 616e 6420 6c65 7420 4d2c 4e20 c, and let M,N │ │ │ │ -0005d960: 6265 0a52 6261 722d 6d6f 6475 6c65 732e be.Rbar-modules. │ │ │ │ -0005d970: 2057 6520 6173 7375 6d65 2074 6861 7420 We assume that │ │ │ │ -0005d980: 7468 6520 7075 7368 466f 7277 6172 6420 the pushForward │ │ │ │ -0005d990: 6f66 204d 2074 6f20 5220 6861 7320 6669 of M to R has fi │ │ │ │ -0005d9a0: 6e69 7465 2066 7265 650a 7265 736f 6c75 nite free.resolu │ │ │ │ -0005d9b0: 7469 6f6e 2e20 5468 6520 7363 7269 7074 tion. The script │ │ │ │ -0005d9c0: 2074 6865 6e20 636f 6d70 7574 6573 2074 then computes t │ │ │ │ -0005d9d0: 6865 2074 6f74 616c 2045 7874 284d 2c4e he total Ext(M,N │ │ │ │ -0005d9e0: 2920 6173 2061 206d 6f64 756c 6520 6f76 ) as a module ov │ │ │ │ -0005d9f0: 6572 2053 203d 0a6b 6b28 735f 312e 2e73 er S =.kk(s_1..s │ │ │ │ -0005da00: 5f63 2c67 656e 7320 5229 2c20 7573 696e _c,gens R), usin │ │ │ │ -0005da10: 6720 4569 7365 6e62 7564 5368 616d 6173 g EisenbudShamas │ │ │ │ -0005da20: 6854 6f74 616c 2e0a 0a49 6620 4368 6563 hTotal...If Chec │ │ │ │ -0005da30: 6b20 3d3e 2074 7275 652c 2074 6865 6e20 k => true, then │ │ │ │ -0005da40: 7468 6520 7265 7375 6c74 2069 7320 636f the result is co │ │ │ │ -0005da50: 6d70 6172 6564 2077 6974 6820 7468 6520 mpared with the │ │ │ │ -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 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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 ---------------- │ │ │ │ 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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 │ │ 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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 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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 │ │ │ │ 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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 │ │ │ │ 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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 │ │ │ │ -00063f40: 3820 3339 2030 2030 2030 2030 2030 2030 8 39 0 0 0 0 0 0 │ │ │ │ -00063f50: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063f80: 2020 207b 302c 2030 7d20 2020 7c20 2d31 {0, 0} | -1 │ │ │ │ -00063f90: 3620 3231 2030 2030 2030 2030 2030 2030 6 21 0 0 0 0 0 0 │ │ │ │ -00063fa0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063fd0: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00063fe0: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00063ff0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064020: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064030: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064040: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064060: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064070: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064080: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064090: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000640a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000640b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000640c0: 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2020 2020 2020 2020 207c | │ │ │ │ +00063f20: 0a7c 6f33 3320 3d20 7b30 2c20 307d 2020 .|o33 = {0, 0} │ │ │ │ +00063f30: 207c 202d 3338 2033 3920 3020 3020 3020 | -38 39 0 0 0 │ │ │ │ +00063f40: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063f60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063f70: 0a7c 2020 2020 2020 7b30 2c20 307d 2020 .| {0, 0} │ │ │ │ +00063f80: 207c 202d 3136 2032 3120 3020 3020 3020 | -16 21 0 0 0 │ │ │ │ +00063f90: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063fb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063fc0: 0a7c 2020 2020 2020 7b2d 322c 202d 337d .| {-2, -3} │ │ │ │ +00063fd0: 207c 2030 2020 2030 2020 3020 3020 3020 | 0 0 0 0 0 │ │ │ │ +00063fe0: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064000: 2020 2020 2020 2020 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2020 2020 7c0a 7c6f 3333 |.|o33 │ │ │ │ -00064200: 203a 204d 6174 7269 7820 4553 203c 2d2d : Matrix ES <-- │ │ │ │ -00064210: 2045 4527 2020 2020 2020 2020 2020 2020 EE' │ │ │ │ +000641e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000641f0: 0a7c 6f33 3320 3a20 4d61 7472 6978 2045 .|o33 : Matrix E │ │ │ │ +00064200: 5320 3c2d 2d20 4545 2720 2020 2020 2020 S <-- EE' │ │ │ │ +00064210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064240: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00064230: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00064240: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00064250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064290: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620 ----------+..If │ │ │ │ -000642a0: 4545 2061 6e64 2045 5320 7765 7265 2069 EE and ES were i │ │ │ │ -000642b0: 736f 6d6f 7270 6869 632c 2077 6520 776f somorphic, we wo │ │ │ │ -000642c0: 756c 6420 6578 7065 6374 2063 6f6b 6572 uld expect coker │ │ │ │ -000642d0: 2066 2074 6f20 6265 2030 2c20 616e 6420 f to be 0, and │ │ │ │ -000642e0: 6974 2773 206e 6f74 2e0a 7072 756e 6520 it's not..prune │ │ │ │ -000642f0: 636f 6b65 7220 660a 0a53 6565 2061 6c73 coker f..See als │ │ │ │ -00064300: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00064310: 2a6e 6f74 6520 4578 743a 2028 4d61 6361 *note Ext: (Maca │ │ │ │ -00064320: 756c 6179 3244 6f63 2945 7874 2c20 2d2d ulay2Doc)Ext, -- │ │ │ │ -00064330: 2063 6f6d 7075 7465 2061 6e20 4578 7420 compute an Ext │ │ │ │ -00064340: 6d6f 6475 6c65 0a20 202a 202a 6e6f 7465 module. * *note │ │ │ │ -00064350: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00064360: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00064370: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00064380: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00064390: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -000643a0: 7874 0a0a 5761 7973 2074 6f20 7573 6520 xt..Ways to use │ │ │ │ -000643b0: 6e65 7745 7874 3a0a 3d3d 3d3d 3d3d 3d3d newExt:.======== │ │ │ │ -000643c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -000643d0: 2022 6e65 7745 7874 284d 6f64 756c 652c "newExt(Module, │ │ │ │ -000643e0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ -000643f0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00064400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00064410: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00064420: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ -00064430: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -00064440: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -00064450: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361 h options:.(Maca │ │ │ │ -00064460: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00064470: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -00064480: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00064290: 0a0a 4966 2045 4520 616e 6420 4553 2077 ..If EE and ES w │ │ │ │ +000642a0: 6572 6520 6973 6f6d 6f72 7068 6963 2c20 ere isomorphic, │ │ │ │ +000642b0: 7765 2077 6f75 6c64 2065 7870 6563 7420 we would expect │ │ │ │ +000642c0: 636f 6b65 7220 6620 746f 2062 6520 302c coker f to be 0, │ │ │ │ +000642d0: 2061 6e64 2069 7427 7320 6e6f 742e 0a70 and it's not..p │ │ │ │ +000642e0: 7275 6e65 2063 6f6b 6572 2066 0a0a 5365 rune coker f..Se │ │ │ │ +000642f0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00064300: 0a20 202a 202a 6e6f 7465 2045 7874 3a20 . * *note Ext: │ │ │ │ +00064310: 284d 6163 6175 6c61 7932 446f 6329 4578 (Macaulay2Doc)Ex │ │ │ │ +00064320: 742c 202d 2d20 636f 6d70 7574 6520 616e t, -- compute an │ │ │ │ +00064330: 2045 7874 206d 6f64 756c 650a 2020 2a20 Ext module. * │ │ │ │ +00064340: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +00064350: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00064360: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00064370: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00064380: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00064390: 7461 6c20 4578 740a 0a57 6179 7320 746f tal Ext..Ways to │ │ │ │ +000643a0: 2075 7365 206e 6577 4578 743a 0a3d 3d3d use newExt:.=== │ │ │ │ +000643b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000643c0: 0a0a 2020 2a20 226e 6577 4578 7428 4d6f .. * "newExt(Mo │ │ │ │ +000643d0: 6475 6c65 2c4d 6f64 756c 6529 220a 0a46 dule,Module)"..F │ │ │ │ +000643e0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +000643f0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00064400: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00064410: 202a 6e6f 7465 206e 6577 4578 743a 206e *note newExt: n │ │ │ │ +00064420: 6577 4578 742c 2069 7320 6120 2a6e 6f74 ewExt, is a *not │ │ │ │ +00064430: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00064440: 6e20 7769 7468 206f 7074 696f 6e73 3a0a n with options:. │ │ │ │ +00064450: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00064460: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00064470: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00064480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -000644e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -000644f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00064500: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00064510: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ -00064520: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00064530: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -00064540: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00064550: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -00064560: 3a32 3536 343a 302e 0a1f 0a46 696c 653a :2564:0....File: │ │ │ │ -00064570: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -00064580: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -00064590: 2e69 6e66 6f2c 204e 6f64 653a 206f 6464 .info, Node: odd │ │ │ │ -000645a0: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ -000645b0: 204f 7074 696d 6973 6d2c 2050 7265 763a Optimism, Prev: │ │ │ │ -000645c0: 206e 6577 4578 742c 2055 703a 2054 6f70 newExt, Up: Top │ │ │ │ -000645d0: 0a0a 6f64 6445 7874 4d6f 6475 6c65 202d ..oddExtModule - │ │ │ │ -000645e0: 2d20 6f64 6420 7061 7274 206f 6620 4578 - odd part of Ex │ │ │ │ -000645f0: 745e 2a28 4d2c 6b29 206f 7665 7220 6120 t^*(M,k) over a │ │ │ │ -00064600: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00064610: 6374 696f 6e20 6173 206d 6f64 756c 6520 ction as module │ │ │ │ -00064620: 6f76 6572 2043 4920 6f70 6572 6174 6f72 over CI operator │ │ │ │ -00064630: 2072 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a ring.********** │ │ │ │ +000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +000644d0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +000644e0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +000644f0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00064500: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00064510: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ +00064520: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00064530: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +00064540: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00064550: 6e73 2e6d 323a 3235 3634 3a30 2e0a 1f0a ns.m2:2564:0.... │ │ │ │ +00064560: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +00064570: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +00064580: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +00064590: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ +000645a0: 4e65 7874 3a20 4f70 7469 6d69 736d 2c20 Next: Optimism, │ │ │ │ +000645b0: 5072 6576 3a20 6e65 7745 7874 2c20 5570 Prev: newExt, Up │ │ │ │ +000645c0: 3a20 546f 700a 0a6f 6464 4578 744d 6f64 : Top..oddExtMod │ │ │ │ +000645d0: 756c 6520 2d2d 206f 6464 2070 6172 7420 ule -- odd part │ │ │ │ +000645e0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +000645f0: 6572 2061 2063 6f6d 706c 6574 6520 696e er a complete in │ │ │ │ +00064600: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ +00064610: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ +00064620: 7261 746f 7220 7269 6e67 0a2a 2a2a 2a2a rator ring.***** │ │ │ │ +00064630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064690: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -000646a0: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -000646b0: 203d 206f 6464 4578 744d 6f64 756c 6520 = oddExtModule │ │ │ │ -000646c0: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ -000646d0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -000646e0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -000646f0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00064700: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ -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 │ │ │ │ -00064740: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ -00064750: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ -00064760: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -00064770: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ -00064780: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e * OutRing => ... │ │ │ │ -00064790: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000647a0: 300a 2020 2a20 4f75 7470 7574 733a 0a20 0. * Outputs:. │ │ │ │ -000647b0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ -000647c0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -000647d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -000647e0: 206f 7665 7220 6120 706f 6c79 6e6f 6d69 over a polynomi │ │ │ │ -000647f0: 616c 2072 696e 6720 7769 7468 0a20 2020 al ring with. │ │ │ │ -00064800: 2020 2020 2067 656e 7320 696e 2064 6567 gens in deg │ │ │ │ -00064810: 7265 6520 310a 0a44 6573 6372 6970 7469 ree 1..Descripti │ │ │ │ -00064820: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00064830: 4578 7472 6163 7473 2074 6865 206f 6464 Extracts the odd │ │ │ │ -00064840: 2064 6567 7265 6520 7061 7274 2066 726f degree part fro │ │ │ │ -00064850: 6d20 4578 744d 6f64 756c 6520 4d2e 2049 m ExtModule M. I │ │ │ │ -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 │ │ │ │ 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|.|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 │ │ │ │ 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+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 │ │ │ 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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 │ │ │ │ -00065d70: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00065d80: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00065d90: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00065da0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00065db0: 3235 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+ds/M2/Maca │ │ │ │ -00065dc0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00065dd0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00065de0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00065df0: 6d32 3a33 3637 393a 302e 0a1f 0a46 696c m2:3679:0....Fil │ │ │ │ -00065e00: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00065e10: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00065e20: 6e73 2e69 6e66 6f2c 204e 6f64 653a 204f ns.info, Node: O │ │ │ │ -00065e30: 7074 696d 6973 6d2c 204e 6578 743a 204f ptimism, Next: O │ │ │ │ -00065e40: 7574 5269 6e67 2c20 5072 6576 3a20 6f64 utRing, Prev: od │ │ │ │ -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 **************** │ │ │ │ -00065ea0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00065eb0: 653a 200a 2020 2020 2020 2020 6869 6768 e: . high │ │ │ │ -00065ec0: 5379 7a79 6779 284d 2c20 4f70 7469 6d69 Syzygy(M, Optimi │ │ │ │ -00065ed0: 736d 203d 3e20 3129 0a20 202a 2049 6e70 sm => 1). * Inp │ │ │ │ -00065ee0: 7574 733a 0a20 2020 2020 202a 204f 7074 uts:. * Opt │ │ │ │ -00065ef0: 696d 6973 6d2c 2061 6e20 2a6e 6f74 6520 imism, an *note │ │ │ │ -00065f00: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -00065f10: 6179 3244 6f63 295a 5a2c 2c20 0a0a 4465 ay2Doc)ZZ,, ..De │ │ │ │ -00065f20: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00065f30: 3d3d 3d3d 3d0a 0a49 6620 6869 6768 5379 =====..If highSy │ │ │ │ -00065f40: 7a79 6779 284d 2920 6368 6f6f 7365 7320 zygy(M) chooses │ │ │ │ -00065f50: 7468 6520 702d 7468 2073 797a 7967 792c the p-th syzygy, │ │ │ │ -00065f60: 2074 6865 6e20 6869 6768 5379 7a79 6779 then highSyzygy │ │ │ │ -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 │ │ │ │ -00065f90: 292d 7468 2073 797a 7967 792e 2028 506f )-th syzygy. (Po │ │ │ │ -00065fa0: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -00065fb0: 6368 6f6f 7365 7320 6120 6c6f 7765 7220 chooses a lower │ │ │ │ -00065fc0: 2268 6967 6822 2073 797a 7967 792c 0a6e "high" syzygy,.n │ │ │ │ -00065fd0: 6567 6174 6976 6520 4f70 7469 6d69 736d egative Optimism │ │ │ │ -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 │ │ │ │ -00066020: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -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, -- │ │ │ │ -00066070: 2064 6574 6572 6d69 6e65 7320 686f 7720 determines how │ │ │ │ -00066080: 6869 6768 2061 2073 797a 7967 7920 746f high a syzygy to │ │ │ │ -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 │ │ │ │ -00066110: 6620 4578 7428 4d2c 6b29 0a0a 4675 6e63 f Ext(M,k)..Func │ │ │ │ -00066120: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00066130: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00066140: 6564 204f 7074 696d 6973 6d3a 0a3d 3d3d ed Optimism:.=== │ │ │ │ +00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00065d60: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00065d70: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00065d80: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00065d90: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00065da0: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ +00065db0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00065dc0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00065dd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00065de0: 696f 6e73 2e6d 323a 3336 3739 3a30 2e0a ions.m2:3679:0.. │ │ │ │ +00065df0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00065e00: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00065e10: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00065e20: 6465 3a20 4f70 7469 6d69 736d 2c20 4e65 de: Optimism, Ne │ │ │ │ +00065e30: 7874 3a20 4f75 7452 696e 672c 2050 7265 xt: OutRing, Pre │ │ │ │ +00065e40: 763a 206f 6464 4578 744d 6f64 756c 652c v: oddExtModule, │ │ │ │ +00065e50: 2055 703a 2054 6f70 0a0a 4f70 7469 6d69 Up: Top..Optimi │ │ │ │ +00065e60: 736d 202d 2d20 4f70 7469 6f6e 2074 6f20 sm -- Option to │ │ │ │ +00065e70: 6869 6768 5379 7a79 6779 0a2a 2a2a 2a2a highSyzygy.***** │ │ │ │ +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 * │ │ │ │ +00065ef0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00065f00: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00065f10: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ +00065f20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 2068 ==========..If h │ │ │ │ +00065f30: 6967 6853 797a 7967 7928 4d29 2063 686f ighSyzygy(M) cho │ │ │ │ +00065f40: 6f73 6573 2074 6865 2070 2d74 6820 7379 oses the p-th sy │ │ │ │ +00065f50: 7a79 6779 2c20 7468 656e 2068 6967 6853 zygy, then highS │ │ │ │ +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 │ │ │ │ +00065fa0: 6d69 736d 2063 686f 6f73 6573 2061 206c mism chooses a l │ │ │ │ +00065fb0: 6f77 6572 2022 6869 6768 2220 7379 7a79 ower "high" syzy │ │ │ │ +00065fc0: 6779 2c0a 6e65 6761 7469 7665 204f 7074 gy,.negative Opt │ │ │ │ +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 │ │ │ │ +00066000: 6520 7468 6572 6520 6361 7365 7320 7768 e there cases wh │ │ │ │ +00066010: 656e 2070 6f73 6974 6976 6520 4f70 7469 en positive Opti │ │ │ │ +00066020: 6d69 736d 2069 7320 6a75 7374 6966 6965 mism is justifie │ │ │ │ +00066030: 643f 0a0a 5365 6520 616c 736f 0a3d 3d3d d?..See also.=== │ │ │ │ +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 │ │ │ │ +00066130: 7420 6e61 6d65 6420 4f70 7469 6d69 736d t named Optimism │ │ │ │ +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 │ │ │ │ -000661d0: 6574 7572 6e73 2061 2073 797a 7967 7920 eturns a syzygy │ │ │ │ -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 │ │ │ │ -00066270: 7565 6e63 6573 206f 6620 4252 616e 6b73 uences of BRanks │ │ │ │ -00066280: 2066 6f72 2063 6572 7461 696e 2065 7861 for certain exa │ │ │ │ -00066290: 6d70 6c65 730a 0a46 6f72 2074 6865 2070 mples..For the p │ │ │ │ -000662a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000662b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000662c0: 6520 6f62 6a65 6374 202a 6e6f 7465 204f e object *note O │ │ │ │ -000662d0: 7074 696d 6973 6d3a 204f 7074 696d 6973 ptimism: Optimis │ │ │ │ -000662e0: 6d2c 2069 7320 6120 2a6e 6f74 6520 7379 m, is a *note sy │ │ │ │ -000662f0: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932 mbol: (Macaulay2 │ │ │ │ -00066300: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d Doc)Symbol,...-- │ │ │ │ +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 │ │ │ │ +000662c0: 6f74 6520 4f70 7469 6d69 736d 3a20 4f70 ote Optimism: Op │ │ │ │ +000662d0: 7469 6d69 736d 2c20 6973 2061 202a 6e6f timism, is a *no │ │ │ │ +000662e0: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ +000662f0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ +00066300: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 00066310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00066360: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00066370: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00066380: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00066390: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -000663a0: 6179 322d 312e 3235 2e30 362b 6473 2f4d ay2-1.25.06+ds/M │ │ │ │ -000663b0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -000663c0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -000663d0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000663e0: 7469 6f6e 732e 6d32 3a33 3136 353a 302e tions.m2:3165:0. │ │ │ │ -000663f0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -00066400: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -00066410: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -00066420: 6f64 653a 204f 7574 5269 6e67 2c20 4e65 ode: OutRing, Ne │ │ │ │ -00066430: 7874 3a20 7073 694d 6170 732c 2050 7265 xt: psiMaps, Pre │ │ │ │ -00066440: 763a 204f 7074 696d 6973 6d2c 2055 703a v: Optimism, Up: │ │ │ │ -00066450: 2054 6f70 0a0a 4f75 7452 696e 6720 2d2d Top..OutRing -- │ │ │ │ -00066460: 204f 7074 696f 6e20 616c 6c6f 7769 6e67 Option allowing │ │ │ │ -00066470: 2073 7065 6369 6669 6361 7469 6f6e 206f specification o │ │ │ │ -00066480: 6620 7468 6520 7269 6e67 206f 7665 7220 f the ring over │ │ │ │ -00066490: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -000664a0: 2069 7320 6465 6669 6e65 640a 2a2a 2a2a is defined.**** │ │ │ │ +00066350: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00066360: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00066370: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00066380: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00066390: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +000663a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000663b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +000663c0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +000663d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331 esolutions.m2:31 │ │ │ │ +000663e0: 3635 3a30 2e0a 1f0a 4669 6c65 3a20 436f 65:0....File: Co │ │ │ │ +000663f0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +00066400: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +00066410: 666f 2c20 4e6f 6465 3a20 4f75 7452 696e fo, Node: OutRin │ │ │ │ +00066420: 672c 204e 6578 743a 2070 7369 4d61 7073 g, Next: psiMaps │ │ │ │ +00066430: 2c20 5072 6576 3a20 4f70 7469 6d69 736d , Prev: Optimism │ │ │ │ +00066440: 2c20 5570 3a20 546f 700a 0a4f 7574 5269 , Up: Top..OutRi │ │ │ │ +00066450: 6e67 202d 2d20 4f70 7469 6f6e 2061 6c6c ng -- Option all │ │ │ │ +00066460: 6f77 696e 6720 7370 6563 6966 6963 6174 owing specificat │ │ │ │ +00066470: 696f 6e20 6f66 2074 6865 2072 696e 6720 ion of the ring │ │ │ │ +00066480: 6f76 6572 2077 6869 6368 2074 6865 206f over which the o │ │ │ │ +00066490: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +000664a0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 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 │ │ │ │ -00066620: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -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 │ │ │ │ +00066520: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ +00066530: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ +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 3036 2b64 aulay2-1.25.06+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: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+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: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+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 2e30 362b 6473 2f4d 322f 4d61 1.25.06+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 │ │ │ │ -00067890: 6578 742c 2073 6f63 2064 6567 7320 6576 ext, soc degs ev │ │ │ │ -000678a0: 656e 2065 7874 2c20 7265 6720 6f64 6420 en ext, reg odd │ │ │ │ -000678b0: 6578 742c 2073 6f63 2064 6567 7320 6f64 ext, soc degs od │ │ │ │ -000678c0: 6420 6578 747c 0a7c 2020 2020 2020 2020 d ext|.| │ │ │ │ +00067830: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ +00067840: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ +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 │ │ │ │ +00067890: 6773 2065 7665 6e20 6578 742c 2072 6567 gs even ext, reg │ │ │ │ +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, │ │ │ │ -00067910: 2031 2c20 317d 2c20 322c 207b 312c 2031 1, 1}, 2, {1, 1 │ │ │ │ -00067920: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -00067930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067940: 2020 2020 2020 207c 0a7c 7b32 2c20 7b30 |.|{2, {0 │ │ │ │ -00067950: 2c20 302c 2030 2c20 317d 2c20 322c 207b , 0, 0, 1}, 2, { │ │ │ │ -00067960: 302c 2030 2c20 307d 7d20 2020 2020 2020 0, 0, 0}} │ │ │ │ -00067970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067980: 2020 2020 2020 2020 7c0a 7c7b 302c 207b |.|{0, { │ │ │ │ -00067990: 7d2c 2030 2c20 7b7d 7d20 2020 2020 2020 }, 0, {}} │ │ │ │ +000678f0: 2020 2020 2020 2020 2020 207c 0a7c 7b33 |.|{3 │ │ │ │ +00067900: 2c20 7b31 2c20 312c 2031 7d2c 2032 2c20 , {1, 1, 1}, 2, │ │ │ │ +00067910: 7b31 2c20 317d 7d20 2020 2020 2020 2020 {1, 1}} │ │ │ │ +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 ---------------- │ │ │ │ -00067a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -00067a10: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00067a20: 2020 2a20 2a6e 6f74 6520 6170 7072 6f78 * *note approx │ │ │ │ -00067a30: 696d 6174 696f 6e3a 2028 4d43 4d41 7070 imation: (MCMApp │ │ │ │ -00067a40: 726f 7869 6d61 7469 6f6e 7329 6170 7072 roximations)appr │ │ │ │ -00067a50: 6f78 696d 6174 696f 6e2c 202d 2d20 7265 oximation, -- re │ │ │ │ -00067a60: 7475 726e 7320 7061 6972 206f 660a 2020 turns pair of. │ │ │ │ -00067a70: 2020 636f 6d70 6f6e 656e 7473 206f 6620 components of │ │ │ │ -00067a80: 7468 6520 6d61 7020 6672 6f6d 2074 6865 the map from the │ │ │ │ -00067a90: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067aa0: 6f6e 0a20 202a 202a 6e6f 7465 2061 7573 on. * *note aus │ │ │ │ -00067ab0: 6c61 6e64 6572 496e 7661 7269 616e 743a landerInvariant: │ │ │ │ -00067ac0: 2028 4d43 4d41 7070 726f 7869 6d61 7469 (MCMApproximati │ │ │ │ -00067ad0: 6f6e 7329 6175 736c 616e 6465 7249 6e76 ons)auslanderInv │ │ │ │ -00067ae0: 6172 6961 6e74 2c20 2d2d 0a20 2020 206d ariant, --. m │ │ │ │ -00067af0: 6561 7375 7265 7320 6661 696c 7572 6520 easures failure │ │ │ │ -00067b00: 6f66 2073 7572 6a65 6374 6976 6974 7920 of surjectivity │ │ │ │ -00067b10: 6f66 2074 6865 2065 7373 656e 7469 616c of the essential │ │ │ │ -00067b20: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067b30: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520 on..Ways to use │ │ │ │ -00067b40: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ -00067b50: 6365 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ce:.============ │ │ │ │ -00067b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067b70: 3d3d 3d0a 0a20 202a 2022 7265 6775 6c61 ===.. * "regula │ │ │ │ -00067b80: 7269 7479 5365 7175 656e 6365 284c 6973 ritySequence(Lis │ │ │ │ -00067b90: 742c 4d6f 6475 6c65 2922 0a0a 466f 7220 t,Module)"..For │ │ │ │ -00067ba0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00067bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067bc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00067bd0: 6f74 6520 7265 6775 6c61 7269 7479 5365 ote regularitySe │ │ │ │ -00067be0: 7175 656e 6365 3a20 7265 6775 6c61 7269 quence: regulari │ │ │ │ -00067bf0: 7479 5365 7175 656e 6365 2c20 6973 2061 tySequence, is a │ │ │ │ -00067c00: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00067c10: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -00067c20: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00067c30: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00067a00: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00067a10: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2061 ===.. * *note a │ │ │ │ +00067a20: 7070 726f 7869 6d61 7469 6f6e 3a20 284d pproximation: (M │ │ │ │ +00067a30: 434d 4170 7072 6f78 696d 6174 696f 6e73 CMApproximations │ │ │ │ +00067a40: 2961 7070 726f 7869 6d61 7469 6f6e 2c20 )approximation, │ │ │ │ +00067a50: 2d2d 2072 6574 7572 6e73 2070 6169 7220 -- returns pair │ │ │ │ +00067a60: 6f66 0a20 2020 2063 6f6d 706f 6e65 6e74 of. component │ │ │ │ +00067a70: 7320 6f66 2074 6865 206d 6170 2066 726f s of the map fro │ │ │ │ +00067a80: 6d20 7468 6520 4d43 4d20 6170 7072 6f78 m the MCM approx │ │ │ │ +00067a90: 696d 6174 696f 6e0a 2020 2a20 2a6e 6f74 imation. * *not │ │ │ │ +00067aa0: 6520 6175 736c 616e 6465 7249 6e76 6172 e auslanderInvar │ │ │ │ +00067ab0: 6961 6e74 3a20 284d 434d 4170 7072 6f78 iant: (MCMApprox │ │ │ │ +00067ac0: 696d 6174 696f 6e73 2961 7573 6c61 6e64 imations)ausland │ │ │ │ +00067ad0: 6572 496e 7661 7269 616e 742c 202d 2d0a erInvariant, --. │ │ │ │ +00067ae0: 2020 2020 6d65 6173 7572 6573 2066 6169 measures fai │ │ │ │ +00067af0: 6c75 7265 206f 6620 7375 726a 6563 7469 lure of surjecti │ │ │ │ +00067b00: 7669 7479 206f 6620 7468 6520 6573 7365 vity of the esse │ │ │ │ +00067b10: 6e74 6961 6c20 4d43 4d20 6170 7072 6f78 ntial MCM approx │ │ │ │ +00067b20: 696d 6174 696f 6e0a 0a57 6179 7320 746f imation..Ways to │ │ │ │ +00067b30: 2075 7365 2072 6567 756c 6172 6974 7953 use regularityS │ │ │ │ +00067b40: 6571 7565 6e63 653a 0a3d 3d3d 3d3d 3d3d equence:.======= │ │ │ │ +00067b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00067b60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272 ========.. * "r │ │ │ │ +00067b70: 6567 756c 6172 6974 7953 6571 7565 6e63 egularitySequenc │ │ │ │ +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 2e30 362b 6473 2f4d 322f 4d61 6361 25.06+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 3036 2b64 732f 4d32 y2-1.25.06+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 │ │ │ │ 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+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 |.| │ │ │ │ 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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: 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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 │ │ │ │ +0006a670: 706f 6e64 656e 6365 0a20 202a 202a 6e6f pondence. * *no │ │ │ │ +0006a680: 7465 2063 6f68 6f6d 6f6c 6f67 793a 2028 te cohomology: ( │ │ │ │ +0006a690: 4d61 6361 756c 6179 3244 6f63 2963 6f68 Macaulay2Doc)coh │ │ │ │ +0006a6a0: 6f6d 6f6c 6f67 792c 202d 2d20 6765 6e65 omology, -- gene │ │ │ │ +0006a6b0: 7261 6c20 636f 686f 6d6f 6c6f 6779 2066 ral cohomology f │ │ │ │ +0006a6c0: 756e 6374 6f72 0a20 202a 2048 485e 5a5a unctor. * HH^ZZ │ │ │ │ +0006a6d0: 2053 756d 4f66 5477 6973 7473 2028 6d69 SumOfTwists (mi │ │ │ │ +0006a6e0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +0006a6f0: 696f 6e29 0a0a 5761 7973 2074 6f20 7573 ion)..Ways to us │ │ │ │ +0006a700: 6520 5332 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d e S2:.========== │ │ │ │ +0006a710: 3d3d 3d3d 3d0a 0a20 202a 2022 5332 285a =====.. * "S2(Z │ │ │ │ +0006a720: 5a2c 4d6f 6475 6c65 2922 0a0a 466f 7220 Z,Module)"..For │ │ │ │ +0006a730: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0006a740: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006a750: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0006a760: 6f74 6520 5332 3a20 5332 2c20 6973 2061 ote S2: S2, is a │ │ │ │ +0006a770: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +0006a780: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61 nction:.(Macaula │ │ │ │ +0006a790: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0006a7a0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +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 3036 acaulay2-1.25.06 │ │ │ │ -0006a850: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006a860: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006a870: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006a880: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ -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 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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 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---------------- │ │ │ │ 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 │ │ │ │ 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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 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│ │ │ │ -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 . │ │ │ │ -0006c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c020: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0006c030: 2020 2020 313a 202e 202e 2033 2020 3920 1: . . 3 9 │ │ │ │ -0006c040: 2039 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ -0006c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006c060: 7c20 2020 2020 2020 2020 2032 3a20 2e20 | 2: . │ │ │ │ -0006c070: 2e20 2e20 202e 2020 3620 2020 2020 2020 . . . 6 │ │ │ │ -0006c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c090: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bf70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006bf80: 2020 2020 2020 2030 2031 2032 2020 3320 0 1 2 3 │ │ │ │ +0006bf90: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0006bfa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006bfb0: 7c6f 3131 203d 2074 6f74 616c 3a20 3120 |o11 = total: 1 │ │ │ │ +0006bfc0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +0006bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bff0: 2020 303a 2031 2033 2033 2020 3120 202e 0: 1 3 3 1 . │ │ │ │ +0006c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006c020: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0006c030: 3320 2039 2020 3920 2020 2020 2020 2020 3 9 9 │ │ │ │ +0006c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c050: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c060: 323a 202e 202e 202e 2020 2e20 2036 2020 2: . . . . 6 │ │ │ │ +0006c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c080: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006c0d0: 3131 203a 2042 6574 7469 5461 6c6c 7920 11 : BettiTally │ │ │ │ +0006c0c0: 207c 0a7c 6f31 3120 3a20 4265 7474 6954 |.|o11 : BettiT │ │ │ │ +0006c0d0: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 0006c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006c0f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ -0006c140: 203a 2072 696e 6720 4747 2020 2020 2020 : ring GG │ │ │ │ +0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0006c130: 0a7c 6931 3220 3a20 7269 6e67 2047 4720 .|i12 : ring GG │ │ │ │ +0006c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c170: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c160: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c1a0: 2020 2020 2020 2020 7c0a 7c6f 3132 203d |.|o12 = │ │ │ │ -0006c1b0: 2052 3120 2020 2020 2020 2020 2020 2020 R1 │ │ │ │ +0006c190: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006c1a0: 6f31 3220 3d20 5231 2020 2020 2020 2020 o12 = R1 │ │ │ │ +0006c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c1c0: 2020 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ -0006c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c390: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006c380: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006c3d0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a .Caveat.======.. │ │ │ │ -0006c3e0: 4620 6973 2061 7373 756d 6564 2074 6f20 F is assumed to │ │ │ │ -0006c3f0: 6265 2061 2068 6f6d 6f6c 6f67 6963 616c be a homological │ │ │ │ -0006c400: 2063 6f6d 706c 6578 2073 7461 7274 696e complex startin │ │ │ │ -0006c410: 6720 6672 6f6d 2046 5f30 2e20 5468 6520 g from F_0. The │ │ │ │ -0006c420: 6d61 7472 6978 2066 6620 6d75 7374 0a62 matrix ff must.b │ │ │ │ -0006c430: 6520 3178 312e 0a0a 5365 6520 616c 736f e 1x1...See also │ │ │ │ -0006c440: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0006c450: 6e6f 7465 2045 6973 656e 6275 6453 6861 note EisenbudSha │ │ │ │ -0006c460: 6d61 7368 3a20 4569 7365 6e62 7564 5368 mash: EisenbudSh │ │ │ │ -0006c470: 616d 6173 682c 202d 2d20 436f 6d70 7574 amash, -- Comput │ │ │ │ -0006c480: 6573 2074 6865 2045 6973 656e 6275 642d es the Eisenbud- │ │ │ │ -0006c490: 5368 616d 6173 680a 2020 2020 436f 6d70 Shamash. Comp │ │ │ │ -0006c4a0: 6c65 780a 2020 2a20 2a6e 6f74 6520 6d61 lex. * *note ma │ │ │ │ -0006c4b0: 6b65 486f 6d6f 746f 7069 6573 3a20 6d61 keHomotopies: ma │ │ │ │ -0006c4c0: 6b65 486f 6d6f 746f 7069 6573 2c20 2d2d keHomotopies, -- │ │ │ │ -0006c4d0: 2072 6574 7572 6e73 2061 2073 7973 7465 returns a syste │ │ │ │ -0006c4e0: 6d20 6f66 2068 6967 6865 720a 2020 2020 m of higher. │ │ │ │ -0006c4f0: 686f 6d6f 746f 7069 6573 0a0a 5761 7973 homotopies..Ways │ │ │ │ -0006c500: 2074 6f20 7573 6520 5368 616d 6173 683a to use Shamash: │ │ │ │ -0006c510: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006c520: 3d3d 3d3d 3d0a 0a20 202a 2022 5368 616d =====.. * "Sham │ │ │ │ -0006c530: 6173 6828 4d61 7472 6978 2c43 6f6d 706c ash(Matrix,Compl │ │ │ │ -0006c540: 6578 2c5a 5a29 220a 2020 2a20 2253 6861 ex,ZZ)". * "Sha │ │ │ │ -0006c550: 6d61 7368 2852 696e 672c 436f 6d70 6c65 mash(Ring,Comple │ │ │ │ -0006c560: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -0006c570: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0006c580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0006c590: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0006c5a0: 5368 616d 6173 683a 2053 6861 6d61 7368 Shamash: Shamash │ │ │ │ -0006c5b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0006c5c0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0006c5d0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0006c5e0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0006c3c0: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +0006c3d0: 3d3d 3d0a 0a46 2069 7320 6173 7375 6d65 ===..F is assume │ │ │ │ +0006c3e0: 6420 746f 2062 6520 6120 686f 6d6f 6c6f d to be a homolo │ │ │ │ +0006c3f0: 6769 6361 6c20 636f 6d70 6c65 7820 7374 gical complex st │ │ │ │ +0006c400: 6172 7469 6e67 2066 726f 6d20 465f 302e arting from F_0. │ │ │ │ +0006c410: 2054 6865 206d 6174 7269 7820 6666 206d The matrix ff m │ │ │ │ +0006c420: 7573 740a 6265 2031 7831 2e0a 0a53 6565 ust.be 1x1...See │ │ │ │ +0006c430: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +0006c440: 2020 2a20 2a6e 6f74 6520 4569 7365 6e62 * *note Eisenb │ │ │ │ +0006c450: 7564 5368 616d 6173 683a 2045 6973 656e udShamash: Eisen │ │ │ │ +0006c460: 6275 6453 6861 6d61 7368 2c20 2d2d 2043 budShamash, -- C │ │ │ │ +0006c470: 6f6d 7075 7465 7320 7468 6520 4569 7365 omputes the Eise │ │ │ │ +0006c480: 6e62 7564 2d53 6861 6d61 7368 0a20 2020 nbud-Shamash. │ │ │ │ +0006c490: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ +0006c4a0: 7465 206d 616b 6548 6f6d 6f74 6f70 6965 te makeHomotopie │ │ │ │ +0006c4b0: 733a 206d 616b 6548 6f6d 6f74 6f70 6965 s: makeHomotopie │ │ │ │ +0006c4c0: 732c 202d 2d20 7265 7475 726e 7320 6120 s, -- returns a │ │ │ │ +0006c4d0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ +0006c4e0: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ +0006c4f0: 0a57 6179 7320 746f 2075 7365 2053 6861 .Ways to use Sha │ │ │ │ +0006c500: 6d61 7368 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d mash:.========== │ │ │ │ +0006c510: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0006c520: 2253 6861 6d61 7368 284d 6174 7269 782c "Shamash(Matrix, │ │ │ │ +0006c530: 436f 6d70 6c65 782c 5a5a 2922 0a20 202a Complex,ZZ)". * │ │ │ │ +0006c540: 2022 5368 616d 6173 6828 5269 6e67 2c43 "Shamash(Ring,C │ │ │ │ +0006c550: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +0006c560: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0006c570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006c580: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0006c590: 6e6f 7465 2053 6861 6d61 7368 3a20 5368 note Shamash: Sh │ │ │ │ +0006c5a0: 616d 6173 682c 2069 7320 6120 2a6e 6f74 amash, is a *not │ │ │ │ +0006c5b0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0006c5c0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0006c5d0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0006c5e0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0006c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0006c640: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0006c650: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0006c660: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0006c670: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0006c680: 6179 322d 312e 3235 2e30 362b 6473 2f4d ay2-1.25.06+ds/M │ │ │ │ -0006c690: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0006c6a0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0006c6b0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0006c6c0: 7469 6f6e 732e 6d32 3a34 3736 303a 302e tions.m2:4760:0. │ │ │ │ -0006c6d0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0006c6e0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006c6f0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0006c700: 6f64 653a 2073 706c 6974 7469 6e67 732c ode: splittings, │ │ │ │ -0006c710: 204e 6578 743a 2073 7461 626c 6548 6f6d Next: stableHom │ │ │ │ -0006c720: 2c20 5072 6576 3a20 5368 616d 6173 682c , Prev: Shamash, │ │ │ │ -0006c730: 2055 703a 2054 6f70 0a0a 7370 6c69 7474 Up: Top..splitt │ │ │ │ -0006c740: 696e 6773 202d 2d20 636f 6d70 7574 6520 ings -- compute │ │ │ │ -0006c750: 7468 6520 7370 6c69 7474 696e 6773 206f the splittings o │ │ │ │ -0006c760: 6620 6120 7370 6c69 7420 7269 6768 7420 f a split right │ │ │ │ -0006c770: 6578 6163 7420 7365 7175 656e 6365 0a2a exact sequence.* │ │ │ │ +0006c630: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0006c640: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0006c650: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0006c660: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0006c670: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ +0006c680: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0006c690: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0006c6a0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0006c6b0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3437 esolutions.m2:47 │ │ │ │ +0006c6c0: 3630 3a30 2e0a 1f0a 4669 6c65 3a20 436f 60:0....File: Co │ │ │ │ +0006c6d0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006c6e0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0006c6f0: 666f 2c20 4e6f 6465 3a20 7370 6c69 7474 fo, Node: splitt │ │ │ │ +0006c700: 696e 6773 2c20 4e65 7874 3a20 7374 6162 ings, Next: stab │ │ │ │ +0006c710: 6c65 486f 6d2c 2050 7265 763a 2053 6861 leHom, Prev: Sha │ │ │ │ +0006c720: 6d61 7368 2c20 5570 3a20 546f 700a 0a73 mash, Up: Top..s │ │ │ │ +0006c730: 706c 6974 7469 6e67 7320 2d2d 2063 6f6d plittings -- com │ │ │ │ +0006c740: 7075 7465 2074 6865 2073 706c 6974 7469 pute the splitti │ │ │ │ +0006c750: 6e67 7320 6f66 2061 2073 706c 6974 2072 ngs of a split r │ │ │ │ +0006c760: 6967 6874 2065 7861 6374 2073 6571 7565 ight exact seque │ │ │ │ +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 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+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 │ │ │ │ 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│ │ │ │ -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 | 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006d840: 2020 2020 2020 2020 2032 3a20 3220 2e20 2: 2 . │ │ │ │ -0006d850: 2020 2020 2032 3a20 3420 2e20 2020 2020 2: 4 . │ │ │ │ +0006d830: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ +0006d840: 2032 202e 2020 2020 2020 323a 2034 202e 2 . 2: 4 . │ │ │ │ +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 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0a57 6179 7320 746f 2075 7365 2073 +..Ways to use s │ │ │ │ -0006d980: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d plittings:.===== │ │ │ │ -0006d990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -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.======== │ │ │ │ -0006d9e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0006d9f0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c object *note spl │ │ │ │ -0006da00: 6974 7469 6e67 733a 2073 706c 6974 7469 ittings: splitti │ │ │ │ -0006da10: 6e67 732c 2069 7320 6120 2a6e 6f74 6520 ngs, is a *note │ │ │ │ -0006da20: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -0006da30: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -0006da40: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -0006da50: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0006d960: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ +0006d970: 7573 6520 7370 6c69 7474 696e 6773 3a0a use splittings:. │ │ │ │ +0006d980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006d990: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7370 =======.. * "sp │ │ │ │ +0006d9a0: 6c69 7474 696e 6773 284d 6174 7269 782c littings(Matrix, │ │ │ │ +0006d9b0: 4d61 7472 6978 2922 0a0a 466f 7220 7468 Matrix)"..For th │ │ │ │ +0006d9c0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +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 │ │ │ │ +0006da20: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0006da30: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0006da40: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ -0006dab0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0006dac0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0006dad0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0006dae0: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+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 │ │ │ │ -0006db30: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -0006db40: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -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: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+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 │ │ │ │ +0006db20: 3a33 3932 353a 302e 0a1f 0a46 696c 653a :3925:0....File: │ │ │ │ +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 │ │ │ │ -0006dda0: 6572 206f 6620 4e20 286f 722c 2065 7175 er of N (or, equ │ │ │ │ -0006ddb0: 6976 616c 656e 746c 792c 2074 6872 6f75 ivalently, throu │ │ │ │ -0006ddc0: 6768 2061 6e79 2070 726f 6a65 6374 6976 gh any projectiv │ │ │ │ -0006ddd0: 6529 0a0a 5365 6520 616c 736f 0a3d 3d3d e)..See also.=== │ │ │ │ -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 │ │ │ │ -0006de20: 7565 2069 6620 7468 6520 6d61 7020 676f ue if the map go │ │ │ │ -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 │ │ │ │ -0006de90: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220 e,Module)"..For │ │ │ │ -0006dea0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -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 │ │ │ │ -0006df10: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0006df20: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006dc00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0006dc10: 7361 6765 3a20 0a20 2020 2020 2020 2070 sage: . p │ │ │ │ +0006dc20: 203d 2073 7461 626c 6548 6f6d 284d 2c4e = stableHom(M,N │ │ │ │ +0006dc30: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006dc40: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0006dc50: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0006dc60: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0006dc70: 0a20 2020 2020 202a 204e 2c20 6120 2a6e . * N, a *n │ │ │ │ +0006dc80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ +0006dc90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ +0006dca0: 2c2c 200a 2020 2a20 4f75 7470 7574 733a ,, . * Outputs: │ │ │ │ +0006dcb0: 0a20 2020 2020 202a 2070 2c20 6120 2a6e . * p, a *n │ │ │ │ +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 │ │ │ │ +0006dcf0: 6f6d 2048 6f6d 284d 2c4e 2920 746f 0a20 om Hom(M,N) to. │ │ │ │ +0006dd00: 2020 2020 2020 2074 6865 2073 7461 626c the stabl │ │ │ │ +0006dd10: 6520 486f 6d0a 0a44 6573 6372 6970 7469 e Hom..Descripti │ │ │ │ +0006dd20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0006dd30: 5468 6520 7374 6162 6c65 2048 6f6d 2069 The stable Hom i │ │ │ │ +0006dd40: 7320 486f 6d28 4d2c 4e29 2f54 2077 6865 s Hom(M,N)/T whe │ │ │ │ +0006dd50: 7265 2054 2069 7320 7468 6520 7375 626d re T is the subm │ │ │ │ +0006dd60: 6f64 756c 6520 6f66 2068 6f6d 6f6d 6f72 odule of homomor │ │ │ │ +0006dd70: 7068 6973 6d73 2074 6861 740a 6661 6374 phisms that.fact │ │ │ │ +0006dd80: 6f72 2074 6872 6f75 6768 2061 2066 7265 or through a fre │ │ │ │ +0006dd90: 6520 636f 7665 7220 6f66 204e 2028 6f72 e cover of N (or │ │ │ │ +0006dda0: 2c20 6571 7569 7661 6c65 6e74 6c79 2c20 , equivalently, │ │ │ │ +0006ddb0: 7468 726f 7567 6820 616e 7920 7072 6f6a through any proj │ │ │ │ +0006ddc0: 6563 7469 7665 290a 0a53 6565 2061 6c73 ective)..See als │ │ │ │ +0006ddd0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +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 │ │ │ │ +0006de50: 6162 6c65 486f 6d3a 0a3d 3d3d 3d3d 3d3d ableHom:.======= │ │ │ │ +0006de60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006de70: 0a20 202a 2022 7374 6162 6c65 486f 6d28 . * "stableHom( │ │ │ │ +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 │ │ │ │ +0006dec0: 6374 202a 6e6f 7465 2073 7461 626c 6548 ct *note stableH │ │ │ │ +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 │ │ │ │ +0006df10: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0006df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ -0006df80: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0006df90: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -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: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+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 │ │ │ │ -0006e000: 3a34 3634 393a 302e 0a1f 0a46 696c 653a :4649:0....File: │ │ │ │ -0006e010: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -0006e020: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -0006e030: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d .info, Node: sum │ │ │ │ -0006e040: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65 TwoMonomials, Ne │ │ │ │ -0006e050: 7874 3a20 5461 7465 5265 736f 6c75 7469 xt: TateResoluti │ │ │ │ -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 │ │ │ │ -0006e0a0: 6e63 6573 206f 6620 4252 616e 6b73 2066 nces of BRanks f │ │ │ │ -0006e0b0: 6f72 2063 6572 7461 696e 2065 7861 6d70 or certain examp │ │ │ │ -0006e0c0: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a les.************ │ │ │ │ +0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006df70: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +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 3036 2b64 732f 4d32 2f4d -1.25.06+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 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ -0006e370: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0006e360: 6931 203a 2073 6574 5261 6e64 6f6d 5365 i1 : setRandomSe │ │ │ │ +0006e370: 6564 2030 2020 2020 2020 2020 2020 2020 ed 0 │ │ │ │ 0006e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3a0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -0006e3b0: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -0006e3c0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -0006e3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006e390: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +0006e3a0: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ +0006e3b0: 6565 6420 746f 2030 2020 2020 2020 2020 eed to 0 │ │ │ │ +0006e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e3d0: 2020 2020 207c 0a7c 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 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0006e420: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +0006e410: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +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 2020 │ │ │ │ -0006e450: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006e440: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006e490: 7c69 3220 3a20 7375 6d54 776f 4d6f 6e6f |i2 : sumTwoMono │ │ │ │ -0006e4a0: 6d69 616c 7328 322c 3329 2020 2020 2020 mials(2,3) │ │ │ │ +0006e480: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ +0006e490: 2073 756d 5477 6f4d 6f6e 6f6d 6961 6c73 sumTwoMonomials │ │ │ │ +0006e4a0: 2832 2c33 2920 2020 2020 2020 2020 2020 (2,3) │ │ │ │ 0006e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e4c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0006e4d0: 7573 6564 2030 2e36 3236 3233 3273 2028 used 0.626232s ( │ │ │ │ -0006e4e0: 6370 7529 3b20 302e 3431 3734 3533 7320 cpu); 0.417453s │ │ │ │ -0006e4f0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0006e500: 2920 2020 7c0a 7c20 2d2d 2075 7365 6420 ) |.| -- used │ │ │ │ -0006e510: 302e 3230 3438 3573 2028 6370 7529 3b20 0.20485s (cpu); │ │ │ │ -0006e520: 302e 3133 3633 3337 7320 2874 6872 6561 0.136337s (threa │ │ │ │ -0006e530: 6429 3b20 3073 2028 6763 2920 2020 207c d); 0s (gc) | │ │ │ │ -0006e540: 0a7c 202d 2d20 7573 6564 2030 2e30 3030 .| -- used 0.000 │ │ │ │ -0006e550: 3136 3031 3173 2028 6370 7529 3b20 332e 16011s (cpu); 3. │ │ │ │ -0006e560: 3230 3665 2d30 3673 2028 7468 7265 6164 206e-06s (thread │ │ │ │ -0006e570: 293b 2030 7320 2867 6329 7c0a 7c32 2020 ); 0s (gc)|.|2 │ │ │ │ +0006e4c0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ +0006e4d0: 2030 2e37 3236 3839 3973 2028 6370 7529 0.726899s (cpu) │ │ │ │ +0006e4e0: 3b20 302e 3433 3536 3535 7320 2874 6872 ; 0.435655s (thr │ │ │ │ +0006e4f0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0006e500: 207c 0a7c 202d 2d20 7573 6564 2030 2e32 |.| -- used 0.2 │ │ │ │ +0006e510: 3833 3931 3573 2028 6370 7529 3b20 302e 83915s (cpu); 0. │ │ │ │ +0006e520: 3135 3430 3031 7320 2874 6872 6561 6429 154001s (thread) │ │ │ │ +0006e530: 3b20 3073 2028 6763 2920 2020 207c 0a7c ; 0s (gc) |.| │ │ │ │ +0006e540: 202d 2d20 7573 6564 2030 2e30 3030 3135 -- used 0.00015 │ │ │ │ +0006e550: 3139 3833 7320 2863 7075 293b 2032 2e35 1983s (cpu); 2.5 │ │ │ │ +0006e560: 3432 652d 3036 7320 2874 6872 6561 6429 42e-06s (thread) │ │ │ │ +0006e570: 3b20 3073 2028 6763 297c 0a7c 3220 2020 ; 0s (gc)|.|2 │ │ │ │ 0006e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5b0: 2020 2020 207c 0a7c 5461 6c6c 797b 7b7b |.|Tally{{{ │ │ │ │ 0006e5c0: 322c 2032 7d2c 207b 312c 2032 7d7d 203d 2, 2}, {1, 2}} = │ │ │ │ 0006e5d0: 3e20 337d 2020 2020 2020 2020 2020 2020 > 3} │ │ │ │ 0006e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006e5f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e620: 2020 2020 2020 2020 2020 207c 0a7c 3320 |.|3 │ │ │ │ -0006e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e620: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006e630: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0006e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e660: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ -0006e670: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ -0006e680: 3d3e 2031 7d20 2020 2020 2020 2020 2020 => 1} │ │ │ │ +0006e660: 2020 2020 2020 2020 207c 0a7c 5461 6c6c |.|Tall │ │ │ │ +0006e670: 797b 7b7b 322c 2032 7d2c 207b 312c 2032 y{{{2, 2}, {1, 2 │ │ │ │ +0006e680: 7d7d 203d 3e20 317d 2020 2020 2020 2020 }} => 1} │ │ │ │ 0006e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006e6a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0006e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6d0: 2020 2020 2020 2020 2020 2020 7c0a 7c34 |.|4 │ │ │ │ -0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e6e0: 207c 0a7c 3420 2020 2020 2020 2020 2020 |.|4 │ │ │ │ 0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e710: 2020 2020 2020 207c 0a7c 5461 6c6c 797b |.|Tally{ │ │ │ │ -0006e720: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0006e710: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006e720: 5461 6c6c 797b 7d20 2020 2020 2020 2020 Tally{} │ │ │ │ 0006e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e750: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0006e750: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0006e790: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0006e7a0: 3d0a 0a20 202a 202a 6e6f 7465 2074 776f =.. * *note two │ │ │ │ -0006e7b0: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f Monomials: twoMo │ │ │ │ -0006e7c0: 6e6f 6d69 616c 732c 202d 2d20 7461 6c6c nomials, -- tall │ │ │ │ -0006e7d0: 7920 7468 6520 7365 7175 656e 6365 7320 y the sequences │ │ │ │ -0006e7e0: 6f66 2042 5261 6e6b 7320 666f 720a 2020 of BRanks for. │ │ │ │ -0006e7f0: 2020 6365 7274 6169 6e20 6578 616d 706c certain exampl │ │ │ │ -0006e800: 6573 0a0a 5761 7973 2074 6f20 7573 6520 es..Ways to use │ │ │ │ -0006e810: 7375 6d54 776f 4d6f 6e6f 6d69 616c 733a sumTwoMonomials: │ │ │ │ -0006e820: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006e830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0006e840: 202a 2022 7375 6d54 776f 4d6f 6e6f 6d69 * "sumTwoMonomi │ │ │ │ -0006e850: 616c 7328 5a5a 2c5a 5a29 220a 0a46 6f72 als(ZZ,ZZ)"..For │ │ │ │ -0006e860: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0006e870: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006e880: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0006e890: 6e6f 7465 2073 756d 5477 6f4d 6f6e 6f6d note sumTwoMonom │ │ │ │ -0006e8a0: 6961 6c73 3a20 7375 6d54 776f 4d6f 6e6f ials: sumTwoMono │ │ │ │ -0006e8b0: 6d69 616c 732c 2069 7320 6120 2a6e 6f74 mials, is a *not │ │ │ │ -0006e8c0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -0006e8d0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ -0006e8e0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -0006e8f0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0006e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e790: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ +0006e7a0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0006e7b0: 6e6f 7465 2074 776f 4d6f 6e6f 6d69 616c note twoMonomial │ │ │ │ +0006e7c0: 733a 2074 776f 4d6f 6e6f 6d69 616c 732c s: twoMonomials, │ │ │ │ +0006e7d0: 202d 2d20 7461 6c6c 7920 7468 6520 7365 -- tally the se │ │ │ │ +0006e7e0: 7175 656e 6365 7320 6f66 2042 5261 6e6b quences of BRank │ │ │ │ +0006e7f0: 7320 666f 720a 2020 2020 6365 7274 6169 s for. certai │ │ │ │ +0006e800: 6e20 6578 616d 706c 6573 0a0a 5761 7973 n examples..Ways │ │ │ │ +0006e810: 2074 6f20 7573 6520 7375 6d54 776f 4d6f to use sumTwoMo │ │ │ │ +0006e820: 6e6f 6d69 616c 733a 0a3d 3d3d 3d3d 3d3d nomials:.======= │ │ │ │ +0006e830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006e840: 3d3d 3d3d 3d0a 0a20 202a 2022 7375 6d54 =====.. * "sumT │ │ │ │ +0006e850: 776f 4d6f 6e6f 6d69 616c 7328 5a5a 2c5a woMonomials(ZZ,Z │ │ │ │ +0006e860: 5a29 220a 0a46 6f72 2074 6865 2070 726f Z)"..For the pro │ │ │ │ +0006e870: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0006e880: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0006e890: 6f62 6a65 6374 202a 6e6f 7465 2073 756d object *note sum │ │ │ │ +0006e8a0: 5477 6f4d 6f6e 6f6d 6961 6c73 3a20 7375 TwoMonomials: su │ │ │ │ +0006e8b0: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2069 mTwoMonomials, i │ │ │ │ +0006e8c0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0006e8d0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ +0006e8e0: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0006e8f0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ 0006e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e940: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0006e950: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0006e960: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0006e970: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0006e980: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ -0006e990: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006e9a0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006e9b0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006e9c0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3435 esolutions.m2:45 │ │ │ │ -0006e9d0: 3132 3a30 2e0a 1f0a 4669 6c65 3a20 436f 12:0....File: Co │ │ │ │ -0006e9e0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006e9f0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -0006ea00: 666f 2c20 4e6f 6465 3a20 5461 7465 5265 fo, Node: TateRe │ │ │ │ -0006ea10: 736f 6c75 7469 6f6e 2c20 4e65 7874 3a20 solution, Next: │ │ │ │ -0006ea20: 7465 6e73 6f72 5769 7468 436f 6d70 6f6e tensorWithCompon │ │ │ │ -0006ea30: 656e 7473 2c20 5072 6576 3a20 7375 6d54 ents, Prev: sumT │ │ │ │ -0006ea40: 776f 4d6f 6e6f 6d69 616c 732c 2055 703a woMonomials, Up: │ │ │ │ -0006ea50: 2054 6f70 0a0a 5461 7465 5265 736f 6c75 Top..TateResolu │ │ │ │ -0006ea60: 7469 6f6e 202d 2d20 5461 7465 5265 736f tion -- TateReso │ │ │ │ -0006ea70: 6c75 7469 6f6e 206f 6620 6120 6d6f 6475 lution of a modu │ │ │ │ -0006ea80: 6c65 206f 7665 7220 616e 2065 7874 6572 le over an exter │ │ │ │ -0006ea90: 696f 7220 616c 6765 6272 610a 2a2a 2a2a ior algebra.**** │ │ │ │ -0006eaa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0006e940: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006e950: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0006e960: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0006e970: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0006e980: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0006e990: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ +0006e9a0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0006e9b0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +0006e9c0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006e9d0: 6e73 2e6d 323a 3435 3132 3a30 2e0a 1f0a ns.m2:4512:0.... │ │ │ │ +0006e9e0: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +0006e9f0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +0006ea00: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +0006ea10: 3a20 5461 7465 5265 736f 6c75 7469 6f6e : TateResolution │ │ │ │ +0006ea20: 2c20 4e65 7874 3a20 7465 6e73 6f72 5769 , Next: tensorWi │ │ │ │ +0006ea30: 7468 436f 6d70 6f6e 656e 7473 2c20 5072 thComponents, Pr │ │ │ │ +0006ea40: 6576 3a20 7375 6d54 776f 4d6f 6e6f 6d69 ev: sumTwoMonomi │ │ │ │ +0006ea50: 616c 732c 2055 703a 2054 6f70 0a0a 5461 als, Up: Top..Ta │ │ │ │ +0006ea60: 7465 5265 736f 6c75 7469 6f6e 202d 2d20 teResolution -- │ │ │ │ +0006ea70: 5461 7465 5265 736f 6c75 7469 6f6e 206f TateResolution o │ │ │ │ +0006ea80: 6620 6120 6d6f 6475 6c65 206f 7665 7220 f a module over │ │ │ │ +0006ea90: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ +0006eaa0: 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a bra.************ │ │ │ │ 0006eab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006eac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006ead0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006eae0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ -0006eaf0: 2020 2020 2020 2046 203d 2054 6174 6552 F = TateR │ │ │ │ -0006eb00: 6573 6f6c 7574 696f 6e28 4d2c 6c6f 7765 esolution(M,lowe │ │ │ │ -0006eb10: 722c 7570 7065 7229 0a20 202a 2049 6e70 r,upper). * Inp │ │ │ │ -0006eb20: 7574 733a 0a20 2020 2020 202a 204d 2c20 uts:. * M, │ │ │ │ -0006eb30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ -0006eb40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ -0006eb50: 6475 6c65 2c2c 200a 2020 2020 2020 2a20 dule,, . * │ │ │ │ -0006eb60: 6c6f 7765 722c 2061 6e20 2a6e 6f74 6520 lower, an *note │ │ │ │ -0006eb70: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -0006eb80: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020 ay2Doc)ZZ,, . │ │ │ │ -0006eb90: 2020 202a 2075 7070 6572 2c20 616e 202a * upper, an * │ │ │ │ -0006eba0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ -0006ebb0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ -0006ebc0: 206c 6f77 6572 2061 6e64 2075 7070 6572 lower and upper │ │ │ │ -0006ebd0: 2062 6f75 6e64 7320 666f 720a 2020 2020 bounds for. │ │ │ │ -0006ebe0: 2020 2020 7468 6520 7265 736f 6c75 7469 the resoluti │ │ │ │ -0006ebf0: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a on. * Outputs:. │ │ │ │ -0006ec00: 2020 2020 2020 2a20 462c 2061 202a 6e6f * F, a *no │ │ │ │ -0006ec10: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ -0006ec20: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ -0006ec30: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ -0006ec40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f 726d ==========..Form │ │ │ │ -0006ec50: 7320 616e 2069 6e74 6572 7661 6c2c 206c s an interval, l │ │ │ │ -0006ec60: 6f77 6572 2e2e 7570 7065 722c 206f 6620 ower..upper, of │ │ │ │ -0006ec70: 6120 646f 7562 6c79 2069 6e66 696e 6974 a doubly infinit │ │ │ │ -0006ec80: 6520 6672 6565 2072 6573 6f6c 7574 696f e free resolutio │ │ │ │ -0006ec90: 6e20 6f66 2061 2061 0a43 6f68 656e 2d4d n of a a.Cohen-M │ │ │ │ -0006eca0: 6163 6175 6c61 7920 6d6f 6475 6c65 206f acaulay module o │ │ │ │ -0006ecb0: 7665 7220 6120 476f 7265 6e73 7465 696e ver a Gorenstein │ │ │ │ -0006ecc0: 2072 696e 672c 2073 7563 6820 6173 2061 ring, such as a │ │ │ │ -0006ecd0: 6e79 206d 6f64 756c 6520 6f76 6572 2061 ny module over a │ │ │ │ -0006ece0: 6e0a 6578 7465 7269 6f72 2061 6c67 6562 n.exterior algeb │ │ │ │ -0006ecf0: 7261 2028 6163 7475 616c 6c79 2c20 616e ra (actually, an │ │ │ │ -0006ed00: 7920 6d6f 6475 6c65 206f 7665 7220 616e y module over an │ │ │ │ -0006ed10: 7920 7269 6e67 2e29 0a0a 2b2d 2d2d 2d2d y ring.)..+----- │ │ │ │ -0006ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006eae0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0006eaf0: 7361 6765 3a20 0a20 2020 2020 2020 2046 sage: . F │ │ │ │ +0006eb00: 203d 2054 6174 6552 6573 6f6c 7574 696f = TateResolutio │ │ │ │ +0006eb10: 6e28 4d2c 6c6f 7765 722c 7570 7065 7229 n(M,lower,upper) │ │ │ │ +0006eb20: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0006eb30: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +0006eb40: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0006eb50: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a y2Doc)Module,, . │ │ │ │ +0006eb60: 2020 2020 2020 2a20 6c6f 7765 722c 2061 * lower, a │ │ │ │ +0006eb70: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ +0006eb80: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ +0006eb90: 5a2c 2c20 0a20 2020 2020 202a 2075 7070 Z,, . * upp │ │ │ │ +0006eba0: 6572 2c20 616e 202a 6e6f 7465 2069 6e74 er, an *note int │ │ │ │ +0006ebb0: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ +0006ebc0: 446f 6329 5a5a 2c2c 206c 6f77 6572 2061 Doc)ZZ,, lower a │ │ │ │ +0006ebd0: 6e64 2075 7070 6572 2062 6f75 6e64 7320 nd upper bounds │ │ │ │ +0006ebe0: 666f 720a 2020 2020 2020 2020 7468 6520 for. the │ │ │ │ +0006ebf0: 7265 736f 6c75 7469 6f6e 0a20 202a 204f resolution. * O │ │ │ │ +0006ec00: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +0006ec10: 462c 2061 202a 6e6f 7465 2063 6f6d 706c F, a *note compl │ │ │ │ +0006ec20: 6578 3a20 2843 6f6d 706c 6578 6573 2943 ex: (Complexes)C │ │ │ │ +0006ec30: 6f6d 706c 6578 2c2c 200a 0a44 6573 6372 omplex,, ..Descr │ │ │ │ +0006ec40: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +0006ec50: 3d3d 0a0a 466f 726d 7320 616e 2069 6e74 ==..Forms an int │ │ │ │ +0006ec60: 6572 7661 6c2c 206c 6f77 6572 2e2e 7570 erval, lower..up │ │ │ │ +0006ec70: 7065 722c 206f 6620 6120 646f 7562 6c79 per, of a doubly │ │ │ │ +0006ec80: 2069 6e66 696e 6974 6520 6672 6565 2072 infinite free r │ │ │ │ +0006ec90: 6573 6f6c 7574 696f 6e20 6f66 2061 2061 esolution of a a │ │ │ │ +0006eca0: 0a43 6f68 656e 2d4d 6163 6175 6c61 7920 .Cohen-Macaulay │ │ │ │ +0006ecb0: 6d6f 6475 6c65 206f 7665 7220 6120 476f module over a Go │ │ │ │ +0006ecc0: 7265 6e73 7465 696e 2072 696e 672c 2073 renstein ring, s │ │ │ │ +0006ecd0: 7563 6820 6173 2061 6e79 206d 6f64 756c uch as any modul │ │ │ │ +0006ece0: 6520 6f76 6572 2061 6e0a 6578 7465 7269 e over an.exteri │ │ │ │ +0006ecf0: 6f72 2061 6c67 6562 7261 2028 6163 7475 or algebra (actu │ │ │ │ +0006ed00: 616c 6c79 2c20 616e 7920 6d6f 6475 6c65 ally, any module │ │ │ │ +0006ed10: 206f 7665 7220 616e 7920 7269 6e67 2e29 over any ring.) │ │ │ │ +0006ed20: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 0006ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0006ed70: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63 E = ZZ/101[a,b,c │ │ │ │ -0006ed80: 2c20 536b 6577 436f 6d6d 7574 6174 6976 , SkewCommutativ │ │ │ │ -0006ed90: 653d 3e74 7275 655d 2020 2020 2020 2020 e=>true] │ │ │ │ +0006ed60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006ed70: 2b0a 7c69 3120 3a20 4520 3d20 5a5a 2f31 +.|i1 : E = ZZ/1 │ │ │ │ +0006ed80: 3031 5b61 2c62 2c63 2c20 536b 6577 436f 01[a,b,c, SkewCo │ │ │ │ +0006ed90: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ 0006eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006edc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee00: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0006ee10: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +0006ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee10: 7c0a 7c6f 3120 3d20 4520 2020 2020 2020 |.|o1 = E │ │ │ │ 0006ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eea0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -0006eeb0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -0006eec0: 3320 736b 6577 2063 6f6d 6d75 7461 7469 3 skew commutati │ │ │ │ -0006eed0: 7665 2076 6172 6961 626c 6528 7329 2020 ve variable(s) │ │ │ │ -0006eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eef0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0006ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006eeb0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ +0006eec0: 616c 5269 6e67 2c20 3320 736b 6577 2063 alRing, 3 skew c │ │ │ │ +0006eed0: 6f6d 6d75 7461 7469 7665 2076 6172 6961 ommutative varia │ │ │ │ +0006eee0: 626c 6528 7329 2020 2020 2020 2020 2020 ble(s) │ │ │ │ +0006eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ef00: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0006ef50: 4d20 3d20 636f 6b65 7220 6d61 7028 455e M = coker map(E^ │ │ │ │ -0006ef60: 322c 2045 5e7b 2d31 7d2c 206d 6174 7269 2, E^{-1}, matri │ │ │ │ -0006ef70: 7822 6162 3b62 6322 2920 2020 2020 2020 x"ab;bc") │ │ │ │ -0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006ef50: 2b0a 7c69 3220 3a20 4d20 3d20 636f 6b65 +.|i2 : M = coke │ │ │ │ +0006ef60: 7220 6d61 7028 455e 322c 2045 5e7b 2d31 r map(E^2, E^{-1 │ │ │ │ +0006ef70: 7d2c 206d 6174 7269 7822 6162 3b62 6322 }, matrix"ab;bc" │ │ │ │ +0006ef80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006efa0: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efe0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -0006eff0: 636f 6b65 726e 656c 207c 2061 6220 7c20 cokernel | ab | │ │ │ │ -0006f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006eff0: 7c0a 7c6f 3220 3d20 636f 6b65 726e 656c |.|o2 = cokernel │ │ │ │ +0006f000: 207c 2061 6220 7c20 2020 2020 2020 2020 | ab | │ │ │ │ 0006f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f040: 2020 2020 2020 2020 207c 2062 6320 7c20 | bc | │ │ │ │ -0006f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f040: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006f050: 207c 2062 6320 7c20 2020 2020 2020 2020 | bc | │ │ │ │ 0006f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f090: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0f0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0006f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f0e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 0006f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f120: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -0006f130: 452d 6d6f 6475 6c65 2c20 7175 6f74 6965 E-module, quotie │ │ │ │ -0006f140: 6e74 206f 6620 4520 2020 2020 2020 2020 nt of E │ │ │ │ +0006f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f130: 7c0a 7c6f 3220 3a20 452d 6d6f 6475 6c65 |.|o2 : E-module │ │ │ │ +0006f140: 2c20 7175 6f74 6965 6e74 206f 6620 4520 , quotient of E │ │ │ │ 0006f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0006f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f180: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -0006f1d0: 7072 6573 656e 7461 7469 6f6e 204d 2020 presentation M │ │ │ │ -0006f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006f1d0: 2b0a 7c69 3320 3a20 7072 6573 656e 7461 +.|i3 : presenta │ │ │ │ +0006f1e0: 7469 6f6e 204d 2020 2020 2020 2020 2020 tion M │ │ │ │ 0006f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f210: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f220: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f260: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -0006f270: 7c20 6162 207c 2020 2020 2020 2020 2020 | ab | │ │ │ │ +0006f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f270: 7c0a 7c6f 3320 3d20 7c20 6162 207c 2020 |.|o3 = | ab | │ │ │ │ 0006f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f2c0: 7c20 6263 207c 2020 2020 2020 2020 2020 | bc | │ │ │ │ +0006f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f2c0: 7c0a 7c20 2020 2020 7c20 6263 207c 2020 |.| | bc | │ │ │ │ 0006f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f300: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f310: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f350: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f360: 2020 2020 2020 2020 3220 2020 2020 2031 2 1 │ │ │ │ -0006f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f360: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006f370: 3220 2020 2020 2031 2020 2020 2020 2020 2 1 │ │ │ │ 0006f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f3a0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -0006f3b0: 4d61 7472 6978 2045 2020 3c2d 2d20 4520 Matrix E <-- E │ │ │ │ -0006f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f3b0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2045 |.|o3 : Matrix E │ │ │ │ +0006f3c0: 2020 3c2d 2d20 4520 2020 2020 2020 2020 <-- E │ │ │ │ 0006f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f3f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0006f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f400: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f440: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -0006f450: 5461 7465 5265 736f 6c75 7469 6f6e 284d TateResolution(M │ │ │ │ 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source │ │ │ │ +0006fa70: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0006fa80: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0006fa90: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0006faa0: 6d61 6361 756c 6179 322d 312e 3235 2e30 macaulay2-1.25.0 │ │ │ │ +0006fab0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +0006fac0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +0006fad0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0006fae0: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +0006faf0: 3439 393a 302e 0a1f 0a46 696c 653a 2043 499:0....File: C │ │ │ │ +0006fb00: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0006fb10: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +0006fb20: 6e66 6f2c 204e 6f64 653a 2074 656e 736f nfo, Node: tenso │ │ │ │ +0006fb30: 7257 6974 6843 6f6d 706f 6e65 6e74 732c rWithComponents, │ │ │ 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2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0006fc20: 2020 2020 5420 3d20 7465 6e73 6f72 284d T = tensor(M │ │ │ │ -0006fc30: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a ,N). * Inputs:. │ │ │ │ -0006fc40: 2020 2020 2020 2a20 4d2c 2061 202a 6e6f * M, a *no │ │ │ │ -0006fc50: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -0006fc60: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -0006fc70: 2c20 0a20 2020 2020 202a 204e 2c20 6120 , . * N, a │ │ │ │ -0006fc80: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0006fc90: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0006fca0: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574 le,, . * Output │ │ │ │ -0006fcb0: 733a 0a20 2020 2020 202a 2054 2c20 6120 s:. * T, a │ │ │ │ -0006fcc0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0006fcd0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0006fce0: 6c65 2c2c 200a 0a44 6573 6372 6970 7469 le,, ..Descripti │ │ │ │ -0006fcf0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0006fd00: 4966 204d 2061 6e64 2f6f 7220 4e20 6172 If M and/or N ar │ │ │ │ -0006fd10: 6520 6469 7265 6374 2073 756d 206d 6f64 e direct sum mod │ │ │ │ -0006fd20: 756c 6573 2028 6973 4469 7265 6374 5375 ules (isDirectSu │ │ │ │ -0006fd30: 6d20 4d20 3d3d 2074 7275 6529 2074 6865 m M == true) the │ │ │ │ -0006fd40: 6e20 5420 6973 2074 6865 0a64 6972 6563 n T is the.direc │ │ │ │ -0006fd50: 7420 7375 6d20 6f66 2074 6865 2074 656e t sum of the ten │ │ │ │ -0006fd60: 736f 7220 7072 6f64 7563 7473 2062 6574 sor products bet │ │ │ │ -0006fd70: 7765 656e 2074 6865 2063 6f6d 706f 6e65 ween the compone │ │ │ │ -0006fd80: 6e74 732e 2054 6869 7320 5348 4f55 4c44 nts. This SHOULD │ │ │ │ -0006fd90: 2062 6520 6275 696c 740a 696e 746f 204d be built.into M │ │ │ │ -0006fda0: 2a2a 4e2c 2062 7574 2069 736e 2774 2061 **N, but isn't a │ │ │ │ -0006fdb0: 7320 6f66 204d 322c 2076 2e20 312e 370a s of M2, v. 1.7. │ │ │ │ -0006fdc0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0006fdd0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 486f ==.. * *note Ho │ │ │ │ -0006fde0: 6d57 6974 6843 6f6d 706f 6e65 6e74 733a mWithComponents: │ │ │ │ -0006fdf0: 2048 6f6d 5769 7468 436f 6d70 6f6e 656e HomWithComponen │ │ │ │ -0006fe00: 7473 2c20 2d2d 2063 6f6d 7075 7465 7320 ts, -- computes │ │ │ │ -0006fe10: 486f 6d2c 2070 7265 7365 7276 696e 670a Hom, preserving. │ │ │ │ -0006fe20: 2020 2020 6469 7265 6374 2073 756d 2069 direct sum i │ │ │ │ -0006fe30: 6e66 6f72 6d61 7469 6f6e 0a20 202a 202a nformation. * * │ │ │ │ -0006fe40: 6e6f 7465 2064 7561 6c57 6974 6843 6f6d note dualWithCom │ │ │ │ -0006fe50: 706f 6e65 6e74 733a 2064 7561 6c57 6974 ponents: dualWit │ │ │ │ -0006fe60: 6843 6f6d 706f 6e65 6e74 732c 202d 2d20 hComponents, -- │ │ │ │ -0006fe70: 6475 616c 206d 6f64 756c 6520 7072 6573 dual module pres │ │ │ │ -0006fe80: 6572 7669 6e67 0a20 2020 2064 6972 6563 erving. direc │ │ │ │ -0006fe90: 7420 7375 6d20 696e 666f 726d 6174 696f t sum informatio │ │ │ │ -0006fea0: 6e0a 0a57 6179 7320 746f 2075 7365 2074 n..Ways to use t │ │ │ │ -0006feb0: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ -0006fec0: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d nts:.=========== │ │ │ │ +0006fc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0006fc10: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ +0006fc20: 653a 200a 2020 2020 2020 2020 5420 3d20 e: . T = │ │ │ │ +0006fc30: 7465 6e73 6f72 284d 2c4e 290a 2020 2a20 tensor(M,N). * │ │ │ │ +0006fc40: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +0006fc50: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +0006fc60: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0006fc70: 294d 6f64 756c 652c 2c20 0a20 2020 2020 )Module,, . │ │ │ │ +0006fc80: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f * N, a *note mo │ │ │ │ +0006fc90: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +0006fca0: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020 Doc)Module,, . │ │ │ │ +0006fcb0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0006fcc0: 202a 2054 2c20 6120 2a6e 6f74 6520 6d6f * T, a *note mo │ │ │ │ +0006fcd0: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +0006fce0: 446f 6329 4d6f 6475 6c65 2c2c 200a 0a44 Doc)Module,, ..D │ │ │ │ +0006fcf0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0006fd00: 3d3d 3d3d 3d3d 0a0a 4966 204d 2061 6e64 ======..If M and │ │ │ │ +0006fd10: 2f6f 7220 4e20 6172 6520 6469 7265 6374 /or N are direct │ │ │ │ +0006fd20: 2073 756d 206d 6f64 756c 6573 2028 6973 sum modules (is │ │ │ │ +0006fd30: 4469 7265 6374 5375 6d20 4d20 3d3d 2074 DirectSum M == t │ │ │ │ +0006fd40: 7275 6529 2074 6865 6e20 5420 6973 2074 rue) then T is t │ │ │ │ +0006fd50: 6865 0a64 6972 6563 7420 7375 6d20 6f66 he.direct sum of │ │ │ │ +0006fd60: 2074 6865 2074 656e 736f 7220 7072 6f64 the tensor prod │ │ │ │ +0006fd70: 7563 7473 2062 6574 7765 656e 2074 6865 ucts between the │ │ │ │ +0006fd80: 2063 6f6d 706f 6e65 6e74 732e 2054 6869 components. Thi │ │ │ │ +0006fd90: 7320 5348 4f55 4c44 2062 6520 6275 696c s SHOULD be buil │ │ │ │ +0006fda0: 740a 696e 746f 204d 2a2a 4e2c 2062 7574 t.into M**N, but │ │ │ │ +0006fdb0: 2069 736e 2774 2061 7320 6f66 204d 322c isn't as of M2, │ │ │ │ +0006fdc0: 2076 2e20 312e 370a 0a53 6565 2061 6c73 v. 1.7..See als │ │ │ │ +0006fdd0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0006fde0: 2a6e 6f74 6520 486f 6d57 6974 6843 6f6d *note HomWithCom │ │ │ │ +0006fdf0: 706f 6e65 6e74 733a 2048 6f6d 5769 7468 ponents: HomWith │ │ │ │ +0006fe00: 436f 6d70 6f6e 656e 7473 2c20 2d2d 2063 Components, -- c │ │ │ │ +0006fe10: 6f6d 7075 7465 7320 486f 6d2c 2070 7265 omputes Hom, pre │ │ │ │ +0006fe20: 7365 7276 696e 670a 2020 2020 6469 7265 serving. dire │ │ │ │ +0006fe30: 6374 2073 756d 2069 6e66 6f72 6d61 7469 ct sum informati │ │ │ │ +0006fe40: 6f6e 0a20 202a 202a 6e6f 7465 2064 7561 on. * *note dua │ │ │ │ +0006fe50: 6c57 6974 6843 6f6d 706f 6e65 6e74 733a lWithComponents: │ │ │ │ +0006fe60: 2064 7561 6c57 6974 6843 6f6d 706f 6e65 dualWithCompone │ │ │ │ +0006fe70: 6e74 732c 202d 2d20 6475 616c 206d 6f64 nts, -- dual mod │ │ │ │ +0006fe80: 756c 6520 7072 6573 6572 7669 6e67 0a20 ule preserving. │ │ │ │ +0006fe90: 2020 2064 6972 6563 7420 7375 6d20 696e direct sum in │ │ │ │ +0006fea0: 666f 726d 6174 696f 6e0a 0a57 6179 7320 formation..Ways │ │ │ │ +0006feb0: 746f 2075 7365 2074 656e 736f 7257 6974 to use tensorWit │ │ │ │ +0006fec0: 6843 6f6d 706f 6e65 6e74 733a 0a3d 3d3d hComponents:.=== │ │ │ │ 0006fed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006fee0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274 656e ======.. * "ten │ │ │ │ -0006fef0: 736f 7257 6974 6843 6f6d 706f 6e65 6e74 sorWithComponent │ │ │ │ -0006ff00: 7328 4d6f 6475 6c65 2c4d 6f64 756c 6529 s(Module,Module) │ │ │ │ -0006ff10: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -0006ff20: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -0006ff30: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -0006ff40: 6a65 6374 202a 6e6f 7465 2074 656e 736f ject *note tenso │ │ │ │ -0006ff50: 7257 6974 6843 6f6d 706f 6e65 6e74 733a rWithComponents: │ │ │ │ -0006ff60: 2074 656e 736f 7257 6974 6843 6f6d 706f tensorWithCompo │ │ │ │ -0006ff70: 6e65 6e74 732c 2069 7320 6120 2a6e 6f74 nents, is a *not │ │ │ │ -0006ff80: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -0006ff90: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ -0006ffa0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -0006ffb0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0006fee0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0006fef0: 2020 2a20 2274 656e 736f 7257 6974 6843 * "tensorWithC │ │ │ │ +0006ff00: 6f6d 706f 6e65 6e74 7328 4d6f 6475 6c65 omponents(Module │ │ │ │ +0006ff10: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074 ,Module)"..For t │ │ │ │ +0006ff20: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +0006ff30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006ff40: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +0006ff50: 7465 2074 656e 736f 7257 6974 6843 6f6d te tensorWithCom │ │ │ │ +0006ff60: 706f 6e65 6e74 733a 2074 656e 736f 7257 ponents: tensorW │ │ │ │ +0006ff70: 6974 6843 6f6d 706f 6e65 6e74 732c 2069 ithComponents, i │ │ │ │ +0006ff80: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0006ff90: 0a66 756e 6374 696f 6e3a 2028 4d61 6361 .function: (Maca │ │ │ │ +0006ffa0: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0006ffb0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ 0006ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ffe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006fff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070000: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00070010: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00070020: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00070030: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00070040: 6163 6175 6c61 7932 2d31 2e32 352e 3036 acaulay2-1.25.06 │ │ │ │ -00070050: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00070060: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -00070070: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -00070080: 6573 6f6c 7574 696f 6e73 2e6d 323a 3236 esolutions.m2:26 │ │ │ │ -00070090: 3637 3a30 2e0a 1f0a 4669 6c65 3a20 436f 67:0....File: Co │ │ │ │ -000700a0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -000700b0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -000700c0: 666f 2c20 4e6f 6465 3a20 746f 4172 7261 fo, Node: toArra │ │ │ │ -000700d0: 792c 204e 6578 743a 2074 776f 4d6f 6e6f y, Next: twoMono │ │ │ │ -000700e0: 6d69 616c 732c 2050 7265 763a 2074 656e mials, Prev: ten │ │ │ │ -000700f0: 736f 7257 6974 6843 6f6d 706f 6e65 6e74 sorWithComponent │ │ │ │ -00070100: 732c 2055 703a 2054 6f70 0a0a 746f 4172 s, Up: Top..toAr │ │ │ │ -00070110: 7261 7920 2d2d 206d 616b 6573 2061 6e20 ray -- makes an │ │ │ │ -00070120: 6172 7261 7920 6672 6f6d 2061 204c 6973 array from a Lis │ │ │ │ -00070130: 7420 6f72 2066 726f 6d20 6120 7369 6e67 t or from a sing │ │ │ │ -00070140: 6c65 2069 6e74 6567 6572 0a2a 2a2a 2a2a le integer.***** │ │ │ │ -00070150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070000: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00070010: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00070020: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00070030: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00070040: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00070050: 2d31 2e32 352e 3036 2b64 732f 4d32 2f4d -1.25.06+ds/M2/M │ │ │ │ +00070060: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00070070: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +00070080: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00070090: 6e73 2e6d 323a 3236 3637 3a30 2e0a 1f0a ns.m2:2667:0.... │ │ │ │ +000700a0: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +000700b0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +000700c0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +000700d0: 3a20 746f 4172 7261 792c 204e 6578 743a : toArray, Next: │ │ │ │ +000700e0: 2074 776f 4d6f 6e6f 6d69 616c 732c 2050 twoMonomials, P │ │ │ │ +000700f0: 7265 763a 2074 656e 736f 7257 6974 6843 rev: tensorWithC │ │ │ │ +00070100: 6f6d 706f 6e65 6e74 732c 2055 703a 2054 omponents, Up: T │ │ │ │ +00070110: 6f70 0a0a 746f 4172 7261 7920 2d2d 206d op..toArray -- m │ │ │ │ +00070120: 616b 6573 2061 6e20 6172 7261 7920 6672 akes an array fr │ │ │ │ +00070130: 6f6d 2061 204c 6973 7420 6f72 2066 726f om a List or fro │ │ │ │ +00070140: 6d20 6120 7369 6e67 6c65 2069 6e74 6567 m a single integ │ │ │ │ +00070150: 6572 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a er.************* │ │ │ │ 00070160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070180: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00070190: 7361 6765 3a20 0a20 2020 2020 2020 2061 sage: . a │ │ │ │ -000701a0: 7272 203d 2074 6f41 7272 6179 204c 0a20 rr = toArray L. │ │ │ │ -000701b0: 2020 2020 2020 2061 7272 203d 2074 6f41 arr = toA │ │ │ │ -000701c0: 7272 6179 206e 0a20 202a 2049 6e70 7574 rray n. * Input │ │ │ │ -000701d0: 733a 0a20 2020 2020 202a 204c 2c20 6120 s:. * L, a │ │ │ │ -000701e0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -000701f0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -00070200: 200a 2020 2020 2020 2a20 6e2c 2061 6e20 . * n, an │ │ │ │ -00070210: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00070220: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00070230: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ -00070240: 2020 2020 2020 2a20 6172 722c 2061 6e20 * arr, an │ │ │ │ -00070250: 2a6e 6f74 6520 6172 7261 793a 2028 4d61 *note array: (Ma │ │ │ │ -00070260: 6361 756c 6179 3244 6f63 2941 7272 6179 caulay2Doc)Array │ │ │ │ -00070270: 2c2c 200a 0a57 6179 7320 746f 2075 7365 ,, ..Ways to use │ │ │ │ -00070280: 2074 6f41 7272 6179 3a0a 3d3d 3d3d 3d3d toArray:.====== │ │ │ │ -00070290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -000702a0: 2020 2a20 2274 6f41 7272 6179 284c 6973 * "toArray(Lis │ │ │ │ -000702b0: 7429 220a 2020 2a20 2274 6f41 7272 6179 t)". * "toArray │ │ │ │ -000702c0: 285a 5a29 220a 0a46 6f72 2074 6865 2070 (ZZ)"..For the p │ │ │ │ -000702d0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000702e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000702f0: 6520 6f62 6a65 6374 202a 6e6f 7465 2074 e object *note t │ │ │ │ -00070300: 6f41 7272 6179 3a20 746f 4172 7261 792c oArray: toArray, │ │ │ │ -00070310: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -00070320: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61 od function:.(Ma │ │ │ │ -00070330: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00070340: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ -00070350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070180: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070190: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +000701a0: 2020 2020 2020 2061 7272 203d 2074 6f41 arr = toA │ │ │ │ +000701b0: 7272 6179 204c 0a20 2020 2020 2020 2061 rray L. a │ │ │ │ +000701c0: 7272 203d 2074 6f41 7272 6179 206e 0a20 rr = toArray n. │ │ │ │ +000701d0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000701e0: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69 * L, a *note li │ │ │ │ +000701f0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +00070200: 6329 4c69 7374 2c2c 200a 2020 2020 2020 c)List,, . │ │ │ │ +00070210: 2a20 6e2c 2061 6e20 2a6e 6f74 6520 696e * n, an *note in │ │ │ │ +00070220: 7465 6765 723a 2028 4d61 6361 756c 6179 teger: (Macaulay │ │ │ │ +00070230: 3244 6f63 295a 5a2c 2c20 0a20 202a 204f 2Doc)ZZ,, . * O │ │ │ │ +00070240: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00070250: 6172 722c 2061 6e20 2a6e 6f74 6520 6172 arr, an *note ar │ │ │ │ +00070260: 7261 793a 2028 4d61 6361 756c 6179 3244 ray: (Macaulay2D │ │ │ │ +00070270: 6f63 2941 7272 6179 2c2c 200a 0a57 6179 oc)Array,, ..Way │ │ │ │ +00070280: 7320 746f 2075 7365 2074 6f41 7272 6179 s to use toArray │ │ │ │ +00070290: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +000702a0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274 6f41 ======.. * "toA │ │ │ │ +000702b0: 7272 6179 284c 6973 7429 220a 2020 2a20 rray(List)". * │ │ │ │ +000702c0: 2274 6f41 7272 6179 285a 5a29 220a 0a46 "toArray(ZZ)"..F │ │ │ │ +000702d0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +000702e0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +000702f0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00070300: 202a 6e6f 7465 2074 6f41 7272 6179 3a20 *note toArray: │ │ │ │ +00070310: 746f 4172 7261 792c 2069 7320 6120 2a6e toArray, is a *n │ │ │ │ +00070320: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +00070330: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ +00070340: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00070350: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ 00070360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -000703a0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000703b0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -000703c0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -000703d0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -000703e0: 7932 2d31 2e32 352e 3036 2b64 732f 4d32 y2-1.25.06+ds/M2 │ │ │ │ -000703f0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00070400: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ -00070410: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00070420: 696f 6e73 2e6d 323a 3339 3433 3a30 2e0a ions.m2:3943:0.. │ │ │ │ -00070430: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ -00070440: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -00070450: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ -00070460: 6465 3a20 7477 6f4d 6f6e 6f6d 6961 6c73 de: twoMonomials │ │ │ │ -00070470: 2c20 5072 6576 3a20 746f 4172 7261 792c , Prev: toArray, │ │ │ │ -00070480: 2055 703a 2054 6f70 0a0a 7477 6f4d 6f6e Up: Top..twoMon │ │ │ │ -00070490: 6f6d 6961 6c73 202d 2d20 7461 6c6c 7920 omials -- tally │ │ │ │ -000704a0: 7468 6520 7365 7175 656e 6365 7320 6f66 the sequences of │ │ │ │ -000704b0: 2042 5261 6e6b 7320 666f 7220 6365 7274 BRanks for cert │ │ │ │ -000704c0: 6169 6e20 6578 616d 706c 6573 0a2a 2a2a ain examples.*** │ │ │ │ -000704d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000703a0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +000703b0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +000703c0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +000703d0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +000703e0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ +000703f0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +00070400: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +00070410: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +00070420: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +00070430: 3339 3433 3a30 2e0a 1f0a 4669 6c65 3a20 3943:0....File: │ │ │ │ +00070440: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00070450: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +00070460: 696e 666f 2c20 4e6f 6465 3a20 7477 6f4d info, Node: twoM │ │ │ │ +00070470: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20 onomials, Prev: │ │ │ │ +00070480: 746f 4172 7261 792c 2055 703a 2054 6f70 toArray, Up: Top │ │ │ │ +00070490: 0a0a 7477 6f4d 6f6e 6f6d 6961 6c73 202d ..twoMonomials - │ │ │ │ +000704a0: 2d20 7461 6c6c 7920 7468 6520 7365 7175 - tally the sequ │ │ │ │ +000704b0: 656e 6365 7320 6f66 2042 5261 6e6b 7320 ences of BRanks │ │ │ │ +000704c0: 666f 7220 6365 7274 6169 6e20 6578 616d for certain exam │ │ │ │ +000704d0: 706c 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ples.*********** │ │ │ │ 000704e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00070510: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00070520: 2020 2020 2054 203d 2054 776f 4d6f 6e6f T = TwoMono │ │ │ │ -00070530: 6d69 616c 7328 632c 6429 0a20 202a 2049 mials(c,d). * I │ │ │ │ -00070540: 6e70 7574 733a 0a20 2020 2020 202a 2063 nputs:. * c │ │ │ │ -00070550: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ -00070560: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ -00070570: 6329 5a5a 2c2c 2063 6f64 696d 656e 7369 c)ZZ,, codimensi │ │ │ │ -00070580: 6f6e 2069 6e20 7768 6963 6820 746f 2077 on in which to w │ │ │ │ -00070590: 6f72 6b0a 2020 2020 2020 2a20 642c 2061 ork. * d, a │ │ │ │ -000705a0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -000705b0: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ -000705c0: 5a2c 2c20 6465 6772 6565 206f 6620 7468 Z,, degree of th │ │ │ │ -000705d0: 6520 6d6f 6e6f 6d69 616c 7320 746f 2074 e monomials to t │ │ │ │ -000705e0: 616b 650a 2020 2a20 2a6e 6f74 6520 4f70 ake. * *note Op │ │ │ │ -000705f0: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -00070600: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -00070610: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -00070620: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -00070630: 732c 3a0a 2020 2020 2020 2a20 4f70 7469 s,:. * Opti │ │ │ │ -00070640: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566 mism => ..., def │ │ │ │ -00070650: 6175 6c74 2076 616c 7565 2030 0a20 202a ault value 0. * │ │ │ │ -00070660: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00070670: 2a20 542c 2061 202a 6e6f 7465 2074 616c * T, a *note tal │ │ │ │ -00070680: 6c79 3a20 284d 6163 6175 6c61 7932 446f ly: (Macaulay2Do │ │ │ │ -00070690: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363 c)Tally,, ..Desc │ │ │ │ -000706a0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -000706b0: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865 ===..tallies the │ │ │ │ -000706c0: 2073 6571 7565 6e63 6573 206f 6620 422d sequences of B- │ │ │ │ -000706d0: 7261 6e6b 7320 7468 6174 206f 6363 7572 ranks that occur │ │ │ │ -000706e0: 2066 6f72 2069 6465 616c 7320 6765 6e65 for ideals gene │ │ │ │ -000706f0: 7261 7465 6420 6279 2070 6169 7273 206f rated by pairs o │ │ │ │ -00070700: 660a 6d6f 6e6f 6d69 616c 7320 696e 2052 f.monomials in R │ │ │ │ -00070710: 203d 2053 2f28 642d 7468 2070 6f77 6572 = S/(d-th power │ │ │ │ -00070720: 7320 6f66 2074 6865 2076 6172 6961 626c s of the variabl │ │ │ │ -00070730: 6573 292c 2077 6974 6820 6675 6c6c 2063 es), with full c │ │ │ │ -00070740: 6f6d 706c 6578 6974 7920 283d 6329 3b0a omplexity (=c);. │ │ │ │ -00070750: 7468 6174 2069 732c 2066 6f72 2061 6e20 that is, for an │ │ │ │ -00070760: 6170 7072 6f70 7269 6174 6520 7379 7a79 appropriate syzy │ │ │ │ -00070770: 6779 204d 206f 6620 4d30 203d 2052 2f28 gy M of M0 = R/( │ │ │ │ -00070780: 6d31 2c20 6d32 2920 7768 6572 6520 6d31 m1, m2) where m1 │ │ │ │ -00070790: 2061 6e64 206d 3220 6172 650a 6d6f 6e6f and m2 are.mono │ │ │ │ -000707a0: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ -000707b0: 6520 6465 6772 6565 2e0a 0a2b 2d2d 2d2d e degree...+---- │ │ │ │ -000707c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070510: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +00070520: 6765 3a20 0a20 2020 2020 2020 2054 203d ge: . T = │ │ │ │ +00070530: 2054 776f 4d6f 6e6f 6d69 616c 7328 632c TwoMonomials(c, │ │ │ │ +00070540: 6429 0a20 202a 2049 6e70 7574 733a 0a20 d). * Inputs:. │ │ │ │ +00070550: 2020 2020 202a 2063 2c20 616e 202a 6e6f * c, an *no │ │ │ │ +00070560: 7465 2069 6e74 6567 6572 3a20 284d 6163 te integer: (Mac │ │ │ │ +00070570: 6175 6c61 7932 446f 6329 5a5a 2c2c 2063 aulay2Doc)ZZ,, c │ │ │ │ +00070580: 6f64 696d 656e 7369 6f6e 2069 6e20 7768 odimension in wh │ │ │ │ +00070590: 6963 6820 746f 2077 6f72 6b0a 2020 2020 ich to work. │ │ │ │ +000705a0: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520 * d, an *note │ │ │ │ +000705b0: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ +000705c0: 6179 3244 6f63 295a 5a2c 2c20 6465 6772 ay2Doc)ZZ,, degr │ │ │ │ +000705d0: 6565 206f 6620 7468 6520 6d6f 6e6f 6d69 ee of the monomi │ │ │ │ +000705e0: 616c 7320 746f 2074 616b 650a 2020 2a20 als to take. * │ │ │ │ +000705f0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ +00070600: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ +00070610: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ +00070620: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +00070630: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +00070640: 2020 2a20 4f70 7469 6d69 736d 203d 3e20 * Optimism => │ │ │ │ +00070650: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +00070660: 7565 2030 0a20 202a 204f 7574 7075 7473 ue 0. * Outputs │ │ │ │ +00070670: 3a0a 2020 2020 2020 2a20 542c 2061 202a :. * T, a * │ │ │ │ +00070680: 6e6f 7465 2074 616c 6c79 3a20 284d 6163 note tally: (Mac │ │ │ │ +00070690: 6175 6c61 7932 446f 6329 5461 6c6c 792c aulay2Doc)Tally, │ │ │ │ +000706a0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +000706b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a74 616c ===========..tal │ │ │ │ +000706c0: 6c69 6573 2074 6865 2073 6571 7565 6e63 lies the sequenc │ │ │ │ +000706d0: 6573 206f 6620 422d 7261 6e6b 7320 7468 es of B-ranks th │ │ │ │ +000706e0: 6174 206f 6363 7572 2066 6f72 2069 6465 at occur for ide │ │ │ │ +000706f0: 616c 7320 6765 6e65 7261 7465 6420 6279 als generated by │ │ │ │ +00070700: 2070 6169 7273 206f 660a 6d6f 6e6f 6d69 pairs of.monomi │ │ │ │ +00070710: 616c 7320 696e 2052 203d 2053 2f28 642d als in R = S/(d- │ │ │ │ +00070720: 7468 2070 6f77 6572 7320 6f66 2074 6865 th powers of the │ │ │ │ +00070730: 2076 6172 6961 626c 6573 292c 2077 6974 variables), wit │ │ │ │ +00070740: 6820 6675 6c6c 2063 6f6d 706c 6578 6974 h full complexit │ │ │ │ +00070750: 7920 283d 6329 3b0a 7468 6174 2069 732c y (=c);.that is, │ │ │ │ +00070760: 2066 6f72 2061 6e20 6170 7072 6f70 7269 for an appropri │ │ │ │ +00070770: 6174 6520 7379 7a79 6779 204d 206f 6620 ate syzygy M of │ │ │ │ +00070780: 4d30 203d 2052 2f28 6d31 2c20 6d32 2920 M0 = R/(m1, m2) │ │ │ │ +00070790: 7768 6572 6520 6d31 2061 6e64 206d 3220 where m1 and m2 │ │ │ │ +000707a0: 6172 650a 6d6f 6e6f 6d69 616c 7320 6f66 are.monomials of │ │ │ │ +000707b0: 2074 6865 2073 616d 6520 6465 6772 6565 the same degree │ │ │ │ +000707c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 000707d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000707e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707f0: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64 -+.|i1 : setRand │ │ │ │ -00070800: 6f6d 5365 6564 2030 2020 2020 2020 2020 omSeed 0 │ │ │ │ +000707f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +00070800: 2073 6574 5261 6e64 6f6d 5365 6564 2030 setRandomSeed 0 │ │ │ │ 00070810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070820: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00070830: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ -00070840: 6565 6420 746f 2030 2020 2020 2020 2020 eed to 0 │ │ │ │ +00070820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070830: 207c 0a7c 202d 2d20 7365 7474 696e 6720 |.| -- setting │ │ │ │ +00070840: 7261 6e64 6f6d 2073 6565 6420 746f 2030 random seed to 0 │ │ │ │ 00070850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00070860: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00070870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070890: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -000708a0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00070890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000708a0: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ 000708b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000708c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000708d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000708d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000708f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070900: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00070910: 2074 776f 4d6f 6e6f 6d69 616c 7328 322c twoMonomials(2, │ │ │ │ -00070920: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ +00070900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070910: 2d2b 0a7c 6932 203a 2074 776f 4d6f 6e6f -+.|i2 : twoMono │ │ │ │ +00070920: 6d69 616c 7328 322c 3329 2020 2020 2020 mials(2,3) │ │ │ │ 00070930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070940: 207c 0a7c 202d 2d20 7573 6564 2030 2e39 |.| -- used 0.9 │ │ │ │ -00070950: 3830 3632 3173 2028 6370 7529 3b20 302e 80621s (cpu); 0. │ │ │ │ -00070960: 3638 3135 3173 2028 7468 7265 6164 293b 68151s (thread); │ │ │ │ -00070970: 2030 7320 2867 6329 207c 0a7c 3220 2020 0s (gc) |.|2 │ │ │ │ -00070980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070940: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00070950: 7573 6564 2031 2e33 3938 3431 7320 2863 used 1.39841s (c │ │ │ │ +00070960: 7075 293b 2030 2e38 3037 3238 3273 2028 pu); 0.807282s ( │ │ │ │ +00070970: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00070980: 207c 0a7c 3220 2020 2020 2020 2020 2020 |.|2 │ │ │ │ 00070990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000709a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709b0: 207c 0a7c 5461 6c6c 797b 7b7b 312c 2031 |.|Tally{{{1, 1 │ │ │ │ -000709c0: 7d7d 203d 3e20 3220 2020 2020 2020 207d }} => 2 } │ │ │ │ -000709d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000709f0: 2020 7b7b 322c 2032 7d2c 207b 312c 2032 {{2, 2}, {1, 2 │ │ │ │ -00070a00: 7d7d 203d 3e20 3420 2020 2020 2020 2020 }} => 4 │ │ │ │ +000709b0: 2020 2020 2020 2020 207c 0a7c 5461 6c6c |.|Tall │ │ │ │ +000709c0: 797b 7b7b 312c 2031 7d7d 203d 3e20 3220 y{{{1, 1}} => 2 │ │ │ │ +000709d0: 2020 2020 2020 207d 2020 2020 2020 2020 } │ │ │ 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070c20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00070c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c50: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -00070c60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00070c70: 2074 776f 4d6f 6e6f 6d69 616c 733a 2074 twoMonomials: t │ │ │ │ -00070c80: 776f 4d6f 6e6f 6d69 616c 732c 202d 2d20 woMonomials, -- │ │ │ │ -00070c90: 7461 6c6c 7920 7468 6520 7365 7175 656e tally the sequen │ │ │ │ -00070ca0: 6365 7320 6f66 2042 5261 6e6b 7320 666f ces of BRanks fo │ │ │ │ -00070cb0: 720a 2020 2020 6365 7274 6169 6e20 6578 r. certain ex │ │ │ │ -00070cc0: 616d 706c 6573 0a0a 5761 7973 2074 6f20 amples..Ways to │ │ │ │ -00070cd0: 7573 6520 7477 6f4d 6f6e 6f6d 6961 6c73 use twoMonomials │ │ │ │ -00070ce0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00070cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00070d00: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 285a "twoMonomials(Z │ │ │ │ -00070d10: 5a2c 5a5a 2922 0a0a 466f 7220 7468 6520 Z,ZZ)"..For the │ │ │ │ -00070d20: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00070d30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00070d40: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00070d50: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 twoMonomials: tw │ │ │ │ -00070d60: 6f4d 6f6e 6f6d 6961 6c73 2c20 6973 2061 oMonomials, is a │ │ │ │ -00070d70: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ -00070d80: 6e63 7469 6f6e 2077 6974 680a 6f70 7469 nction with.opti │ │ │ │ -00070d90: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -00070da0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00070db0: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ -00070dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070c50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ +00070c60: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +00070c70: 202a 202a 6e6f 7465 2074 776f 4d6f 6e6f * *note twoMono │ │ │ │ +00070c80: 6d69 616c 733a 2074 776f 4d6f 6e6f 6d69 mials: twoMonomi │ │ │ │ +00070c90: 616c 732c 202d 2d20 7461 6c6c 7920 7468 als, -- tally th │ │ │ │ +00070ca0: 6520 7365 7175 656e 6365 7320 6f66 2042 e sequences of B │ │ │ │ +00070cb0: 5261 6e6b 7320 666f 720a 2020 2020 6365 Ranks for. ce │ │ │ │ +00070cc0: 7274 6169 6e20 6578 616d 706c 6573 0a0a rtain examples.. │ │ │ │ +00070cd0: 5761 7973 2074 6f20 7573 6520 7477 6f4d Ways to use twoM │ │ │ │ +00070ce0: 6f6e 6f6d 6961 6c73 3a0a 3d3d 3d3d 3d3d onomials:.====== │ │ │ │ +00070cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00070d00: 3d3d 3d0a 0a20 202a 2022 7477 6f4d 6f6e ===.. * "twoMon │ │ │ │ +00070d10: 6f6d 6961 6c73 285a 5a2c 5a5a 2922 0a0a omials(ZZ,ZZ)".. │ │ │ │ +00070d20: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +00070d30: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +00070d40: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +00070d50: 7420 2a6e 6f74 6520 7477 6f4d 6f6e 6f6d t *note twoMonom │ │ │ │ +00070d60: 6961 6c73 3a20 7477 6f4d 6f6e 6f6d 6961 ials: twoMonomia │ │ │ │ +00070d70: 6c73 2c20 6973 2061 202a 6e6f 7465 206d ls, is a *note m │ │ │ │ +00070d80: 6574 686f 6420 6675 6e63 7469 6f6e 2077 ethod function w │ │ │ │ +00070d90: 6974 680a 6f70 7469 6f6e 733a 2028 4d61 ith.options: (Ma │ │ │ │ +00070da0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +00070db0: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ +00070dc0: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ 00070dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070de0: 2d2d 2d2d 2d2d 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2.89707s el │ │ │ │ +00009750: 7c20 2d2d 2032 2e34 3334 3036 7320 656c | -- 2.43406s 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 2e33 3937 3931 7320 656c | -- 5.39791s el │ │ │ │ +00009840: 7c20 2d2d 2034 2e32 3037 3234 7320 656c | -- 4.20724s 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 ---------------- │ │ │ │ @@ -4559,16 +4559,16 @@ │ │ │ │ 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 3930 3539 7320 656c |.| -- .9059s el │ │ │ │ -00011d60: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3438 3232 3731 7320 |.| -- .482271s │ │ │ │ +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 │ │ │ │ 00011dd0: 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 2e33 3939 3332 7320 |.| -- 1.39932s │ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e31 3030 3037 7320 |.| -- 1.10007s │ │ │ │ 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 3932 3137 3235 7320 |.| -- .921725s │ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3832 3339 3537 7320 |.| -- .823957s │ │ │ │ 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 3335 |.| -- .435 │ │ │ │ -00013ad0: 3773 2065 6c61 7073 6564 2020 2020 2020 7s elapsed │ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e33 3834 |.| -- .384 │ │ │ │ +00013ad0: 3337 3973 2065 6c61 7073 6564 2020 2020 379s 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 3134 |.| -- .714 │ │ │ │ -00013bc0: 3236 3673 2065 6c61 7073 6564 2020 2020 266s elapsed │ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e36 3539 |.| -- .659 │ │ │ │ +00013bc0: 3935 3673 2065 6c61 7073 6564 2020 2020 956s 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 3437 3236 3873 2065 6c61 -- .247268s ela │ │ │ │ +00015460: 202d 2d20 2e31 3939 3536 3373 2065 6c61 -- .199563s ela │ │ │ │ 00015470: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 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 2e32 3836 3239 3273 2065 6c61 -- .286292s ela │ │ │ │ +00015550: 202d 2d20 2e32 3138 3133 3773 2065 6c61 -- .218137s 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 3036 3039 3230 3773 2028 ed 0.00609207s ( │ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3430 3635 3234 cpu); 0.00406524 │ │ │ │ +00000990: 6564 2030 2e30 3034 3030 3135 3473 2028 ed 0.00400154s ( │ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3435 3037 3036 cpu); 0.00450706 │ │ │ │ 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 │ │ │ │ @@ -322,18 +322,18 @@ │ │ │ │ 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 3331 3536 3373 2028 6370 ed 0.131563s (cp │ │ │ │ -00001490: 7529 3b20 302e 3038 3330 3338 3373 2028 u); 0.0830383s ( │ │ │ │ -000014a0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000014b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00001480: 6564 2030 2e30 3433 3932 3331 7320 2863 ed 0.0439231s (c │ │ │ │ +00001490: 7075 293b 2030 2e30 3437 3031 3138 7320 pu); 0.0470118s │ │ │ │ +000014a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +000014b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000014d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ │ │ 00001510: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ 00001520: 6465 616c 2028 7820 7820 202d 2078 2078 deal (x x - x x │ │ │ │ @@ -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 3031 3937 7320 2863 ed 0.0270197s (c │ │ │ │ -000018a0: 7075 293b 2030 2e30 3237 3633 3833 7320 pu); 0.0276383s │ │ │ │ +00001890: 6564 2030 2e30 3332 3031 3533 7320 2863 ed 0.0320153s (c │ │ │ │ +000018a0: 7075 293b 2030 2e30 3331 3730 3132 7320 pu); 0.0317012s │ │ │ │ 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 │ │ │ │ @@ -412,16 +412,16 @@ │ │ │ │ 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 2e35 3338 3538 3373 2028 6370 ed 0.538583s (cp │ │ │ │ -00001a30: 7529 3b20 302e 3433 3036 3438 7320 2874 u); 0.430648s (t │ │ │ │ +00001a20: 6564 2030 2e36 3034 3534 3173 2028 6370 ed 0.604541s (cp │ │ │ │ +00001a30: 7529 3b20 302e 3437 3333 3236 7320 2874 u); 0.473326s (t │ │ │ │ 00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00001a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -447,18 +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 3534 3932 3433 7320 2863 ed 0.0549243s (c │ │ │ │ -00001c60: 7075 293b 2030 2e30 3538 3138 3639 7320 pu); 0.0581869s │ │ │ │ -00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00001c80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00001c50: 6564 2030 2e31 3537 3933 3673 2028 6370 ed 0.157936s (cp │ │ │ │ +00001c60: 7529 3b20 302e 3039 3138 3639 3573 2028 u); 0.0918695s ( │ │ │ │ +00001c70: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00001c80: 2020 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 692c 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 3030 3530 3034 3038 7320 ed 0.000500408s │ │ │ │ +00001e80: 6564 2030 2e30 3030 3135 3733 3337 7320 ed 0.000157337s │ │ │ │ 00001e90: 2863 7075 2020 2020 2020 2020 2020 2020 (cpu │ │ │ │ 00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -567,16 +567,16 @@ │ │ │ │ 00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 446f 6d69 6e61 -------|.|Domina │ │ │ │ 00002380: 6e74 3d3e 4a29 2020 2020 2020 2020 2020 nt=>J) │ │ │ │ 00002390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30 |.|); 0.0 │ │ │ │ -000023d0: 3032 3135 3732 3173 2028 7468 7265 6164 0215721s (thread │ │ │ │ -000023e0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +000023d0: 3032 3338 3432 7320 2874 6872 6561 6429 023842s (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 2e35 3733 3139 3373 2028 6370 ed 0.573193s (cp │ │ │ │ -00003470: 7529 3b20 302e 3435 3338 3232 7320 2874 u); 0.453822s (t │ │ │ │ +00003460: 6564 2030 2e34 3533 3931 3973 2028 6370 ed 0.453919s (cp │ │ │ │ +00003470: 7529 3b20 302e 3435 3332 3338 7320 2874 u); 0.453238s (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 3132 3030 3237 7320 2863 ed 0.0120027s (c │ │ │ │ -00004640: 7075 293b 2030 2e30 3039 3639 3236 3573 pu); 0.00969265s │ │ │ │ -00004650: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00004660: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +00004630: 6564 2030 2e30 3132 3034 3537 7320 2863 ed 0.0120457s (c │ │ │ │ +00004640: 7075 293b 2030 2e30 3131 3837 3239 7320 pu); 0.0118729s │ │ │ │ +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 │ │ │ │ @@ -1142,16 +1142,16 @@ │ │ │ │ 00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ 00004770: 7469 6d65 2064 6567 7265 654d 6170 2070 time degreeMap p │ │ │ │ 00004780: 7369 2020 2020 2020 2020 2020 2020 2020 si │ │ │ │ 00004790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -000047c0: 6564 2030 2e32 3235 3735 3473 2028 6370 ed 0.225754s (cp │ │ │ │ -000047d0: 7529 3b20 302e 3136 3332 3038 7320 2874 u); 0.163208s (t │ │ │ │ +000047c0: 6564 2030 2e33 3439 3337 3473 2028 6370 ed 0.349374s (cp │ │ │ │ +000047d0: 7529 3b20 302e 3232 3932 3936 7320 2874 u); 0.229296s (t │ │ │ │ 000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000047f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00004810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1167,16 +1167,16 @@ │ │ │ │ 000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ 00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544 time projectiveD │ │ │ │ 00004910: 6567 7265 6573 2070 7369 2020 2020 2020 egrees psi │ │ │ │ 00004920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004950: 6564 2035 2e32 3038 3239 7320 2863 7075 ed 5.20829s (cpu │ │ │ │ -00004960: 293b 2034 2e35 3039 3338 7320 2874 6872 ); 4.50938s (thr │ │ │ │ +00004950: 6564 2035 2e32 3037 3834 7320 2863 7075 ed 5.20784s (cpu │ │ │ │ +00004960: 293b 2034 2e38 3932 3631 7320 2874 6872 ); 4.89261s (thr │ │ │ │ 00004970: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00004980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000049a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1213,18 +1213,18 @@ │ │ │ │ 00004bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869 .|i12 : time phi │ │ │ │ 00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d = rationalMap m │ │ │ │ 00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b inors(3,matrix{{ │ │ │ │ 00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e t_0..t_4},{t_1.. │ │ │ │ 00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c t_5},{t_2..t_6 | │ │ │ │ -00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3030 .| -- used 0.000 │ │ │ │ -00004c40: 3635 3930 3636 7320 2863 7075 293b 2030 659066s (cpu); 0 │ │ │ │ -00004c50: 2e30 3032 3037 3636 3273 2028 7468 7265 .00207662s (thre │ │ │ │ -00004c60: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032 .| -- used 0.002 │ │ │ │ +00004c40: 3439 3239 3773 2028 6370 7529 3b20 302e 49297s (cpu); 0. │ │ │ │ +00004c50: 3030 3234 3630 3239 7320 2874 6872 6561 00246029s (threa │ │ │ │ +00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00004c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00004c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00004cd0: 0a7c 6f31 3220 3d20 2d2d 2072 6174 696f .|o12 = -- ratio │ │ │ │ @@ -1493,17 +1493,17 @@ │ │ │ │ 00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869 .|i13 : time phi │ │ │ │ 00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870 = rationalMap(p │ │ │ │ 00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920 hi,Dominant=>2) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3530 .| -- used 0.050 │ │ │ │ -00005dc0: 3832 3136 7320 2863 7075 293b 2030 2e30 8216s (cpu); 0.0 │ │ │ │ -00005dd0: 3530 3333 3134 7320 2874 6872 6561 6429 503314s (thread) │ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3633 .| -- used 0.063 │ │ │ │ +00005dc0: 3930 3733 7320 2863 7075 293b 2030 2e30 9073s (cpu); 0.0 │ │ │ │ +00005dd0: 3630 3837 3839 7320 2874 6872 6561 6429 608789s (thread) │ │ │ │ 00005de0: 3b20 3073 2028 6763 2920 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 2e36 3637 .| -- used 0.667 │ │ │ │ -00008700: 3934 3173 2028 6370 7529 3b20 302e 3531 941s (cpu); 0.51 │ │ │ │ -00008710: 3532 3131 7320 2874 6872 6561 6429 3b20 5211s (thread); │ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3431 .| -- used 0.541 │ │ │ │ +00008700: 3930 3373 2028 6370 7529 3b20 302e 3437 903s (cpu); 0.47 │ │ │ │ +00008710: 3237 3531 7320 2874 6872 6561 6429 3b20 2751s (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 3932 .| -- used 0.392 │ │ │ │ -0000a9b0: 3133 3973 2028 6370 7529 3b20 302e 3331 139s (cpu); 0.31 │ │ │ │ -0000a9c0: 3738 3431 7320 2874 6872 6561 6429 3b20 7841s (thread); │ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3433 .| -- used 0.443 │ │ │ │ +0000a9b0: 3038 3973 2028 6370 7529 3b20 302e 3336 089s (cpu); 0.36 │ │ │ │ +0000a9c0: 3635 3833 7320 2874 6872 6561 6429 3b20 6583s (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 3135 .| -- used 0.015 │ │ │ │ -0000abe0: 3738 3333 7320 2863 7075 293b 2030 2e30 7833s (cpu); 0.0 │ │ │ │ -0000abf0: 3137 3233 3832 7320 2874 6872 6561 6429 172382s (thread) │ │ │ │ -0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3533 .| -- used 0.053 │ │ │ │ +0000abe0: 3734 3873 2028 6370 7529 3b20 302e 3032 748s (cpu); 0.02 │ │ │ │ +0000abf0: 3036 3234 3673 2028 7468 7265 6164 293b 06246s (thread); │ │ │ │ +0000ac00: 2030 7320 2867 6329 2020 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 3031 .| -- used 0.001 │ │ │ │ -0000ae10: 3930 3036 3573 2028 6370 7529 3b20 302e 90065s (cpu); 0. │ │ │ │ -0000ae20: 3030 3239 3832 3637 7320 2874 6872 6561 00298267s (threa │ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3033 .| -- used 0.003 │ │ │ │ +0000ae10: 3231 3835 3673 2028 6370 7529 3b20 302e 21856s (cpu); 0. │ │ │ │ +0000ae20: 3030 3335 3432 3538 7320 2874 6872 6561 00354258s (threa │ │ │ │ 0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2843,18 +2843,18 @@ │ │ │ │ 0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573 .|i18 : time des │ │ │ │ 0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020 cribe phi^(-1) │ │ │ │ 0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3036 .| -- used 0.006 │ │ │ │ -0000b220: 3734 3933 3173 2028 6370 7529 3b20 302e 74931s (cpu); 0. │ │ │ │ -0000b230: 3030 3938 3135 3337 7320 2874 6872 6561 00981537s (threa │ │ │ │ -0000b240: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3130 .| -- used 0.010 │ │ │ │ +0000b220: 3733 3033 7320 2863 7075 293b 2030 2e30 7303s (cpu); 0.0 │ │ │ │ +0000b230: 3131 3333 3432 7320 2874 6872 6561 6429 113342s (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 3038 .| -- used 0.008 │ │ │ │ -0000b720: 3832 3536 3873 2028 6370 7529 3b20 302e 82568s (cpu); 0. │ │ │ │ -0000b730: 3030 3934 3538 3133 7320 2874 6872 6561 00945813s (threa │ │ │ │ -0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3037 .| -- used 0.007 │ │ │ │ +0000b720: 3331 3435 3673 2028 6370 7529 3b20 302e 31456s (cpu); 0. │ │ │ │ +0000b730: 3031 3034 3638 7320 2874 6872 6561 6429 010468s (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 │ │ │ │ @@ -2958,17 +2958,17 @@ │ │ │ │ 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 2e33 3336 .| -- used 1.336 │ │ │ │ -0000b950: 3932 7320 2863 7075 293b 2030 2e39 3430 92s (cpu); 0.940 │ │ │ │ -0000b960: 3537 3573 2028 7468 7265 6164 293b 2030 575s (thread); 0 │ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e31 3934 .| -- used 1.194 │ │ │ │ +0000b950: 3035 7320 2863 7075 293b 2030 2e39 3333 05s (cpu); 0.933 │ │ │ │ +0000b960: 3737 3573 2028 7468 7265 6164 293b 2030 775s (thread); 0 │ │ │ │ 0000b970: 7320 2867 6329 2020 2020 2020 2020 2020 s (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 | │ │ │ │ @@ -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 2030 2e30 3030 .| -- used 0.000 │ │ │ │ -0000bb80: 3138 3938 3637 7320 2863 7075 293b 2031 189867s (cpu); 1 │ │ │ │ -0000bb90: 2e34 3837 3765 2d30 3573 2028 7468 7265 .4877e-05s (thre │ │ │ │ -0000bba0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2039 2e39 3531 .| -- used 9.951 │ │ │ │ +0000bb80: 3765 2d30 3573 2028 6370 7529 3b20 312e 7e-05s (cpu); 1. │ │ │ │ +0000bb90: 3630 3731 652d 3035 7320 2874 6872 6561 6071e-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 │ │ │ │ @@ -3018,17 +3018,17 @@ │ │ │ │ 0000bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573 .|i23 : time des │ │ │ │ 0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020 cribe f │ │ │ │ 0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000bd00: 0a7c 202d 2d20 7573 6564 2039 2e32 3337 .| -- used 9.237 │ │ │ │ -0000bd10: 3465 2d30 3573 2028 6370 7529 3b20 302e 4e-05s (cpu); 0. │ │ │ │ -0000bd20: 3030 3135 3139 3837 7320 2874 6872 6561 00151987s (threa │ │ │ │ +0000bd00: 0a7c 202d 2d20 7573 6564 2039 2e31 3734 .| -- used 9.174 │ │ │ │ +0000bd10: 3265 2d30 3573 2028 6370 7529 3b20 302e 2e-05s (cpu); 0. │ │ │ │ +0000bd20: 3030 3136 3838 3133 7320 2874 6872 6561 00168813s (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: 3332 3936 3332 7320 2863 7075 293b 2030 329632s (cpu); 0 │ │ │ │ -000124b0: 2e30 3030 3339 3734 3036 7320 2874 6872 .000397406s (thr │ │ │ │ +000124a0: 3031 3532 3431 3973 2028 6370 7529 3b20 0152419s (cpu); │ │ │ │ +000124b0: 302e 3030 3034 3332 3336 7320 2874 6872 0.00043236s (thr │ │ │ │ 000124c0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 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 │ │ │ │ @@ -4746,16 +4746,16 @@ │ │ │ │ 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 2e32 3034 3431 3173 - used 0.204411s │ │ │ │ -00012910: 2028 6370 7529 3b20 302e 3133 3732 3533 (cpu); 0.137253 │ │ │ │ +00012900: 2d20 7573 6564 2030 2e32 3136 3533 3673 - used 0.216536s │ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3135 3638 3237 (cpu); 0.156827 │ │ │ │ 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 │ │ │ │ @@ -4765,17 +4765,17 @@ │ │ │ │ 000129c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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: 3033 3535 3573 2028 6370 7529 3b20 302e 03555s (cpu); 0. │ │ │ │ -00012a50: 3336 3339 3933 7320 2874 6872 6561 6429 363993s (thread) │ │ │ │ +00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e34 |.| -- used 0.4 │ │ │ │ +00012a40: 3235 3539 3373 2028 6370 7529 3b20 302e 25593s (cpu); 0. │ │ │ │ +00012a50: 3336 3137 3638 7320 2874 6872 6561 6429 361768s (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 -- │ │ │ │ @@ -5189,17 +5189,17 @@ │ │ │ │ 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 3134 3231 3639 7320 2863 7075 d 0.142169s (cpu │ │ │ │ -000144c0: 293b 2030 2e30 3631 3538 3835 7320 2874 ); 0.0615885s (t │ │ │ │ -000144d0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +000144b0: 6420 302e 3034 3830 3835 3173 2028 6370 d 0.0480851s (cp │ │ │ │ +000144c0: 7529 3b20 302e 3034 3831 3433 3873 2028 u); 0.0481438s ( │ │ │ │ +000144d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 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 │ │ │ │ 00014540: 2020 7c0a 7c6f 3134 203d 202d 2d20 7261 |.|o14 = -- ra │ │ │ │ @@ -5261,48248 +5261,48248 @@ │ │ │ │ 000148c0: 7468 6520 6162 7374 7261 6374 206d 6170 the abstract map │ │ │ │ 000148d0: 2054 2063 616e 2062 6520 6f62 7461 696e T can be obtain │ │ │ │ 000148e0: 6564 2062 7920 7468 650a 666f 6c6c 6f77 ed by the.follow │ │ │ │ 000148f0: 696e 6720 636f 6d6d 616e 643a 0a0a 2b2d ing command:..+- │ │ │ │ 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 3234 3331 3773 2028 6370 sed 3.24317s (cp │ │ │ │ -00014980: 7529 3b20 312e 3736 3131 3273 2028 7468 u); 1.76112s (th │ │ │ │ -00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ -000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00014930: 2d2b 0a7c 6931 3520 3a20 7469 6d65 2070 -+.|i15 : time p │ │ │ │ +00014940: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ +00014950: 2854 2c32 2920 2020 2020 2020 2020 2020 (T,2) │ │ │ │ +00014960: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +00014970: 6420 332e 3730 3232 7320 2863 7075 293b d 3.7022s (cpu); │ │ │ │ +00014980: 2032 2e31 3138 3336 7320 2874 6872 6561 2.11836s (threa │ │ │ │ +00014990: 6429 3b20 3073 2028 6763 297c 0a7c 2020 d); 0s (gc)|.| │ │ │ │ +000149a0: 2020 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 │ │ │ │ +000149d0: 7c0a 7c6f 3135 203d 2033 2020 2020 2020 |.|o15 = 3 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00014a00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00014a10: 2d2d 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: 2b0a 0a57 6520 7665 7269 6679 2074 6861 +..We verify tha │ │ │ │ -00014a50: 7420 7468 6520 636f 6d70 6f73 6974 696f t the compositio │ │ │ │ -00014a60: 6e20 6f66 2054 2077 6974 6820 6974 7365 n of T with itse │ │ │ │ -00014a70: 6c66 2069 7320 6465 6669 6e65 6420 6279 lf is defined by │ │ │ │ -00014a80: 206c 696e 6561 7220 666f 726d 733a 0a0a linear forms:.. │ │ │ │ -00014a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520 ----------+..We │ │ │ │ +00014a40: 7665 7269 6679 2074 6861 7420 7468 6520 verify that the │ │ │ │ +00014a50: 636f 6d70 6f73 6974 696f 6e20 6f66 2054 composition of T │ │ │ │ +00014a60: 2077 6974 6820 6974 7365 6c66 2069 7320 with itself is │ │ │ │ +00014a70: 6465 6669 6e65 6420 6279 206c 696e 6561 defined by linea │ │ │ │ +00014a80: 7220 666f 726d 733a 0a0a 2b2d 2d2d 2d2d r forms:..+----- │ │ │ │ +00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 │ │ │ │ +00014ac0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7469 -----+.|i16 : ti │ │ │ │ +00014ad0: 6d65 2054 3220 3d20 5420 2a20 5420 2020 me T2 = T * T │ │ │ │ +00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014b10: 7365 6420 302e 3030 3032 3037 3033 3973 sed 0.000207039s │ │ │ │ -00014b20: 2028 6370 7529 3b20 322e 3733 3231 652d (cpu); 2.7321e- │ │ │ │ -00014b30: 3035 7320 2874 6872 6561 6429 3b20 3073 05s (thread); 0s │ │ │ │ -00014b40: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ +00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ +00014b10: 3030 3031 3831 3439 3773 2028 6370 7529 000181497s (cpu) │ │ │ │ +00014b20: 3b20 322e 3834 3432 652d 3035 7320 2874 ; 2.8442e-05s (t │ │ │ │ +00014b30: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ +00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 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 -- │ │ │ │ +00014b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00014b80: 3136 203d 202d 2d20 7261 7469 6f6e 616c 16 = -- rational │ │ │ │ +00014b90: 206d 6170 202d 2d20 2020 2020 2020 2020 map -- │ │ │ │ 00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00014bd0: 2020 2020 2020 205a 5a20 2020 2020 2020 ZZ │ │ │ │ +00014bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014bd0: 205a 5a20 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ 00014be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014c00: 2020 2020 2073 6f75 7263 653a 2050 726f source: Pro │ │ │ │ -00014c10: 6a28 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 j(-----[x , x , │ │ │ │ -00014c20: 7820 2c20 7820 5d29 2020 2020 2020 2020 x , x ]) │ │ │ │ -00014c30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00014c40: 2020 2020 2020 2020 2020 2020 2020 2036 6 │ │ │ │ -00014c50: 3535 3231 2020 3020 2020 3120 2020 3220 5521 0 1 2 │ │ │ │ -00014c60: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00014c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00014c80: 2020 2020 2020 2020 2020 2020 2020 5a5a ZZ │ │ │ │ +00014bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2073 |.| s │ │ │ │ +00014c00: 6f75 7263 653a 2050 726f 6a28 2d2d 2d2d ource: Proj(---- │ │ │ │ +00014c10: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820 -[x , x , x , x │ │ │ │ +00014c20: 5d29 2020 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ +00014c30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00014c40: 2020 2020 2020 2020 2036 3535 3231 2020 65521 │ │ │ │ +00014c50: 3020 2020 3120 2020 3220 2020 3320 2020 0 1 2 3 │ │ │ │ +00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014c70: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00014c80: 2020 2020 2020 2020 5a5a 2020 2020 2020 ZZ │ │ │ │ 00014c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cb0: 2020 207c 0a7c 2020 2020 2020 7461 7267 |.| targ │ │ │ │ -00014cc0: 6574 3a20 5072 6f6a 282d 2d2d 2d2d 5b78 et: Proj(-----[x │ │ │ │ -00014cd0: 202c 2078 202c 2078 202c 2078 205d 2920 , x , x , x ]) │ │ │ │ -00014ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00014d00: 2020 2020 2020 3635 3532 3120 2030 2020 65521 0 │ │ │ │ -00014d10: 2031 2020 2032 2020 2033 2020 2020 2020 1 2 3 │ │ │ │ -00014d20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00014d30: 2020 2020 2020 6465 6669 6e69 6e67 2066 defining f │ │ │ │ -00014d40: 6f72 6d73 3a20 6769 7665 6e20 6279 2061 orms: given by a │ │ │ │ -00014d50: 2066 756e 6374 696f 6e20 2020 2020 2020 function │ │ │ │ -00014d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00014ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00014cb0: 2020 2020 2020 7461 7267 6574 3a20 5072 target: Pr │ │ │ │ +00014cc0: 6f6a 282d 2d2d 2d2d 5b78 202c 2078 202c oj(-----[x , x , │ │ │ │ +00014cd0: 2078 202c 2078 205d 2920 2020 2020 2020 x , x ]) │ │ │ │ +00014ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014d00: 3635 3532 3120 2030 2020 2031 2020 2032 65521 0 1 2 │ │ │ │ +00014d10: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00014d20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00014d30: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20 defining forms: │ │ │ │ +00014d40: 6769 7665 6e20 6279 2061 2066 756e 6374 given by a funct │ │ │ │ +00014d50: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ +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 207c 0a7c 6f31 3620 3a20 |.|o16 : │ │ │ │ -00014db0: 4162 7374 7261 6374 5261 7469 6f6e 616c AbstractRational │ │ │ │ -00014dc0: 4d61 7020 2872 6174 696f 6e61 6c20 6d61 Map (rational ma │ │ │ │ -00014dd0: 7020 6672 6f6d 2050 505e 3320 746f 2050 p from PP^3 to P │ │ │ │ -00014de0: 505e 3329 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d P^3)|.+--------- │ │ │ │ +00014da0: 207c 0a7c 6f31 3620 3a20 4162 7374 7261 |.|o16 : Abstra │ │ │ │ +00014db0: 6374 5261 7469 6f6e 616c 4d61 7020 2872 ctRationalMap (r │ │ │ │ +00014dc0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d ational map from │ │ │ │ +00014dd0: 2050 505e 3320 746f 2050 505e 3329 7c0a PP^3 to PP^3)|. │ │ │ │ +00014de0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e20: 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 352e 3338 3738 | -- used 5.3878 │ │ │ │ -00014e70: 3573 2028 6370 7529 3b20 322e 3933 3633 5s (cpu); 2.9363 │ │ │ │ -00014e80: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ -00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020 (gc) |.| │ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00014e20: 3720 3a20 7469 6d65 2070 726f 6a65 6374 7 : time project │ │ │ │ +00014e30: 6976 6544 6567 7265 6573 2854 322c 3229 iveDegrees(T2,2) │ │ │ │ +00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014e50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +00014e60: 7365 6420 362e 3231 3331 3173 2028 6370 sed 6.21311s (cp │ │ │ │ +00014e70: 7529 3b20 332e 3337 3936 3373 2028 7468 u); 3.37963s (th │ │ │ │ +00014e80: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +00014e90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00014ed0: 2020 7c0a 7c6f 3137 203d 2031 2020 2020 |.|o17 = 1 │ │ │ │ +00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00014f00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00014f10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014f50: 2d2d 2b0a 0a57 6520 7665 7269 6679 2074 --+..We verify t │ │ │ │ -00014f60: 6861 7420 7468 6520 636f 6d70 6f73 6974 hat the composit │ │ │ │ -00014f70: 696f 6e20 6f66 2054 2077 6974 6820 6974 ion of T with it │ │ │ │ -00014f80: 7365 6c66 206c 6561 7665 7320 6120 7261 self leaves a ra │ │ │ │ -00014f90: 6e64 6f6d 2070 6f69 6e74 2066 6978 6564 ndom point fixed │ │ │ │ -00014fa0: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d :..+------------ │ │ │ │ +00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 ------------+..W │ │ │ │ +00014f50: 6520 7665 7269 6679 2074 6861 7420 7468 e verify that th │ │ │ │ +00014f60: 6520 636f 6d70 6f73 6974 696f 6e20 6f66 e composition of │ │ │ │ +00014f70: 2054 2077 6974 6820 6974 7365 6c66 206c T with itself l │ │ │ │ +00014f80: 6561 7665 7320 6120 7261 6e64 6f6d 2070 eaves a random p │ │ │ │ +00014f90: 6f69 6e74 2066 6978 6564 3a0a 0a2b 2d2d oint fixed:..+-- │ │ │ │ +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 2b0a --------------+. │ │ │ │ -00014fd0: 7c69 3138 203a 2070 203d 2061 7070 6c79 |i18 : p = apply │ │ │ │ -00014fe0: 2833 2c69 2d3e 7261 6e64 6f6d 285a 5a2f (3,i->random(ZZ/ │ │ │ │ -00014ff0: 3635 3532 3129 297c 7b31 7d7c 0a7c 2020 65521))|{1}|.| │ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ +00014fd0: 2070 203d 2061 7070 6c79 2833 2c69 2d3e p = apply(3,i-> │ │ │ │ +00014fe0: 7261 6e64 6f6d 285a 5a2f 3635 3532 3129 random(ZZ/65521) │ │ │ │ +00014ff0: 297c 7b31 7d7c 0a7c 2020 2020 2020 2020 )|{1}|.| │ │ │ │ 00015000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015020: 2020 2020 2020 2020 7c0a 7c6f 3138 203d |.|o18 = │ │ │ │ -00015030: 207b 2d36 3634 382c 202d 3233 3339 362c {-6648, -23396, │ │ │ │ -00015040: 202d 3132 3331 312c 2031 7d20 2020 2020 -12311, 1} │ │ │ │ -00015050: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00015020: 2020 7c0a 7c6f 3138 203d 207b 2d36 3634 |.|o18 = {-664 │ │ │ │ +00015030: 382c 202d 3233 3339 362c 202d 3132 3331 8, -23396, -1231 │ │ │ │ +00015040: 312c 2031 7d20 2020 2020 2020 2020 207c 1, 1} | │ │ │ │ +00015050: 0a7c 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 7c0a 7c6f 3138 203a 204c 6973 7420 |.|o18 : List │ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00015080: 3138 203a 204c 6973 7420 2020 2020 2020 18 : List │ │ │ │ 00015090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000150a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000150b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000150a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000150b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000150c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -000150e0: 3139 203a 2071 203d 2054 2070 2020 2020 19 : q = T p │ │ │ │ +000150d0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2071 ------+.|i19 : q │ │ │ │ +000150e0: 203d 2054 2070 2020 2020 2020 2020 2020 = T p │ │ │ │ 000150f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015100: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00015100: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015130: 2020 2020 2020 7c0a 7c6f 3139 203d 207b |.|o19 = { │ │ │ │ -00015140: 2d39 3633 342c 2032 3037 3034 2c20 2d32 -9634, 20704, -2 │ │ │ │ -00015150: 3530 3134 2c20 317d 2020 2020 2020 2020 5014, 1} │ │ │ │ -00015160: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015130: 7c0a 7c6f 3139 203d 207b 2d39 3633 342c |.|o19 = {-9634, │ │ │ │ +00015140: 2032 3037 3034 2c20 2d32 3530 3134 2c20 20704, -25014, │ │ │ │ +00015150: 317d 2020 2020 2020 2020 2020 207c 0a7c 1} |.| │ │ │ │ +00015160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015190: 7c0a 7c6f 3139 203a 204c 6973 7420 2020 |.|o19 : List │ │ │ │ +00015180: 2020 2020 2020 2020 2020 7c0a 7c6f 3139 |.|o19 │ │ │ │ +00015190: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000151b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000151b0: 2020 2020 2020 207c 0a2b 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 2b0a 7c69 3230 ----------+.|i20 │ │ │ │ -000151f0: 203a 2054 2071 2020 2020 2020 2020 2020 : T q │ │ │ │ +000151e0: 2d2d 2d2d 2b0a 7c69 3230 203a 2054 2071 ----+.|i20 : T q │ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00015210: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015240: 2020 2020 7c0a 7c6f 3230 203d 207b 2d36 |.|o20 = {-6 │ │ │ │ -00015250: 3634 382c 202d 3233 3339 362c 202d 3132 648, -23396, -12 │ │ │ │ -00015260: 3331 312c 2031 7d20 2020 2020 2020 2020 311, 1} │ │ │ │ -00015270: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015240: 7c6f 3230 203d 207b 2d36 3634 382c 202d |o20 = {-6648, - │ │ │ │ +00015250: 3233 3339 362c 202d 3132 3331 312c 2031 23396, -12311, 1 │ │ │ │ +00015260: 7d20 2020 2020 2020 2020 207c 0a7c 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 7c0a |. │ │ │ │ -000152a0: 7c6f 3230 203a 204c 6973 7420 2020 2020 |o20 : List │ │ │ │ +00015290: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ +000152a0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 000152b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000152c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000152c0: 2020 2020 207c 0a2b 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 2d2d 2d2d 2b0a 0a57 6520 6e6f --------+..We no │ │ │ │ -00015300: 7720 636f 6d70 7574 6520 7468 6520 636f w compute the co │ │ │ │ -00015310: 6e63 7265 7465 2072 6174 696f 6e61 6c20 ncrete rational │ │ │ │ -00015320: 6d61 7020 636f 7272 6573 706f 6e64 696e map correspondin │ │ │ │ -00015330: 6720 746f 2054 3a0a 0a2b 2d2d 2d2d 2d2d g to T:..+------ │ │ │ │ +000152f0: 2d2d 2b0a 0a57 6520 6e6f 7720 636f 6d70 --+..We now comp │ │ │ │ +00015300: 7574 6520 7468 6520 636f 6e63 7265 7465 ute the concrete │ │ │ │ +00015310: 2072 6174 696f 6e61 6c20 6d61 7020 636f rational map co │ │ │ │ +00015320: 7272 6573 706f 6e64 696e 6720 746f 2054 rresponding to T │ │ │ │ +00015330: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d :..+------------ │ │ │ │ 00015340: 2d2d 2d2d 2d2d 2d2d 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 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 │ │ │ │ +00015370: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 7469 -----+.|i21 : ti │ │ │ │ +00015380: 6d65 2066 203d 2072 6174 696f 6e61 6c4d me f = rationalM │ │ │ │ +00015390: 6170 2054 2020 2020 2020 2020 2020 2020 ap 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 2034 2e33 3830 .| -- used 4.380 │ │ │ │ -000153d0: 3537 7320 2863 7075 293b 2032 2e34 3735 57s (cpu); 2.475 │ │ │ │ -000153e0: 3937 7320 2874 6872 6561 6429 3b20 3073 97s (thread); 0s │ │ │ │ -000153f0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ -00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000153b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +000153c0: 7573 6564 2034 2e39 3836 3739 7320 2863 used 4.98679s (c │ │ │ │ +000153d0: 7075 293b 2032 2e39 3337 3935 7320 2874 pu); 2.93795s (t │ │ │ │ +000153e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +000153f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 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 │ │ │ │ -00015460: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ +00015440: 207c 0a7c 6f32 3120 3d20 2d2d 2072 6174 |.|o21 = -- rat │ │ │ │ +00015450: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020 ional map -- │ │ │ │ +00015460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015480: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154a0: 2020 205a 5a20 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00015480: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00015490: 2020 2020 2020 2020 2020 2020 205a 5a20 ZZ │ │ │ │ +000154a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154d0: 0a7c 2020 2020 2020 736f 7572 6365 3a20 .| source: │ │ │ │ -000154e0: 5072 6f6a 282d 2d2d 2d2d 5b78 202c 2078 Proj(-----[x , x │ │ │ │ -000154f0: 202c 2078 202c 2078 205d 2920 2020 2020 , x , x ]) │ │ │ │ -00015500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015510: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015520: 2020 2020 2020 2020 2036 3535 3231 2020 65521 │ │ │ │ -00015530: 3020 2020 3120 2020 3220 2020 3320 2020 0 1 2 3 │ │ │ │ +000154c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000154d0: 2020 736f 7572 6365 3a20 5072 6f6a 282d source: Proj(- │ │ │ │ +000154e0: 2d2d 2d2d 5b78 202c 2078 202c 2078 202c ----[x , x , x , │ │ │ │ +000154f0: 2078 205d 2920 2020 2020 2020 2020 2020 x ]) │ │ │ │ +00015500: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015520: 2020 2036 3535 3231 2020 3020 2020 3120 65521 0 1 │ │ │ │ +00015530: 2020 3220 2020 3320 2020 2020 2020 2020 2 3 │ │ │ │ 00015540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015550: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 205a Z │ │ │ │ -00015570: 5a20 2020 2020 2020 2020 2020 2020 2020 Z │ │ │ │ +00015550: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015560: 2020 2020 2020 2020 205a 5a20 2020 2020 ZZ │ │ │ │ +00015570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015590: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000155a0: 2020 2020 7461 7267 6574 3a20 5072 6f6a target: Proj │ │ │ │ -000155b0: 282d 2d2d 2d2d 5b78 202c 2078 202c 2078 (-----[x , x , x │ │ │ │ -000155c0: 202c 2078 205d 2920 2020 2020 2020 2020 , x ]) │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000155f0: 2020 2020 2036 3535 3231 2020 3020 2020 65521 0 │ │ │ │ -00015600: 3120 2020 3220 2020 3320 2020 2020 2020 1 2 3 │ 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│ │ │ 00015ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ae0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00015af0: 3120 3a20 5261 7469 6f6e 616c 4d61 7020 1 : RationalMap │ │ │ │ -00015b00: 2863 7562 6963 2072 6174 696f 6e61 6c20 (cubic rational │ │ │ │ -00015b10: 6d61 7020 6672 6f6d 2050 505e 3320 746f map from PP^3 to │ │ │ │ -00015b20: 2050 505e 3329 2020 2020 2020 2020 207c PP^3) | │ │ │ │ -00015b30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015ae0: 2020 2020 207c 0a7c 6f32 3120 3a20 5261 |.|o21 : Ra │ │ │ │ +00015af0: 7469 6f6e 616c 4d61 7020 2863 7562 6963 tionalMap (cubic │ │ │ │ +00015b00: 2072 6174 696f 6e61 6c20 6d61 7020 6672 rational map fr │ │ │ │ +00015b10: 6f6d 2050 505e 3320 746f 2050 505e 3329 om PP^3 to PP^3) │ │ │ │ +00015b20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00015b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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-00015bf0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00015c00: 3220 3d20 7261 7469 6f6e 616c 206d 6170 2 = rational map │ │ │ │ -00015c10: 2064 6566 696e 6564 2062 7920 666f 726d defined by form │ │ │ │ -00015c20: 7320 6f66 2064 6567 7265 6520 3320 2020 s of degree 3 │ │ │ │ -00015c30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015c40: 0a7c 2020 2020 2020 736f 7572 6365 2076 .| source v │ │ │ │ -00015c50: 6172 6965 7479 3a20 5050 5e33 2020 2020 ariety: PP^3 │ │ │ │ +00015bf0: 2020 2020 207c 0a7c 6f32 3220 3d20 7261 |.|o22 = ra │ │ │ │ +00015c00: 7469 6f6e 616c 206d 6170 2064 6566 696e tional map defin │ │ │ │ +00015c10: 6564 2062 7920 666f 726d 7320 6f66 2064 ed by forms of d │ │ │ │ +00015c20: 6567 7265 6520 3320 2020 2020 2020 2020 egree 3 │ │ │ │ +00015c30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00015c40: 2020 736f 7572 6365 2076 6172 6965 7479 source variety │ │ │ │ +00015c50: 3a20 5050 5e33 2020 2020 2020 2020 2020 : PP^3 │ │ │ │ 00015c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c80: 2020 207c 0a7c 2020 2020 2020 7461 7267 |.| targ │ │ │ │ -00015c90: 6574 2076 6172 6965 7479 3a20 5050 5e33 et variety: PP^3 │ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00015c80: 2020 2020 2020 7461 7267 6574 2076 6172 target var │ │ │ │ +00015c90: 6965 7479 3a20 5050 5e33 2020 2020 2020 iety: PP^3 │ │ │ │ 00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00015cd0: 646f 6d69 6e61 6e63 653a 2074 7275 6520 dominance: true │ │ │ │ +00015cc0: 207c 0a7c 2020 2020 2020 646f 6d69 6e61 |.| domina │ │ │ │ +00015cd0: 6e63 653a 2074 7275 6520 2020 2020 2020 nce: true │ │ │ │ 00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d00: 2020 2020 2020 2020 2020 207c 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-00015ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00015e50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00015e60: 2020 636f 6566 6669 6369 656e 7420 7269 coefficient ri │ │ │ │ +00015e70: 6e67 3a20 5a5a 2f36 3535 3231 2020 2020 ng: ZZ/65521 │ │ │ │ +00015e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e90: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ee0: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 -------+..Caveat │ │ │ │ -00015ef0: 0a3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973 .======..This is │ │ │ │ -00015f00: 2075 6e64 6572 2064 6576 656c 6f70 6d65 under developme │ │ │ │ -00015f10: 6e74 2079 6574 2e0a 0a57 6179 7320 746f nt yet...Ways to 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-00015f90: 6e67 2c46 756e 6374 696f 6e43 6c6f 7375 ng,FunctionClosu │ │ │ │ -00015fa0: 7265 2922 0a20 202a 2022 6162 7374 7261 re)". * "abstra │ │ │ │ -00015fb0: 6374 5261 7469 6f6e 616c 4d61 7028 506f ctRationalMap(Po │ │ │ │ -00015fc0: 6c79 6e6f 6d69 616c 5269 6e67 2c50 6f6c lynomialRing,Pol │ │ │ │ -00015fd0: 796e 6f6d 6961 6c52 696e 672c 4675 6e63 ynomialRing,Func │ │ │ │ -00015fe0: 7469 6f6e 436c 6f73 7572 652c 5a5a 2922 tionClosure,ZZ)" │ │ │ │ -00015ff0: 0a20 202a 2022 6162 7374 7261 6374 5261 . * "abstractRa │ │ │ │ -00016000: 7469 6f6e 616c 4d61 7028 5261 7469 6f6e tionalMap(Ration │ │ │ │ -00016010: 616c 4d61 7029 220a 0a46 6f72 2074 6865 alMap)"..For the │ │ │ │ -00016020: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00016030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00016040: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00016050: 2061 6273 7472 6163 7452 6174 696f 6e61 abstractRationa │ │ │ │ -00016060: 6c4d 6170 3a20 6162 7374 7261 6374 5261 lMap: abstractRa │ │ │ │ -00016070: 7469 6f6e 616c 4d61 702c 2069 7320 6120 tionalMap, is a │ │ │ │ -00016080: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ -00016090: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ -000160a0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -000160b0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +00015f50: 3d3d 3d3d 0a0a 2020 2a20 2261 6273 7472 ====.. * "abstr │ │ │ │ +00015f60: 6163 7452 6174 696f 6e61 6c4d 6170 2850 actRationalMap(P │ │ │ │ +00015f70: 6f6c 796e 6f6d 6961 6c52 696e 672c 506f olynomialRing,Po │ │ │ │ +00015f80: 6c79 6e6f 6d69 616c 5269 6e67 2c46 756e lynomialRing,Fun │ │ │ │ +00015f90: 6374 696f 6e43 6c6f 7375 7265 2922 0a20 ctionClosure)". │ │ │ │ +00015fa0: 202a 2022 6162 7374 7261 6374 5261 7469 * "abstractRati │ │ │ │ +00015fb0: 6f6e 616c 4d61 7028 506f 6c79 6e6f 6d69 onalMap(Polynomi │ │ │ │ +00015fc0: 616c 5269 6e67 2c50 6f6c 796e 6f6d 6961 alRing,Polynomia │ │ │ │ +00015fd0: 6c52 696e 672c 4675 6e63 7469 6f6e 436c lRing,FunctionCl │ │ │ │ +00015fe0: 6f73 7572 652c 5a5a 2922 0a20 202a 2022 osure,ZZ)". * " │ │ │ │ +00015ff0: 6162 7374 7261 6374 5261 7469 6f6e 616c abstractRational │ │ │ │ +00016000: 4d61 7028 5261 7469 6f6e 616c 4d61 7029 Map(RationalMap) │ │ │ │ +00016010: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +00016020: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00016030: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00016040: 6a65 6374 202a 6e6f 7465 2061 6273 7472 ject *note abstr │ │ │ │ +00016050: 6163 7452 6174 696f 6e61 6c4d 6170 3a20 actRationalMap: │ │ │ │ +00016060: 6162 7374 7261 6374 5261 7469 6f6e 616c abstractRational │ │ │ │ +00016070: 4d61 702c 2069 7320 6120 2a6e 6f74 6520 Map, is a *note │ │ │ │ +00016080: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ +00016090: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000160a0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +000160b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 000160c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000160d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016100: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00016110: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00016120: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00016130: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00016140: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00016150: 352e 3036 2b64 732f 4d32 2f4d 6163 6175 5.06+ds/M2/Macau │ │ │ │ -00016160: 6c61 7932 2f70 6163 6b61 6765 732f 4372 lay2/packages/Cr │ │ │ │ -00016170: 656d 6f6e 612f 0a64 6f63 756d 656e 7461 emona/.documenta │ │ │ │ -00016180: 7469 6f6e 2e6d 323a 3131 3032 3a30 2e0a tion.m2:1102:0.. │ │ │ │ -00016190: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e ..File: Cremona. │ │ │ │ -000161a0: 696e 666f 2c20 4e6f 6465 3a20 6170 7072 info, Node: appr │ │ │ │ -000161b0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ -000161c0: 702c 204e 6578 743a 2042 6c6f 7755 7053 p, Next: BlowUpS │ │ │ │ -000161d0: 7472 6174 6567 792c 2050 7265 763a 2061 trategy, Prev: a │ │ │ │ -000161e0: 6273 7472 6163 7452 6174 696f 6e61 6c4d bstractRationalM │ │ │ │ -000161f0: 6170 2c20 5570 3a20 546f 700a 0a61 7070 ap, Up: Top..app │ │ │ │ -00016200: 726f 7869 6d61 7465 496e 7665 7273 654d roximateInverseM │ │ │ │ -00016210: 6170 202d 2d20 7261 6e64 6f6d 206d 6170 ap -- random map │ │ │ │ -00016220: 2072 656c 6174 6564 2074 6f20 7468 6520 related to the │ │ │ │ -00016230: 696e 7665 7273 6520 6f66 2061 2062 6972 inverse of a bir │ │ │ │ -00016240: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a ational map.**** │ │ │ │ +00016100: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00016110: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00016120: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00016130: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00016140: 6175 6c61 7932 2d31 2e32 352e 3036 2b64 aulay2-1.25.06+d │ │ │ │ +00016150: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00016160: 6163 6b61 6765 732f 4372 656d 6f6e 612f ackages/Cremona/ │ │ │ │ +00016170: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d .documentation.m │ │ │ │ +00016180: 323a 3131 3032 3a30 2e0a 1f0a 4669 6c65 2:1102:0....File │ │ │ │ +00016190: 3a20 4372 656d 6f6e 612e 696e 666f 2c20 : Cremona.info, │ │ │ │ +000161a0: 4e6f 6465 3a20 6170 7072 6f78 696d 6174 Node: approximat │ │ │ │ +000161b0: 6549 6e76 6572 7365 4d61 702c 204e 6578 eInverseMap, Nex │ │ │ │ +000161c0: 743a 2042 6c6f 7755 7053 7472 6174 6567 t: BlowUpStrateg │ │ │ │ +000161d0: 792c 2050 7265 763a 2061 6273 7472 6163 y, Prev: abstrac │ │ │ │ +000161e0: 7452 6174 696f 6e61 6c4d 6170 2c20 5570 tRationalMap, Up │ │ │ │ +000161f0: 3a20 546f 700a 0a61 7070 726f 7869 6d61 : Top..approxima │ │ │ │ +00016200: 7465 496e 7665 7273 654d 6170 202d 2d20 teInverseMap -- │ │ │ │ +00016210: 7261 6e64 6f6d 206d 6170 2072 656c 6174 random map relat │ │ │ │ +00016220: 6564 2074 6f20 7468 6520 696e 7665 7273 ed to the invers │ │ │ │ +00016230: 6520 6f66 2061 2062 6972 6174 696f 6e61 e of a birationa │ │ │ │ +00016240: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a l map.********** │ │ │ │ 00016250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016270: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00016280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00016290: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000162a0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000162b0: 6170 7072 6f78 696d 6174 6549 6e76 6572 approximateInver │ │ │ │ -000162c0: 7365 4d61 7020 7068 6920 0a20 2020 2020 seMap phi . │ │ │ │ -000162d0: 2020 2061 7070 726f 7869 6d61 7465 496e approximateIn │ │ │ │ -000162e0: 7665 7273 654d 6170 2870 6869 2c64 290a verseMap(phi,d). │ │ │ │ -000162f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00016300: 2020 2a20 7068 692c 2061 202a 6e6f 7465 * phi, a *note │ │ │ │ -00016310: 2072 6174 696f 6e61 6c20 6d61 703a 2052 rational map: R │ │ │ │ -00016320: 6174 696f 6e61 6c4d 6170 2c2c 2061 2062 ationalMap,, a b │ │ │ │ -00016330: 6972 6174 696f 6e61 6c20 6d61 700a 2020 irational map. │ │ │ │ -00016340: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74 * d, an *not │ │ │ │ -00016350: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ -00016360: 756c 6179 3244 6f63 295a 5a2c 2c20 6f70 ulay2Doc)ZZ,, op │ │ │ │ -00016370: 7469 6f6e 616c 2c20 6275 7420 6974 2073 tional, but it s │ │ │ │ -00016380: 686f 756c 6420 6265 2074 6865 0a20 2020 hould be the. │ │ │ │ -00016390: 2020 2020 2064 6567 7265 6520 6f66 2074 degree of t │ │ │ │ -000163a0: 6865 2066 6f72 6d73 2064 6566 696e 696e he forms definin │ │ │ │ -000163b0: 6720 7468 6520 696e 7665 7273 6520 6f66 g the inverse of │ │ │ │ -000163c0: 2070 6869 0a20 202a 202a 6e6f 7465 204f phi. * *note O │ │ │ │ -000163d0: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ -000163e0: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ -000163f0: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ -00016400: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ -00016410: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f ts,:. * *no │ │ │ │ -00016420: 7465 2043 6572 7469 6679 3a20 4365 7274 te Certify: Cert │ │ │ │ -00016430: 6966 792c 203d 3e20 2e2e 2e2c 2064 6566 ify, => ..., def │ │ │ │ -00016440: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -00016450: 2c20 7768 6574 6865 7220 746f 2065 6e73 , whether to ens │ │ │ │ -00016460: 7572 650a 2020 2020 2020 2020 636f 7272 ure. corr │ │ │ │ -00016470: 6563 746e 6573 7320 6f66 206f 7574 7075 ectness of outpu │ │ │ │ -00016480: 740a 2020 2020 2020 2a20 2a6e 6f74 6520 t. * *note │ │ │ │ -00016490: 436f 6469 6d42 7349 6e76 3a20 436f 6469 CodimBsInv: Codi │ │ │ │ -000164a0: 6d42 7349 6e76 2c20 3d3e 202e 2e2e 2c20 mBsInv, => ..., │ │ │ │ -000164b0: 6465 6661 756c 7420 7661 6c75 6520 6e75 default value nu │ │ │ │ -000164c0: 6c6c 2c20 0a20 2020 2020 202a 202a 6e6f ll, . * *no │ │ │ │ -000164d0: 7465 2056 6572 626f 7365 3a20 696e 7665 te Verbose: inve │ │ │ │ -000164e0: 7273 654d 6170 5f6c 705f 7064 5f70 645f rseMap_lp_pd_pd_ │ │ │ │ -000164f0: 7064 5f63 6d56 6572 626f 7365 3d3e 5f70 pd_cmVerbose=>_p │ │ │ │ -00016500: 645f 7064 5f70 645f 7270 2c20 3d3e 202e d_pd_pd_rp, => . │ │ │ │ -00016510: 2e2e 2c0a 2020 2020 2020 2020 6465 6661 ..,. defa │ │ │ │ -00016520: 756c 7420 7661 6c75 6520 7472 7565 2c0a ult value true,. │ │ │ │ -00016530: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00016540: 2020 202a 2061 202a 6e6f 7465 2072 6174 * a *note rat │ │ │ │ -00016550: 696f 6e61 6c20 6d61 703a 2052 6174 696f ional map: Ratio │ │ │ │ -00016560: 6e61 6c4d 6170 2c2c 2061 2072 616e 646f nalMap,, a rando │ │ │ │ -00016570: 6d20 7261 7469 6f6e 616c 206d 6170 2077 m rational map w │ │ │ │ -00016580: 6869 6368 2069 6e20 736f 6d65 0a20 2020 hich in some. │ │ │ │ -00016590: 2020 2020 2073 656e 7365 2069 7320 7265 sense is re │ │ │ │ -000165a0: 6c61 7465 6420 746f 2074 6865 2069 6e76 lated to the inv │ │ │ │ -000165b0: 6572 7365 206f 6620 7068 6920 2865 2e67 erse of phi (e.g │ │ │ │ -000165c0: 2e2c 2074 6865 7920 7368 6f75 6c64 2068 ., they should h │ │ │ │ -000165d0: 6176 6520 7468 6520 7361 6d65 0a20 2020 ave the same. │ │ │ │ -000165e0: 2020 2020 2062 6173 6520 6c6f 6375 7329 base locus) │ │ │ │ -000165f0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00016600: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2061 =========..The a │ │ │ │ -00016610: 6c67 6f72 6974 686d 2069 7320 746f 2074 lgorithm is to t │ │ │ │ -00016620: 7279 2074 6f20 636f 6e73 7472 7563 7420 ry to construct │ │ │ │ -00016630: 7468 6520 6964 6561 6c20 6f66 2074 6865 the ideal of the │ │ │ │ -00016640: 2062 6173 6520 6c6f 6375 7320 6f66 2074 base locus of t │ │ │ │ -00016650: 6865 2069 6e76 6572 7365 0a62 7920 6c6f he inverse.by lo │ │ │ │ -00016660: 6f6b 696e 6720 666f 7220 7468 6520 696d oking for the im │ │ │ │ -00016670: 6167 6573 2076 6961 2070 6869 206f 6620 ages via phi of │ │ │ │ -00016680: 7261 6e64 6f6d 206c 696e 6561 7220 7365 random linear se │ │ │ │ -00016690: 6374 696f 6e73 206f 6620 7468 6520 736f ctions of the so │ │ │ │ -000166a0: 7572 6365 0a76 6172 6965 7479 2e20 4765 urce.variety. Ge │ │ │ │ -000166b0: 6e65 7261 6c6c 792c 206f 6e65 2063 616e nerally, one can │ │ │ │ -000166c0: 2073 7065 6564 2075 7020 7468 6520 7072 speed up the pr │ │ │ │ -000166d0: 6f63 6573 7320 6279 2070 6173 7369 6e67 ocess by passing │ │ │ │ -000166e0: 2074 6872 6f75 6768 2074 6865 206f 7074 through the opt │ │ │ │ -000166f0: 696f 6e0a 2a6e 6f74 6520 436f 6469 6d42 ion.*note CodimB │ │ │ │ -00016700: 7349 6e76 3a20 436f 6469 6d42 7349 6e76 sInv: CodimBsInv │ │ │ │ -00016710: 2c20 6120 676f 6f64 206c 6f77 6572 2062 , a good lower b │ │ │ │ -00016720: 6f75 6e64 2066 6f72 2074 6865 2063 6f64 ound for the cod │ │ │ │ -00016730: 696d 656e 7369 6f6e 206f 6620 7468 6973 imension of this │ │ │ │ -00016740: 0a62 6173 6520 6c6f 6375 732e 0a0a 2b2d .base locus...+- │ │ │ │ +00016290: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ +000162a0: 200a 2020 2020 2020 2020 6170 7072 6f78 . approx │ │ │ │ +000162b0: 696d 6174 6549 6e76 6572 7365 4d61 7020 imateInverseMap │ │ │ │ +000162c0: 7068 6920 0a20 2020 2020 2020 2061 7070 phi . app │ │ │ │ +000162d0: 726f 7869 6d61 7465 496e 7665 7273 654d roximateInverseM │ │ │ │ +000162e0: 6170 2870 6869 2c64 290a 2020 2a20 496e ap(phi,d). * In │ │ │ │ +000162f0: 7075 7473 3a0a 2020 2020 2020 2a20 7068 puts:. * ph │ │ │ │ +00016300: 692c 2061 202a 6e6f 7465 2072 6174 696f i, a *note ratio │ │ │ │ +00016310: 6e61 6c20 6d61 703a 2052 6174 696f 6e61 nal map: Rationa │ │ │ │ +00016320: 6c4d 6170 2c2c 2061 2062 6972 6174 696f lMap,, a biratio │ │ │ │ +00016330: 6e61 6c20 6d61 700a 2020 2020 2020 2a20 nal map. * │ │ │ │ +00016340: 642c 2061 6e20 2a6e 6f74 6520 696e 7465 d, an *note inte │ │ │ │ +00016350: 6765 723a 2028 4d61 6361 756c 6179 3244 ger: (Macaulay2D │ │ │ │ +00016360: 6f63 295a 5a2c 2c20 6f70 7469 6f6e 616c oc)ZZ,, optional │ │ │ │ +00016370: 2c20 6275 7420 6974 2073 686f 756c 6420 , but it should │ │ │ │ +00016380: 6265 2074 6865 0a20 2020 2020 2020 2064 be the. d │ │ │ │ +00016390: 6567 7265 6520 6f66 2074 6865 2066 6f72 egree of the for │ │ │ │ +000163a0: 6d73 2064 6566 696e 696e 6720 7468 6520 ms defining the │ │ │ │ +000163b0: 696e 7665 7273 6520 6f66 2070 6869 0a20 inverse of phi. │ │ │ │ +000163c0: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ +000163d0: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ +000163e0: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ +000163f0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00016400: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ +00016410: 2020 2020 202a 202a 6e6f 7465 2043 6572 * *note Cer │ │ │ │ +00016420: 7469 6679 3a20 4365 7274 6966 792c 203d tify: Certify, = │ │ │ │ +00016430: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00016440: 616c 7565 2066 616c 7365 2c20 7768 6574 alue false, whet │ │ │ │ +00016450: 6865 7220 746f 2065 6e73 7572 650a 2020 her to ensure. │ │ │ │ +00016460: 2020 2020 2020 636f 7272 6563 746e 6573 correctnes │ │ │ │ +00016470: 7320 6f66 206f 7574 7075 740a 2020 2020 s of output. │ │ │ │ +00016480: 2020 2a20 2a6e 6f74 6520 436f 6469 6d42 * *note CodimB │ │ │ │ +00016490: 7349 6e76 3a20 436f 6469 6d42 7349 6e76 sInv: CodimBsInv │ │ │ │ +000164a0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c , => ..., defaul │ │ │ │ +000164b0: 7420 7661 6c75 6520 6e75 6c6c 2c20 0a20 t value null, . │ │ │ │ +000164c0: 2020 2020 202a 202a 6e6f 7465 2056 6572 * *note Ver │ │ │ │ +000164d0: 626f 7365 3a20 696e 7665 7273 654d 6170 bose: inverseMap │ │ │ │ +000164e0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56 _lp_pd_pd_pd_cmV │ │ │ │ +000164f0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70 erbose=>_pd_pd_p │ │ │ │ +00016500: 645f 7270 2c20 3d3e 202e 2e2e 2c0a 2020 d_rp, => ...,. │ │ │ │ +00016510: 2020 2020 2020 6465 6661 756c 7420 7661 default va │ │ │ │ +00016520: 6c75 6520 7472 7565 2c0a 2020 2a20 4f75 lue true,. * Ou │ │ │ │ +00016530: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00016540: 202a 6e6f 7465 2072 6174 696f 6e61 6c20 *note rational │ │ │ │ +00016550: 6d61 703a 2052 6174 696f 6e61 6c4d 6170 map: RationalMap │ │ │ │ +00016560: 2c2c 2061 2072 616e 646f 6d20 7261 7469 ,, a random rati │ │ │ │ +00016570: 6f6e 616c 206d 6170 2077 6869 6368 2069 onal map which i │ │ │ │ +00016580: 6e20 736f 6d65 0a20 2020 2020 2020 2073 n some. s │ │ │ │ +00016590: 656e 7365 2069 7320 7265 6c61 7465 6420 ense is related │ │ │ │ +000165a0: 746f 2074 6865 2069 6e76 6572 7365 206f to the inverse o │ │ │ │ +000165b0: 6620 7068 6920 2865 2e67 2e2c 2074 6865 f phi (e.g., the │ │ │ │ +000165c0: 7920 7368 6f75 6c64 2068 6176 6520 7468 y should have th │ │ │ │ +000165d0: 6520 7361 6d65 0a20 2020 2020 2020 2062 e same. b │ │ │ │ +000165e0: 6173 6520 6c6f 6375 7329 0a0a 4465 7363 ase locus)..Desc │ │ │ │ +000165f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00016600: 3d3d 3d0a 0a54 6865 2061 6c67 6f72 6974 ===..The algorit │ │ │ │ +00016610: 686d 2069 7320 746f 2074 7279 2074 6f20 hm is to try 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8|.|------- │ │ │ │ 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 7c0a 7c20 ------------|.| │ │ │ │ -00019c20: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ -00019c30: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00019c10: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ +00019c20: 2020 3220 2020 2020 2032 2032 2020 2020 2 2 2 │ │ │ │ +00019c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c0a 7c2d |.|- │ │ │ │ -00019c70: 2034 3778 2078 2078 2020 2d20 3438 7820 47x x x - 48x │ │ │ │ -00019c80: 7820 202b 2034 3878 2078 2078 2078 2020 x + 48x x x x │ │ │ │ -00019c90: 2d20 3438 7820 7820 7820 7820 202d 2034 - 48x x x x - 4 │ │ │ │ -00019ca0: 3878 2078 2078 2078 2020 2b20 3438 7820 8x x x x + 48x │ │ │ │ -00019cb0: 7820 7820 7820 202b 2020 2020 7c0a 7c20 x x x + |.| │ │ │ │ -00019cc0: 2020 2020 3220 3420 3820 2020 2020 2034 2 4 8 4 │ │ │ │ -00019cd0: 2038 2020 2020 2020 3320 3420 3620 3920 8 3 4 6 9 │ │ │ │ -00019ce0: 2020 2020 2032 2035 2036 2039 2020 2020 2 5 6 9 │ │ │ │ -00019cf0: 2020 3120 3320 3720 3920 2020 2020 2030 1 3 7 9 0 │ │ │ │ -00019d00: 2035 2037 2039 2020 2020 2020 7c0a 7c2d 5 7 9 |.|- │ │ │ │ +00019c60: 2020 2020 2020 7c0a 7c2d 2034 3778 2078 |.|- 47x x │ │ │ │ +00019c70: 2078 2020 2d20 3438 7820 7820 202b 2034 x - 48x x + 4 │ │ │ │ +00019c80: 3878 2078 2078 2078 2020 2d20 3438 7820 8x x x x - 48x │ │ │ │ +00019c90: 7820 7820 7820 202d 2034 3878 2078 2078 x x x - 48x x x │ │ │ │ +00019ca0: 2078 2020 2b20 3438 7820 7820 7820 7820 x + 48x x x x │ │ │ │ +00019cb0: 202b 2020 2020 7c0a 7c20 2020 2020 3220 + |.| 2 │ │ │ │ +00019cc0: 3420 3820 2020 2020 2034 2038 2020 2020 4 8 4 8 │ │ │ │ +00019cd0: 2020 3320 3420 3620 3920 2020 2020 2032 3 4 6 9 2 │ │ │ │ +00019ce0: 2035 2036 2039 2020 2020 2020 3120 3320 5 6 9 1 3 │ │ │ │ +00019cf0: 3720 3920 2020 2020 2030 2035 2037 2039 7 9 0 5 7 9 │ │ │ │ +00019d00: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ 00019d10: 2d2d 2d2d 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 ---------------- │ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 ------------|.|4 │ │ │ │ -00019d60: 3878 2078 2078 2078 2020 2d20 3438 7820 8x x x x - 48x │ │ │ │ -00019d70: 7820 7820 7820 2020 2020 2020 2020 2020 x x x │ │ │ │ +00019d50: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078 ------|.|48x x x │ │ │ │ +00019d60: 2078 2020 2d20 3438 7820 7820 7820 7820 x - 48x x x x │ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019db0: 2020 3120 3220 3820 3920 2020 2020 2030 1 2 8 9 0 │ │ │ │ -00019dc0: 2034 2038 2039 2020 2020 2020 2020 2020 4 8 9 │ │ │ │ +00019da0: 2020 2020 2020 7c0a 7c20 2020 3120 3220 |.| 1 2 │ │ │ │ +00019db0: 3820 3920 2020 2020 2030 2034 2038 2039 8 9 0 4 8 9 │ │ │ │ +00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019df0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019df0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00019e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00019e50: 3320 3a20 7469 6d65 2070 7369 203d 2061 3 : time psi = a │ │ │ │ -00019e60: 7070 726f 7869 6d61 7465 496e 7665 7273 pproximateInvers │ │ │ │ -00019e70: 654d 6170 2070 6869 2020 2020 2020 2020 eMap phi │ │ │ │ +00019e40: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469 ------+.|i3 : ti │ │ │ │ +00019e50: 6d65 2070 7369 203d 2061 7070 726f 7869 me psi = approxi │ │ │ │ +00019e60: 6d61 7465 496e 7665 7273 654d 6170 2070 mateInverseMap p │ │ │ │ +00019e70: 6869 2020 2020 2020 2020 2020 2020 2020 hi │ │ │ │ 00019e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019ea0: 2d2d 2075 7365 6420 302e 3337 3533 3337 -- used 0.375337 │ │ │ │ -00019eb0: 7320 2863 7075 293b 2030 2e32 3335 3539 s (cpu); 0.23559 │ │ │ │ -00019ec0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ -00019ed0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00019ef0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -00019f00: 6572 7365 4d61 703a 2073 7465 7020 3120 erseMap: step 1 │ │ │ │ -00019f10: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +00019e90: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +00019ea0: 6420 302e 3337 3738 3038 7320 2863 7075 d 0.377808s (cpu │ │ │ │ +00019eb0: 293b 2030 2e32 3632 3539 3573 2028 7468 ); 0.262595s (th │ │ │ │ +00019ec0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ee0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +00019ef0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +00019f00: 703a 2073 7465 7020 3120 6f66 2031 3020 p: step 1 of 10 │ │ │ │ +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 7c0a 7c2d |.|- │ │ │ │ -00019f40: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -00019f50: 6572 7365 4d61 703a 2073 7465 7020 3220 erseMap: step 2 │ │ │ │ -00019f60: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +00019f30: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +00019f40: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +00019f50: 703a 2073 7465 7020 3220 6f66 2031 3020 p: step 2 of 10 │ │ │ │ +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 7c0a 7c2d |.|- │ │ │ │ -00019f90: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -00019fa0: 6572 7365 4d61 703a 2073 7465 7020 3320 erseMap: step 3 │ │ │ │ -00019fb0: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +00019f80: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +00019f90: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +00019fa0: 703a 2073 7465 7020 3320 6f66 2031 3020 p: step 3 of 10 │ │ │ │ +00019fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00019fe0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -00019ff0: 6572 7365 4d61 703a 2073 7465 7020 3420 erseMap: step 4 │ │ │ │ -0001a000: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +00019fd0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +00019fe0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +00019ff0: 703a 2073 7465 7020 3420 6f66 2031 3020 p: step 4 of 10 │ │ │ │ +0001a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a020: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a030: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a040: 6572 7365 4d61 703a 2073 7465 7020 3520 erseMap: step 5 │ │ │ │ -0001a050: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a020: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a030: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a040: 703a 2073 7465 7020 3520 6f66 2031 3020 p: step 5 of 10 │ │ │ │ +0001a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a070: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a080: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a090: 6572 7365 4d61 703a 2073 7465 7020 3620 erseMap: step 6 │ │ │ │ -0001a0a0: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a070: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a080: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a090: 703a 2073 7465 7020 3620 6f66 2031 3020 p: step 6 of 10 │ │ │ │ +0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a0d0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a0e0: 6572 7365 4d61 703a 2073 7465 7020 3720 erseMap: step 7 │ │ │ │ -0001a0f0: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a0c0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a0d0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a0e0: 703a 2073 7465 7020 3720 6f66 2031 3020 p: step 7 of 10 │ │ │ │ +0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a110: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a120: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a130: 6572 7365 4d61 703a 2073 7465 7020 3820 erseMap: step 8 │ │ │ │ -0001a140: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a110: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a120: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a130: 703a 2073 7465 7020 3820 6f66 2031 3020 p: step 8 of 10 │ │ │ │ +0001a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a180: 6572 7365 4d61 703a 2073 7465 7020 3920 erseMap: step 9 │ │ │ │ -0001a190: 6f66 2031 3020 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a160: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a170: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a180: 703a 2073 7465 7020 3920 6f66 2031 3020 p: step 9 of 10 │ │ │ │ +0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001a1c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001a1d0: 6572 7365 4d61 703a 2073 7465 7020 3130 erseMap: step 10 │ │ │ │ -0001a1e0: 206f 6620 3130 2020 2020 2020 2020 2020 of 10 │ │ │ │ +0001a1b0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001a1c0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001a1d0: 703a 2073 7465 7020 3130 206f 6620 3130 p: step 10 of 10 │ │ │ │ +0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001a200: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 │ │ │ │ -0001a270: 6d61 7020 2d2d 2020 2020 2020 2020 2020 map -- │ │ │ │ +0001a250: 2020 2020 2020 7c0a 7c6f 3320 3d20 2d2d |.|o3 = -- │ │ │ │ +0001a260: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d rational map -- │ │ │ │ +0001a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2c0: 205a 5a20 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +0001a2a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001a2b0: 2020 2020 2020 2020 2020 205a 5a20 2020 ZZ │ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a300: 2020 2020 736f 7572 6365 3a20 5072 6f6a source: Proj │ │ │ │ -0001a310: 282d 2d5b 7420 2c20 7420 2c20 7420 2c20 (--[t , t , t , │ │ │ │ -0001a320: 7420 2c20 7420 2c20 7420 2c20 7420 2c20 t , t , t , t , │ │ │ │ -0001a330: 7420 2c20 7420 5d29 2020 2020 2020 2020 t , t ]) │ │ │ │ -0001a340: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a360: 2039 3720 2030 2020 2031 2020 2032 2020 97 0 1 2 │ │ │ │ -0001a370: 2033 2020 2034 2020 2035 2020 2036 2020 3 4 5 6 │ │ │ │ -0001a380: 2037 2020 2038 2020 2020 2020 2020 2020 7 8 │ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001a2f0: 2020 2020 2020 7c0a 7c20 2020 2020 736f |.| so │ │ │ │ +0001a300: 7572 6365 3a20 5072 6f6a 282d 2d5b 7420 urce: Proj(--[t │ │ │ │ +0001a310: 2c20 7420 2c20 7420 2c20 7420 2c20 7420 , t , t , t , t │ │ │ │ +0001a320: 2c20 7420 2c20 7420 2c20 7420 2c20 7420 , t , t , t , t │ │ │ │ +0001a330: 5d29 2020 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ +0001a340: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001a350: 2020 2020 2020 2020 2020 2039 3720 2030 97 0 │ │ │ │ +0001a360: 2020 2031 2020 2032 2020 2033 2020 2034 1 2 3 4 │ │ │ │ +0001a370: 2020 2035 2020 2036 2020 2037 2020 2038 5 6 7 8 │ │ │ │ +0001a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a390: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a3b0: 2020 2020 2020 2020 2020 2020 2020 205a Z │ │ │ │ -0001a3c0: 5a20 2020 2020 2020 2020 2020 2020 2020 Z │ │ │ │ +0001a3b0: 2020 2020 2020 2020 205a 5a20 2020 2020 ZZ │ │ │ │ +0001a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a3e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a3f0: 2020 2020 7461 7267 6574 3a20 7375 6276 target: subv │ │ │ │ -0001a400: 6172 6965 7479 206f 6620 5072 6f6a 282d ariety of Proj(- │ │ │ │ -0001a410: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820 -[x , x , x , x │ │ │ │ -0001a420: 2c20 7820 2c20 7820 2c20 7820 2c20 7820 , x , x , x , x │ │ │ │ -0001a430: 2c20 7820 2c20 7820 5d29 2020 7c0a 7c20 , x , x ]) |.| │ │ │ │ +0001a3e0: 2020 2020 2020 7c0a 7c20 2020 2020 7461 |.| ta │ │ │ │ +0001a3f0: 7267 6574 3a20 7375 6276 6172 6965 7479 rget: subvariety │ │ │ │ +0001a400: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20 of Proj(--[x , │ │ │ │ +0001a410: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ +0001a420: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ +0001a430: 7820 5d29 2020 7c0a 7c20 2020 2020 2020 x ]) |.| │ │ │ │ 0001a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a450: 2020 2020 2020 2020 2020 2020 2020 2039 9 │ │ │ │ -0001a460: 3720 2030 2020 2031 2020 2032 2020 2033 7 0 1 2 3 │ │ │ │ -0001a470: 2020 2034 2020 2035 2020 2036 2020 2037 4 5 6 7 │ │ │ │ -0001a480: 2020 2038 2020 2039 2020 2020 7c0a 7c20 8 9 |.| │ │ │ │ -0001a490: 2020 2020 2020 2020 2020 2020 7b20 2020 { │ │ │ │ +0001a450: 2020 2020 2020 2020 2039 3720 2030 2020 97 0 │ │ │ │ +0001a460: 2031 2020 2032 2020 2033 2020 2034 2020 1 2 3 4 │ │ │ │ +0001a470: 2035 2020 2036 2020 2037 2020 2038 2020 5 6 7 8 │ │ │ │ +0001a480: 2039 2020 2020 7c0a 7c20 2020 2020 2020 9 |.| │ │ │ │ +0001a490: 2020 2020 2020 7b20 2020 2020 2020 2020 { │ │ │ │ 0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a4e0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001a4f0: 3220 2020 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0001a500: 2020 3220 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0001a4d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001a4e0: 2020 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│ │ 0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +0001f070: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ 0001f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f090: 2020 2020 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 2020 2020 7c0a 7c34 |.|4 │ │ │ │ -0001f0d0: 3578 2078 2078 2078 2020 2b20 3437 7820 5x x x x + 47x │ │ │ │ -0001f0e0: 7820 7820 7820 202d 2032 7820 7820 7820 x x x - 2x x x │ │ │ │ -0001f0f0: 7820 202b 2033 3378 2078 2078 2078 2020 x + 33x x x x │ │ │ │ -0001f100: 2b20 3437 7820 7820 7820 7820 202d 2078 + 47x x x x - x │ │ │ │ -0001f110: 2078 2078 2078 2020 2b20 2020 7c0a 7c20 x x x + |.| │ │ │ │ -0001f120: 2020 3120 3420 3720 3820 2020 2020 2033 1 4 7 8 3 │ │ │ │ -0001f130: 2034 2037 2038 2020 2020 2030 2035 2037 4 7 8 0 5 7 │ │ │ │ -0001f140: 2038 2020 2020 2020 3120 3520 3720 3820 8 1 5 7 8 │ │ │ │ -0001f150: 2020 2020 2032 2035 2037 2038 2020 2020 2 5 7 8 │ │ │ │ -0001f160: 3420 3520 3720 3820 2020 2020 7c0a 7c2d 4 5 7 8 |.|- │ │ │ │ +0001f0c0: 2020 2020 2020 7c0a 7c34 3578 2078 2078 |.|45x x x │ │ │ │ +0001f0d0: 2078 2020 2b20 3437 7820 7820 7820 7820 x + 47x x x x │ │ │ │ +0001f0e0: 202d 2032 7820 7820 7820 7820 202b 2033 - 2x x x x + 3 │ │ │ │ +0001f0f0: 3378 2078 2078 2078 2020 2b20 3437 7820 3x x x x + 47x │ │ │ │ +0001f100: 7820 7820 7820 202d 2078 2078 2078 2078 x x x - x x x x │ │ │ │ +0001f110: 2020 2b20 2020 7c0a 7c20 2020 3120 3420 + |.| 1 4 │ │ │ │ +0001f120: 3720 3820 2020 2020 2033 2034 2037 2038 7 8 3 4 7 8 │ │ │ │ +0001f130: 2020 2020 2030 2035 2037 2038 2020 2020 0 5 7 8 │ │ │ │ +0001f140: 2020 3120 3520 3720 3820 2020 2020 2032 1 5 7 8 2 │ │ │ │ +0001f150: 2035 2037 2038 2020 2020 3420 3520 3720 5 7 8 4 5 7 │ │ │ │ +0001f160: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 8 |.|------- │ │ │ │ 0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0001f1c0: 2032 2032 2020 2020 2020 2020 2020 3220 2 2 2 │ │ │ │ -0001f1d0: 2020 2020 2032 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0001f1e0: 2020 3220 2020 2020 2032 2032 2020 2020 2 2 2 │ │ │ │ -0001f1f0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001f200: 3220 2020 2020 2020 2020 2032 7c0a 7c37 2 2|.|7 │ │ │ │ -0001f210: 7820 7820 202d 2032 3978 2078 2078 2020 x x - 29x x x │ │ │ │ -0001f220: 2b20 3336 7820 7820 202d 2031 3178 2078 + 36x x - 11x x │ │ │ │ -0001f230: 2078 2020 2b20 3438 7820 7820 202b 2032 x + 48x x + 2 │ │ │ │ -0001f240: 7820 7820 7820 202d 2033 3378 2078 2078 x x x - 33x x x │ │ │ │ -0001f250: 2020 2d20 3437 7820 7820 7820 7c0a 7c20 - 47x x x |.| │ │ │ │ -0001f260: 2030 2038 2020 2020 2020 3020 3120 3820 0 8 0 1 8 │ │ │ │ -0001f270: 2020 2020 2031 2038 2020 2020 2020 3020 1 8 0 │ │ │ │ -0001f280: 3220 3820 2020 2020 2032 2038 2020 2020 2 8 2 8 │ │ │ │ -0001f290: 2030 2034 2038 2020 2020 2020 3120 3420 0 4 8 1 4 │ │ │ │ -0001f2a0: 3820 2020 2020 2032 2034 2038 7c0a 7c2d 8 2 4 8|.|- │ │ │ │ +0001f1b0: 2d2d 2d2d 2d2d 7c0a 7c20 2032 2032 2020 ------|.| 2 2 │ │ │ │ +0001f1c0: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ +0001f1d0: 2032 2020 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0001f1e0: 2020 2032 2032 2020 2020 2020 2020 2032 2 2 2 │ │ │ │ +0001f1f0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0001f200: 2020 2020 2032 7c0a 7c37 7820 7820 202d 2|.|7x x - │ │ │ │ +0001f210: 2032 3978 2078 2078 2020 2b20 3336 7820 29x x x + 36x │ │ │ │ +0001f220: 7820 202d 2031 3178 2078 2078 2020 2b20 x - 11x x x + │ │ │ │ +0001f230: 3438 7820 7820 202b 2032 7820 7820 7820 48x x + 2x x x │ │ │ │ +0001f240: 202d 2033 3378 2078 2078 2020 2d20 3437 - 33x x x - 47 │ │ │ │ +0001f250: 7820 7820 7820 7c0a 7c20 2030 2038 2020 x x x |.| 0 8 │ │ │ │ +0001f260: 2020 2020 3020 3120 3820 2020 2020 2031 0 1 8 1 │ │ │ │ +0001f270: 2038 2020 2020 2020 3020 3220 3820 2020 8 0 2 8 │ │ │ │ +0001f280: 2020 2032 2038 2020 2020 2030 2034 2038 2 8 0 4 8 │ │ │ │ +0001f290: 2020 2020 2020 3120 3420 3820 2020 2020 1 4 8 │ │ │ │ +0001f2a0: 2032 2034 2038 7c0a 7c2d 2d2d 2d2d 2d2d 2 4 8|.|------- │ │ │ │ 0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0001f300: 2020 2020 3220 3220 2020 2020 2020 2020 2 2 │ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220 ------|.| 2 │ │ │ │ +0001f300: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 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: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001f350: 2034 3878 2078 2020 2b20 3438 7820 7820 48x x + 48x x │ │ │ │ -0001f360: 7820 7820 202d 2034 3878 2078 2078 2078 x x - 48x x x x │ │ │ │ -0001f370: 2020 2d20 3438 7820 7820 7820 7820 202b - 48x x x x + │ │ │ │ -0001f380: 2034 3878 2078 2078 2078 2020 2b20 3438 48x x x x + 48 │ │ │ │ -0001f390: 7820 7820 7820 7820 202d 2020 7c0a 7c20 x x x x - |.| │ │ │ │ -0001f3a0: 2020 2020 3420 3820 2020 2020 2033 2034 4 8 3 4 │ │ │ │ -0001f3b0: 2036 2039 2020 2020 2020 3220 3520 3620 6 9 2 5 6 │ │ │ │ -0001f3c0: 3920 2020 2020 2031 2033 2037 2039 2020 9 1 3 7 9 │ │ │ │ -0001f3d0: 2020 2020 3020 3520 3720 3920 2020 2020 0 5 7 9 │ │ │ │ -0001f3e0: 2031 2032 2038 2039 2020 2020 7c0a 7c2d 1 2 8 9 |.|- │ │ │ │ +0001f340: 2020 2020 2020 7c0a 7c2d 2034 3878 2078 |.|- 48x x │ │ │ │ +0001f350: 2020 2b20 3438 7820 7820 7820 7820 202d + 48x x x x - │ │ │ │ +0001f360: 2034 3878 2078 2078 2078 2020 2d20 3438 48x x x x - 48 │ │ │ │ +0001f370: 7820 7820 7820 7820 202b 2034 3878 2078 x x x x + 48x x │ │ │ │ +0001f380: 2078 2078 2020 2b20 3438 7820 7820 7820 x x + 48x x x │ │ │ │ +0001f390: 7820 202d 2020 7c0a 7c20 2020 2020 3420 x - |.| 4 │ │ │ │ +0001f3a0: 3820 2020 2020 2033 2034 2036 2039 2020 8 3 4 6 9 │ │ │ │ +0001f3b0: 2020 2020 3220 3520 3620 3920 2020 2020 2 5 6 9 │ │ │ │ +0001f3c0: 2031 2033 2037 2039 2020 2020 2020 3020 1 3 7 9 0 │ │ │ │ +0001f3d0: 3520 3720 3920 2020 2020 2031 2032 2038 5 7 9 1 2 8 │ │ │ │ +0001f3e0: 2039 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 9 |.|------- │ │ │ │ 0001f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 ------------|.|4 │ │ │ │ -0001f440: 3878 2078 2078 2078 2020 2020 2020 2020 8x x x x │ │ │ │ +0001f430: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078 ------|.|48x x x │ │ │ │ +0001f440: 2078 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 0001f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f490: 2020 3020 3420 3820 3920 2020 2020 2020 0 4 8 9 │ │ │ │ +0001f480: 2020 2020 2020 7c0a 7c20 2020 3020 3420 |.| 0 4 │ │ │ │ +0001f490: 3820 3920 2020 2020 2020 2020 2020 2020 8 9 │ │ │ │ 0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001f4d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f530: 3420 3a20 6173 7365 7274 2870 6869 202a 4 : assert(phi * │ │ │ │ -0001f540: 2070 7369 203d 3d20 3120 616e 6420 7073 psi == 1 and ps │ │ │ │ -0001f550: 6920 2a20 7068 6920 3d3d 2031 2920 2020 i * phi == 1) │ │ │ │ +0001f520: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6173 ------+.|i4 : as │ │ │ │ +0001f530: 7365 7274 2870 6869 202a 2070 7369 203d sert(phi * psi = │ │ │ │ +0001f540: 3d20 3120 616e 6420 7073 6920 2a20 7068 = 1 and psi * ph │ │ │ │ +0001f550: 6920 3d3d 2031 2920 2020 2020 2020 2020 i == 1) │ │ │ │ 0001f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f570: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001f570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f5d0: 3520 3a20 7469 6d65 2070 7369 2720 3d20 5 : time psi' = │ │ │ │ -0001f5e0: 6170 7072 6f78 696d 6174 6549 6e76 6572 approximateInver │ │ │ │ -0001f5f0: 7365 4d61 7028 7068 692c 436f 6469 6d42 seMap(phi,CodimB │ │ │ │ -0001f600: 7349 6e76 3d3e 3529 3b20 2020 2020 2020 sInv=>5); │ │ │ │ -0001f610: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f620: 2d2d 2075 7365 6420 302e 3233 3431 3036 -- used 0.234106 │ │ │ │ -0001f630: 7320 2863 7075 293b 2030 2e31 3534 3134 s (cpu); 0.15414 │ │ │ │ -0001f640: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ -0001f650: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ -0001f660: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001f670: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001f680: 6572 7365 4d61 703a 2073 7465 7020 3120 erseMap: step 1 │ │ │ │ -0001f690: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ +0001f5c0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 ------+.|i5 : ti │ │ │ │ +0001f5d0: 6d65 2070 7369 2720 3d20 6170 7072 6f78 me psi' = approx │ │ │ │ +0001f5e0: 696d 6174 6549 6e76 6572 7365 4d61 7028 imateInverseMap( │ │ │ │ +0001f5f0: 7068 692c 436f 6469 6d42 7349 6e76 3d3e phi,CodimBsInv=> │ │ │ │ +0001f600: 3529 3b20 2020 2020 2020 2020 2020 2020 5); │ │ │ │ +0001f610: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +0001f620: 6420 302e 3232 3836 3532 7320 2863 7075 d 0.228652s (cpu │ │ │ │ +0001f630: 293b 2030 2e31 3637 3535 3573 2028 7468 ); 0.167555s (th │ │ │ │ +0001f640: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f660: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001f670: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001f680: 703a 2073 7465 7020 3120 6f66 2033 2020 p: step 1 of 3 │ │ │ │ +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 7c0a 7c2d |.|- │ │ │ │ -0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3220 erseMap: step 2 │ │ │ │ -0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ +0001f6b0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001f6c0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001f6d0: 703a 2073 7465 7020 3220 6f66 2033 2020 p: step 2 of 3 │ │ │ │ +0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -0001f710: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ -0001f720: 6572 7365 4d61 703a 2073 7465 7020 3320 erseMap: step 3 │ │ │ │ -0001f730: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ +0001f700: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ +0001f710: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +0001f720: 703a 2073 7465 7020 3320 6f66 2033 2020 p: step 3 of 3 │ │ │ │ +0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f750: 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001f7b0: 3520 3a20 5261 7469 6f6e 616c 4d61 7020 5 : RationalMap │ │ │ │ -0001f7c0: 2871 7561 6472 6174 6963 2072 6174 696f (quadratic ratio │ │ │ │ -0001f7d0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e nal map from PP^ │ │ │ │ -0001f7e0: 3820 746f 2068 7970 6572 7375 7266 6163 8 to hypersurfac │ │ │ │ -0001f7f0: 6520 696e 2050 505e 3929 2020 7c0a 2b2d e in PP^9) |.+- │ │ │ │ +0001f7a0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5261 |.|o5 : Ra │ │ │ │ +0001f7b0: 7469 6f6e 616c 4d61 7020 2871 7561 6472 tionalMap (quadr │ │ │ │ +0001f7c0: 6174 6963 2072 6174 696f 6e61 6c20 6d61 atic rational ma │ │ │ │ +0001f7d0: 7020 6672 6f6d 2050 505e 3820 746f 2068 p from PP^8 to h │ │ │ │ +0001f7e0: 7970 6572 7375 7266 6163 6520 696e 2050 ypersurface in P │ │ │ │ +0001f7f0: 505e 3929 2020 7c0a 2b2d 2d2d 2d2d 2d2d P^9) |.+------- │ │ │ │ 0001f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f850: 3620 3a20 6173 7365 7274 2870 7369 203d 6 : assert(psi = │ │ │ │ -0001f860: 3d20 7073 6927 2920 2020 2020 2020 2020 = psi') │ │ │ │ +0001f840: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 6173 ------+.|i6 : as │ │ │ │ +0001f850: 7365 7274 2870 7369 203d 3d20 7073 6927 sert(psi == psi' │ │ │ │ +0001f860: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0001f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f890: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001f890: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 ------------+..A │ │ │ │ -0001f8f0: 206d 6f72 6520 636f 6d70 6c69 6361 7465 more complicate │ │ │ │ -0001f900: 6420 6578 616d 706c 6520 6973 2074 6865 d example is the │ │ │ │ -0001f910: 2066 6f6c 6c6f 7769 6e67 2028 6865 7265 following (here │ │ │ │ -0001f920: 202a 6e6f 7465 2069 6e76 6572 7365 4d61 *note inverseMa │ │ │ │ -0001f930: 703a 2069 6e76 6572 7365 4d61 702c 0a74 p: inverseMap,.t │ │ │ │ -0001f940: 616b 6573 2061 206c 6f74 206f 6620 7469 akes a lot of ti │ │ │ │ -0001f950: 6d65 2129 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d me!)...+-------- │ │ │ │ +0001f8e0: 2d2d 2d2d 2d2d 2b0a 0a41 206d 6f72 6520 ------+..A more │ │ │ │ +0001f8f0: 636f 6d70 6c69 6361 7465 6420 6578 616d complicated exam │ │ │ │ +0001f900: 706c 6520 6973 2074 6865 2066 6f6c 6c6f ple is the follo │ │ │ │ +0001f910: 7769 6e67 2028 6865 7265 202a 6e6f 7465 wing (here *note │ │ │ │ +0001f920: 2069 6e76 6572 7365 4d61 703a 2069 6e76 inverseMap: inv │ │ │ │ +0001f930: 6572 7365 4d61 702c 0a74 616b 6573 2061 erseMap,.takes a │ │ │ │ +0001f940: 206c 6f74 206f 6620 7469 6d65 2129 2e0a lot of time!).. │ │ │ │ +0001f950: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9a0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 6869 -----+.|i7 : phi │ │ │ │ -0001f9b0: 203d 2072 6174 696f 6e61 6c4d 6170 206d = rationalMap m │ │ │ │ -0001f9c0: 6170 2850 382c 5a5a 2f39 375b 785f 302e ap(P8,ZZ/97[x_0. │ │ │ │ -0001f9d0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001f9a0: 0a7c 6937 203a 2070 6869 203d 2072 6174 .|i7 : phi = rat │ │ │ │ +0001f9b0: 696f 6e61 6c4d 6170 206d 6170 2850 382c ionalMap map(P8, │ │ │ │ +0001f9c0: 5a5a 2f39 375b 785f 302e 2e20 2020 2020 ZZ/97[x_0.. │ │ │ │ +0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fa30: 2020 2020 2020 2020 2020 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│ │ +00026a70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00026a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ac0: 2020 2020 207c 0a7c 2d20 3237 7820 7820 |.|- 27x x │ │ │ │ -00026ad0: 202c 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00026ac0: 0a7c 2d20 3237 7820 7820 202c 2020 2020 .|- 27x x , │ │ │ │ +00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b10: 2020 2020 207c 0a7c 2020 2020 2035 2031 |.| 5 1 │ │ │ │ -00026b20: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00026b00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00026b10: 0a7c 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│ │ │ 00026be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c00: 2020 2020 207c 0a7c 2020 202d 2031 3778 |.| - 17x │ │ │ │ -00026c10: 2078 2020 2c20 2020 2020 2020 2020 2020 x , │ │ │ │ +00026bf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00026c00: 0a7c 2020 202d 2031 3778 2078 2020 2c20 .| - 17x x , │ │ │ │ +00026c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c50: 2020 2020 207c 0a7c 3131 2020 2020 2020 |.|11 │ │ │ │ -00026c60: 3420 3131 2020 2020 2020 2020 2020 2020 4 11 │ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00026c50: 0a7c 3131 2020 2020 2020 3420 3131 2020 .|11 4 11 │ │ │ │ +00026c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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---------------- │ │ │ │ 00026fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027010: 2d2d 2d2d 2d7c 0a7c 2a78 5f38 2d34 362a -----|.|*x_8-46* │ │ │ │ -00027020: 785f 362a 785f 382d 3137 2a78 5f30 2a78 x_6*x_8-17*x_0*x │ │ │ │ -00027030: 5f39 2b33 322a 785f 312a 785f 392d 382a _9+32*x_1*x_9-8* │ │ │ │ -00027040: 785f 322a 785f 392d 3335 2a78 5f33 2a78 x_2*x_9-35*x_3*x │ │ │ │ -00027050: 5f39 2d34 362a 785f 342a 785f 392b 3236 _9-46*x_4*x_9+26 │ │ │ │ -00027060: 2a78 5f35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d *x_5*|.|-------- │ │ │ │ +00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027010: 0a7c 2a78 5f38 2d34 362a 785f 362a 785f .|*x_8-46*x_6*x_ │ │ │ │ +00027020: 382d 3137 2a78 5f30 2a78 5f39 2b33 322a 8-17*x_0*x_9+32* │ │ │ │ +00027030: 785f 312a 785f 392d 382a 785f 322a 785f x_1*x_9-8*x_2*x_ │ │ │ │ +00027040: 392d 3335 2a78 5f33 2a78 5f39 2d34 362a 9-35*x_3*x_9-46* │ │ │ │ +00027050: 785f 342a 785f 392b 3236 2a78 5f35 2a7c x_4*x_9+26*x_5*| │ │ │ │ +00027060: 0a7c 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270b0: 2d2d 2d2d 2d7c 0a7c 785f 392b 3137 2a78 -----|.|x_9+17*x │ │ │ │ -000270c0: 5f36 2a78 5f39 2b31 352a 785f 302a 785f _6*x_9+15*x_0*x_ │ │ │ │ -000270d0: 3130 2b33 352a 785f 312a 785f 3130 2b33 10+35*x_1*x_10+3 │ │ │ │ -000270e0: 342a 785f 322a 785f 3130 2b32 302a 785f 4*x_2*x_10+20*x_ │ │ │ │ -000270f0: 342a 785f 3130 2b31 342a 785f 302a 785f 4*x_10+14*x_0*x_ │ │ │ │ -00027100: 3131 2b33 367c 0a7c 2d2d 2d2d 2d2d 2d2d 11+36|.|-------- │ │ │ │ +000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000270b0: 0a7c 785f 392b 3137 2a78 5f36 2a78 5f39 .|x_9+17*x_6*x_9 │ │ │ │ +000270c0: 2b31 352a 785f 302a 785f 3130 2b33 352a +15*x_0*x_10+35* │ │ │ │ +000270d0: 785f 312a 785f 3130 2b33 342a 785f 322a x_1*x_10+34*x_2* │ │ │ │ +000270e0: 785f 3130 2b32 302a 785f 342a 785f 3130 x_10+20*x_4*x_10 │ │ │ │ +000270f0: 2b31 342a 785f 302a 785f 3131 2b33 367c +14*x_0*x_11+36| │ │ │ │ +00027100: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027150: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 2a78 5f31 -----|.|*x_1*x_1 │ │ │ │ -00027160: 312b 3335 2a78 5f32 2a78 5f31 312d 3137 1+35*x_2*x_11-17 │ │ │ │ -00027170: 2a78 5f34 2a78 5f31 312c 785f 312a 785f *x_4*x_11,x_1*x_ │ │ │ │ -00027180: 322d 3430 2a78 5f32 2a78 5f33 2b32 382a 2-40*x_2*x_3+28* │ │ │ │ -00027190: 785f 335e 322d 785f 302a 785f 342b 352a x_3^2-x_0*x_4+5* │ │ │ │ -000271a0: 785f 322a 787c 0a7c 2d2d 2d2d 2d2d 2d2d x_2*x|.|-------- │ │ │ │ +00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027150: 0a7c 2a78 5f31 2a78 5f31 312b 3335 2a78 .|*x_1*x_11+35*x │ │ │ │ +00027160: 5f32 2a78 5f31 312d 3137 2a78 5f34 2a78 _2*x_11-17*x_4*x │ │ │ │ +00027170: 5f31 312c 785f 312a 785f 322d 3430 2a78 _11,x_1*x_2-40*x │ │ │ │ +00027180: 5f32 2a78 5f33 2b32 382a 785f 335e 322d _2*x_3+28*x_3^2- │ │ │ │ +00027190: 785f 302a 785f 342b 352a 785f 322a 787c x_0*x_4+5*x_2*x| │ │ │ │ +000271a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000271b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000271c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000271d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000271f0: 2d2d 2d2d 2d7c 0a7c 5f34 2d31 362a 785f -----|.|_4-16*x_ │ │ │ │ -00027200: 332a 785f 342b 352a 785f 302a 785f 352d 3*x_4+5*x_0*x_5- │ │ │ │ -00027210: 3336 2a78 5f32 2a78 5f35 2b33 372a 785f 36*x_2*x_5+37*x_ │ │ │ │ -00027220: 332a 785f 352b 3438 2a78 5f32 2a78 5f36 3*x_5+48*x_2*x_6 │ │ │ │ -00027230: 2d35 2a78 5f31 2a78 5f37 2d35 2a78 5f33 -5*x_1*x_7-5*x_3 │ │ │ │ -00027240: 2a78 5f37 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d *x_7+|.|-------- │ │ │ │ +000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000271f0: 0a7c 5f34 2d31 362a 785f 332a 785f 342b .|_4-16*x_3*x_4+ │ │ │ │ +00027200: 352a 785f 302a 785f 352d 3336 2a78 5f32 5*x_0*x_5-36*x_2 │ │ │ │ +00027210: 2a78 5f35 2b33 372a 785f 332a 785f 352b *x_5+37*x_3*x_5+ │ │ │ │ +00027220: 3438 2a78 5f32 2a78 5f36 2d35 2a78 5f31 48*x_2*x_6-5*x_1 │ │ │ │ +00027230: 2a78 5f37 2d35 2a78 5f33 2a78 5f37 2b7c *x_7-5*x_3*x_7+| │ │ │ │ +00027240: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027290: 2d2d 2d2d 2d7c 0a7c 785f 352a 785f 372b -----|.|x_5*x_7+ │ │ │ │ -000272a0: 3230 2a78 5f36 2a78 5f37 2b31 302a 785f 20*x_6*x_7+10*x_ │ │ │ │ -000272b0: 302a 785f 382b 3334 2a78 5f31 2a78 5f38 0*x_8+34*x_1*x_8 │ │ │ │ -000272c0: 2b34 312a 785f 332a 785f 382d 785f 342a +41*x_3*x_8-x_4* │ │ │ │ -000272d0: 785f 382b 785f 362a 785f 382b 3430 2a78 x_8+x_6*x_8+40*x │ │ │ │ -000272e0: 5f30 2a78 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d _0*x_|.|-------- │ │ │ │ +00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027290: 0a7c 785f 352a 785f 372b 3230 2a78 5f36 .|x_5*x_7+20*x_6 │ │ │ │ +000272a0: 2a78 5f37 2b31 302a 785f 302a 785f 382b *x_7+10*x_0*x_8+ │ │ │ │ +000272b0: 3334 2a78 5f31 2a78 5f38 2b34 312a 785f 34*x_1*x_8+41*x_ │ │ │ │ +000272c0: 332a 785f 382d 785f 342a 785f 382b 785f 3*x_8-x_4*x_8+x_ │ │ │ │ +000272d0: 362a 785f 382b 3430 2a78 5f30 2a78 5f7c 6*x_8+40*x_0*x_| │ │ │ │ +000272e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027330: 2d2d 2d2d 2d7c 0a7c 392d 3332 2a78 5f31 -----|.|9-32*x_1 │ │ │ │ -00027340: 2a78 5f39 2b35 2a78 5f32 2a78 5f39 2d31 *x_9+5*x_2*x_9-1 │ │ │ │ -00027350: 312a 785f 332a 785f 392d 3230 2a78 5f34 1*x_3*x_9-20*x_4 │ │ │ │ -00027360: 2a78 5f39 2b34 352a 785f 352a 785f 392d *x_9+45*x_5*x_9- │ │ │ │ -00027370: 3134 2a78 5f36 2a78 5f39 2d32 352a 785f 14*x_6*x_9-25*x_ │ │ │ │ -00027380: 302a 785f 207c 0a7c 2d2d 2d2d 2d2d 2d2d 0*x_ |.|-------- │ │ │ │ +00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027330: 0a7c 392d 3332 2a78 5f31 2a78 5f39 2b35 .|9-32*x_1*x_9+5 │ │ │ │ +00027340: 2a78 5f32 2a78 5f39 2d31 312a 785f 332a *x_2*x_9-11*x_3* │ │ │ │ +00027350: 785f 392d 3230 2a78 5f34 2a78 5f39 2b34 x_9-20*x_4*x_9+4 │ │ │ │ +00027360: 352a 785f 352a 785f 392d 3134 2a78 5f36 5*x_5*x_9-14*x_6 │ │ │ │ +00027370: 2a78 5f39 2d32 352a 785f 302a 785f 207c *x_9-25*x_0*x_ | │ │ │ │ +00027380: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000273a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000273d0: 2d2d 2d2d 2d7c 0a7c 3130 2b34 352a 785f -----|.|10+45*x_ │ │ │ │ -000273e0: 312a 785f 3130 2d34 312a 785f 322a 785f 1*x_10-41*x_2*x_ │ │ │ │ -000273f0: 3130 2d34 362a 785f 342a 785f 3130 2b38 10-46*x_4*x_10+8 │ │ │ │ -00027400: 2a78 5f36 2a78 5f31 302d 3238 2a78 5f30 *x_6*x_10-28*x_0 │ │ │ │ -00027410: 2a78 5f31 312b 3131 2a78 5f32 2a78 5f31 *x_11+11*x_2*x_1 │ │ │ │ -00027420: 312b 3134 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d 1+14*|.|-------- │ │ │ │ +000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000273d0: 0a7c 3130 2b34 352a 785f 312a 785f 3130 .|10+45*x_1*x_10 │ │ │ │ +000273e0: 2d34 312a 785f 322a 785f 3130 2d34 362a -41*x_2*x_10-46* │ │ │ │ +000273f0: 785f 342a 785f 3130 2b38 2a78 5f36 2a78 x_4*x_10+8*x_6*x │ │ │ │ +00027400: 5f31 302d 3238 2a78 5f30 2a78 5f31 312b _10-28*x_0*x_11+ │ │ │ │ +00027410: 3131 2a78 5f32 2a78 5f31 312b 3134 2a7c 11*x_2*x_11+14*| │ │ │ │ +00027420: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027470: 2d2d 2d2d 2d7c 0a7c 785f 342a 785f 3131 -----|.|x_4*x_11 │ │ │ │ -00027480: 2d38 2a78 5f35 2a78 5f31 3129 2c7b 745f -8*x_5*x_11),{t_ │ │ │ │ -00027490: 345e 322b 745f 302a 745f 352b 745f 312a 4^2+t_0*t_5+t_1* │ │ │ │ -000274a0: 745f 352b 3335 2a74 5f32 2a74 5f35 2b31 t_5+35*t_2*t_5+1 │ │ │ │ -000274b0: 302a 745f 332a 745f 352b 3235 2a74 5f34 0*t_3*t_5+25*t_4 │ │ │ │ -000274c0: 2a74 5f35 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d *t_5-|.|-------- │ │ │ │ +00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027470: 0a7c 785f 342a 785f 3131 2d38 2a78 5f35 .|x_4*x_11-8*x_5 │ │ │ │ +00027480: 2a78 5f31 3129 2c7b 745f 345e 322b 745f *x_11),{t_4^2+t_ │ │ │ │ +00027490: 302a 745f 352b 745f 312a 745f 352b 3335 0*t_5+t_1*t_5+35 │ │ │ │ +000274a0: 2a74 5f32 2a74 5f35 2b31 302a 745f 332a *t_2*t_5+10*t_3* │ │ │ │ +000274b0: 745f 352b 3235 2a74 5f34 2a74 5f35 2d7c t_5+25*t_4*t_5-| │ │ │ │ +000274c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000274d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000274e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027510: 2d2d 2d2d 2d7c 0a7c 352a 745f 355e 322d -----|.|5*t_5^2- │ │ │ │ -00027520: 3134 2a74 5f30 2a74 5f36 2d31 342a 745f 14*t_0*t_6-14*t_ │ │ │ │ -00027530: 312a 745f 362d 352a 745f 322a 745f 362d 1*t_6-5*t_2*t_6- │ │ │ │ -00027540: 3133 2a74 5f34 2a74 5f36 2b33 372a 745f 13*t_4*t_6+37*t_ │ │ │ │ -00027550: 352a 745f 362b 3232 2a74 5f36 5e32 2d33 5*t_6+22*t_6^2-3 │ │ │ │ -00027560: 312a 745f 337c 0a7c 2d2d 2d2d 2d2d 2d2d 1*t_3|.|-------- │ │ │ │ +00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027510: 0a7c 352a 745f 355e 322d 3134 2a74 5f30 .|5*t_5^2-14*t_0 │ │ │ │ +00027520: 2a74 5f36 2d31 342a 745f 312a 745f 362d *t_6-14*t_1*t_6- │ │ │ │ +00027530: 352a 745f 322a 745f 362d 3133 2a74 5f34 5*t_2*t_6-13*t_4 │ │ │ │ +00027540: 2a74 5f36 2b33 372a 745f 352a 745f 362b *t_6+37*t_5*t_6+ │ │ │ │ +00027550: 3232 2a74 5f36 5e32 2d33 312a 745f 337c 22*t_6^2-31*t_3| │ │ │ │ +00027560: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2b32 362a -----|.|*t_7+26* │ │ │ │ -000275c0: 745f 342a 745f 372b 3132 2a74 5f35 2a74 t_4*t_7+12*t_5*t │ │ │ │ -000275d0: 5f37 2d34 352a 745f 362a 745f 372d 3436 _7-45*t_6*t_7-46 │ │ │ │ -000275e0: 2a74 5f33 2a74 5f38 2b33 372a 745f 342a *t_3*t_8+37*t_4* │ │ │ │ -000275f0: 745f 382b 3238 2a74 5f35 2a74 5f38 2b33 t_8+28*t_5*t_8+3 │ │ │ │ -00027600: 332a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d 3*t_6|.|-------- │ │ │ │ +000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000275b0: 0a7c 2a74 5f37 2b32 362a 745f 342a 745f .|*t_7+26*t_4*t_ │ │ │ │ +000275c0: 372b 3132 2a74 5f35 2a74 5f37 2d34 352a 7+12*t_5*t_7-45* │ │ │ │ +000275d0: 745f 362a 745f 372d 3436 2a74 5f33 2a74 t_6*t_7-46*t_3*t │ │ │ │ +000275e0: 5f38 2b33 372a 745f 342a 745f 382b 3238 _8+37*t_4*t_8+28 │ │ │ │ +000275f0: 2a74 5f35 2a74 5f38 2b33 332a 745f 367c *t_5*t_8+33*t_6| │ │ │ │ +00027600: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027650: 2d2d 2d2d 2d7c 0a7c 2a74 5f38 2c74 5f33 -----|.|*t_8,t_3 │ │ │ │ -00027660: 2a74 5f34 2b34 2a74 5f30 2a74 5f35 2b33 *t_4+4*t_0*t_5+3 │ │ │ │ -00027670: 392a 745f 312a 745f 352d 3430 2a74 5f32 9*t_1*t_5-40*t_2 │ │ │ │ -00027680: 2a74 5f35 2b34 302a 745f 332a 745f 352b *t_5+40*t_3*t_5+ │ │ │ │ -00027690: 3236 2a74 5f34 2a74 5f35 2d32 302a 745f 26*t_4*t_5-20*t_ │ │ │ │ -000276a0: 355e 322b 207c 0a7c 2d2d 2d2d 2d2d 2d2d 5^2+ |.|-------- │ │ │ │ +00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027650: 0a7c 2a74 5f38 2c74 5f33 2a74 5f34 2b34 .|*t_8,t_3*t_4+4 │ │ │ │ +00027660: 2a74 5f30 2a74 5f35 2b33 392a 745f 312a *t_0*t_5+39*t_1* │ │ │ │ +00027670: 745f 352d 3430 2a74 5f32 2a74 5f35 2b34 t_5-40*t_2*t_5+4 │ │ │ │ +00027680: 302a 745f 332a 745f 352b 3236 2a74 5f34 0*t_3*t_5+26*t_4 │ │ │ │ +00027690: 2a74 5f35 2d32 302a 745f 355e 322b 207c *t_5-20*t_5^2+ | │ │ │ │ +000276a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276f0: 2d2d 2d2d 2d7c 0a7c 3431 2a74 5f30 2a74 -----|.|41*t_0*t │ │ │ │ -00027700: 5f36 2b33 362a 745f 312a 745f 362d 3232 _6+36*t_1*t_6-22 │ │ │ │ -00027710: 2a74 5f32 2a74 5f36 2b33 362a 745f 342a *t_2*t_6+36*t_4* │ │ │ │ -00027720: 745f 362d 3330 2a74 5f35 2a74 5f36 2d31 t_6-30*t_5*t_6-1 │ │ │ │ -00027730: 332a 745f 365e 322d 3235 2a74 5f33 2a74 3*t_6^2-25*t_3*t │ │ │ │ -00027740: 5f37 2b35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d _7+5*|.|-------- │ │ │ │ +000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000276f0: 0a7c 3431 2a74 5f30 2a74 5f36 2b33 362a .|41*t_0*t_6+36* │ │ │ │ +00027700: 745f 312a 745f 362d 3232 2a74 5f32 2a74 t_1*t_6-22*t_2*t │ │ │ │ +00027710: 5f36 2b33 362a 745f 342a 745f 362d 3330 _6+36*t_4*t_6-30 │ │ │ │ +00027720: 2a74 5f35 2a74 5f36 2d31 332a 745f 365e *t_5*t_6-13*t_6^ │ │ │ │ +00027730: 322d 3235 2a74 5f33 2a74 5f37 2b35 2a7c 2-25*t_3*t_7+5*| │ │ │ │ +00027740: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027790: 2d2d 2d2d 2d7c 0a7c 745f 342a 745f 372d -----|.|t_4*t_7- │ │ │ │ -000277a0: 3335 2a74 5f35 2a74 5f37 2b31 302a 745f 35*t_5*t_7+10*t_ │ │ │ │ -000277b0: 362a 745f 372b 3131 2a74 5f33 2a74 5f38 6*t_7+11*t_3*t_8 │ │ │ │ -000277c0: 2b34 362a 745f 342a 745f 382b 3239 2a74 +46*t_4*t_8+29*t │ │ │ │ -000277d0: 5f35 2a74 5f38 2b32 382a 745f 362a 745f _5*t_8+28*t_6*t_ │ │ │ │ -000277e0: 382c 745f 327c 0a7c 2d2d 2d2d 2d2d 2d2d 8,t_2|.|-------- │ │ │ │ +00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027790: 0a7c 745f 342a 745f 372d 3335 2a74 5f35 .|t_4*t_7-35*t_5 │ │ │ │ +000277a0: 2a74 5f37 2b31 302a 745f 362a 745f 372b *t_7+10*t_6*t_7+ │ │ │ │ +000277b0: 3131 2a74 5f33 2a74 5f38 2b34 362a 745f 11*t_3*t_8+46*t_ │ │ │ │ +000277c0: 342a 745f 382b 3239 2a74 5f35 2a74 5f38 4*t_8+29*t_5*t_8 │ │ │ │ +000277d0: 2b32 382a 745f 362a 745f 382c 745f 327c +28*t_6*t_8,t_2| │ │ │ │ +000277e0: 0a7c 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027830: 2d2d 2d2d 2d7c 0a7c 2a74 5f34 2d35 2a74 -----|.|*t_4-5*t │ │ │ │ -00027840: 5f30 2a74 5f35 2d34 302a 745f 312a 745f _0*t_5-40*t_1*t_ │ │ │ │ -00027850: 352b 3132 2a74 5f32 2a74 5f35 2b34 372a 5+12*t_2*t_5+47* │ │ │ │ -00027860: 745f 332a 745f 352b 3337 2a74 5f34 2a74 t_3*t_5+37*t_4*t │ │ │ │ -00027870: 5f35 2b32 352a 745f 355e 322d 3237 2a74 _5+25*t_5^2-27*t │ │ │ │ -00027880: 5f30 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d _0*t_|.|-------- │ │ │ │ +00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027830: 0a7c 2a74 5f34 2d35 2a74 5f30 2a74 5f35 .|*t_4-5*t_0*t_5 │ │ │ │ +00027840: 2d34 302a 745f 312a 745f 352b 3132 2a74 -40*t_1*t_5+12*t │ │ │ │ +00027850: 5f32 2a74 5f35 2b34 372a 745f 332a 745f _2*t_5+47*t_3*t_ │ │ │ │ +00027860: 352b 3337 2a74 5f34 2a74 5f35 2b32 352a 5+37*t_4*t_5+25* │ │ │ │ +00027870: 745f 355e 322d 3237 2a74 5f30 2a74 5f7c t_5^2-27*t_0*t_| │ │ │ │ +00027880: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000278a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000278d0: 2d2d 2d2d 2d7c 0a7c 362d 3232 2a74 5f31 -----|.|6-22*t_1 │ │ │ │ -000278e0: 2a74 5f36 2b32 372a 745f 322a 745f 362d *t_6+27*t_2*t_6- │ │ │ │ -000278f0: 3233 2a74 5f34 2a74 5f36 2b35 2a74 5f35 23*t_4*t_6+5*t_5 │ │ │ │ -00027900: 2a74 5f36 2d31 332a 745f 365e 322d 3339 *t_6-13*t_6^2-39 │ │ │ │ -00027910: 2a74 5f33 2a74 5f37 2d32 392a 745f 342a *t_3*t_7-29*t_4* │ │ │ │ -00027920: 745f 372b 397c 0a7c 2d2d 2d2d 2d2d 2d2d t_7+9|.|-------- │ │ │ │ +000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000278d0: 0a7c 362d 3232 2a74 5f31 2a74 5f36 2b32 .|6-22*t_1*t_6+2 │ │ │ │ +000278e0: 372a 745f 322a 745f 362d 3233 2a74 5f34 7*t_2*t_6-23*t_4 │ │ │ │ +000278f0: 2a74 5f36 2b35 2a74 5f35 2a74 5f36 2d31 *t_6+5*t_5*t_6-1 │ │ │ │ +00027900: 332a 745f 365e 322d 3339 2a74 5f33 2a74 3*t_6^2-39*t_3*t │ │ │ │ +00027910: 5f37 2d32 392a 745f 342a 745f 372b 397c _7-29*t_4*t_7+9| │ │ │ │ +00027920: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027970: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2a74 5f37 -----|.|*t_5*t_7 │ │ │ │ -00027980: 2b33 392a 745f 362a 745f 372b 3336 2a74 +39*t_6*t_7+36*t │ │ │ │ -00027990: 5f33 2a74 5f38 2b31 332a 745f 342a 745f _3*t_8+13*t_4*t_ │ │ │ │ -000279a0: 382b 3236 2a74 5f35 2a74 5f38 2b33 372a 8+26*t_5*t_8+37* │ │ │ │ -000279b0: 745f 362a 745f 382c 745f 302a 745f 342d t_6*t_8,t_0*t_4- │ │ │ │ -000279c0: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d t_0*t|.|-------- │ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027970: 0a7c 2a74 5f35 2a74 5f37 2b33 392a 745f .|*t_5*t_7+39*t_ │ │ │ │ +00027980: 362a 745f 372b 3336 2a74 5f33 2a74 5f38 6*t_7+36*t_3*t_8 │ │ │ │ +00027990: 2b31 332a 745f 342a 745f 382b 3236 2a74 +13*t_4*t_8+26*t │ │ │ │ +000279a0: 5f35 2a74 5f38 2b33 372a 745f 362a 745f _5*t_8+37*t_6*t_ │ │ │ │ +000279b0: 382c 745f 302a 745f 342d 745f 302a 747c 8,t_0*t_4-t_0*t| │ │ │ │ +000279c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000279e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027a10: 2d2d 2d2d 2d7c 0a7c 5f35 2d38 2a74 5f31 -----|.|_5-8*t_1 │ │ │ │ -00027a20: 2a74 5f35 2d33 352a 745f 322a 745f 352d *t_5-35*t_2*t_5- │ │ │ │ -00027a30: 3130 2a74 5f33 2a74 5f35 2d33 332a 745f 10*t_3*t_5-33*t_ │ │ │ │ -00027a40: 342a 745f 352b 352a 745f 355e 322b 3135 4*t_5+5*t_5^2+15 │ │ │ │ -00027a50: 2a74 5f30 2a74 5f36 2b31 352a 745f 312a *t_0*t_6+15*t_1* │ │ │ │ -00027a60: 745f 362b 357c 0a7c 2d2d 2d2d 2d2d 2d2d t_6+5|.|-------- │ │ │ │ +00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027a10: 0a7c 5f35 2d38 2a74 5f31 2a74 5f35 2d33 .|_5-8*t_1*t_5-3 │ │ │ │ +00027a20: 352a 745f 322a 745f 352d 3130 2a74 5f33 5*t_2*t_5-10*t_3 │ │ │ │ +00027a30: 2a74 5f35 2d33 332a 745f 342a 745f 352b *t_5-33*t_4*t_5+ │ │ │ │ +00027a40: 352a 745f 355e 322b 3135 2a74 5f30 2a74 5*t_5^2+15*t_0*t │ │ │ │ +00027a50: 5f36 2b31 352a 745f 312a 745f 362b 357c _6+15*t_1*t_6+5| │ │ │ │ +00027a60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ab0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f36 -----|.|*t_2*t_6 │ │ │ │ -00027ac0: 2b31 352a 745f 342a 745f 362d 3338 2a74 +15*t_4*t_6-38*t │ │ │ │ -00027ad0: 5f35 2a74 5f36 2d32 322a 745f 365e 322b _5*t_6-22*t_6^2+ │ │ │ │ -00027ae0: 3331 2a74 5f33 2a74 5f37 2d32 352a 745f 31*t_3*t_7-25*t_ │ │ │ │ -00027af0: 342a 745f 372d 3139 2a74 5f35 2a74 5f37 4*t_7-19*t_5*t_7 │ │ │ │ -00027b00: 2b34 372a 747c 0a7c 2d2d 2d2d 2d2d 2d2d +47*t|.|-------- │ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027ab0: 0a7c 2a74 5f32 2a74 5f36 2b31 352a 745f .|*t_2*t_6+15*t_ │ │ │ │ +00027ac0: 342a 745f 362d 3338 2a74 5f35 2a74 5f36 4*t_6-38*t_5*t_6 │ │ │ │ +00027ad0: 2d32 322a 745f 365e 322b 3331 2a74 5f33 -22*t_6^2+31*t_3 │ │ │ │ +00027ae0: 2a74 5f37 2d32 352a 745f 342a 745f 372d *t_7-25*t_4*t_7- │ │ │ │ +00027af0: 3139 2a74 5f35 2a74 5f37 2b34 372a 747c 19*t_5*t_7+47*t| │ │ │ │ +00027b00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027b50: 2d2d 2d2d 2d7c 0a7c 5f36 2a74 5f37 2b34 -----|.|_6*t_7+4 │ │ │ │ -00027b60: 362a 745f 332a 745f 382d 3336 2a74 5f34 6*t_3*t_8-36*t_4 │ │ │ │ -00027b70: 2a74 5f38 2d33 352a 745f 352a 745f 382d *t_8-35*t_5*t_8- │ │ │ │ -00027b80: 3331 2a74 5f36 2a74 5f38 2c74 5f32 2a74 31*t_6*t_8,t_2*t │ │ │ │ -00027b90: 5f33 2d74 5f30 2a74 5f35 2d74 5f31 2a74 _3-t_0*t_5-t_1*t │ │ │ │ -00027ba0: 5f35 2d33 357c 0a7c 2d2d 2d2d 2d2d 2d2d _5-35|.|-------- │ │ │ │ +00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027b50: 0a7c 5f36 2a74 5f37 2b34 362a 745f 332a .|_6*t_7+46*t_3* │ │ │ │ +00027b60: 745f 382d 3336 2a74 5f34 2a74 5f38 2d33 t_8-36*t_4*t_8-3 │ │ │ │ +00027b70: 352a 745f 352a 745f 382d 3331 2a74 5f36 5*t_5*t_8-31*t_6 │ │ │ │ +00027b80: 2a74 5f38 2c74 5f32 2a74 5f33 2d74 5f30 *t_8,t_2*t_3-t_0 │ │ │ │ +00027b90: 2a74 5f35 2d74 5f31 2a74 5f35 2d33 357c *t_5-t_1*t_5-35| │ │ │ │ +00027ba0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027bf0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f35 -----|.|*t_2*t_5 │ │ │ │ -00027c00: 2d31 302a 745f 332a 745f 352d 3333 2a74 -10*t_3*t_5-33*t │ │ │ │ -00027c10: 5f34 2a74 5f35 2b35 2a74 5f35 5e32 2b31 _4*t_5+5*t_5^2+1 │ │ │ │ -00027c20: 342a 745f 302a 745f 362b 3134 2a74 5f31 4*t_0*t_6+14*t_1 │ │ │ │ -00027c30: 2a74 5f36 2b35 2a74 5f32 2a74 5f36 2b31 *t_6+5*t_2*t_6+1 │ │ │ │ -00027c40: 342a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d 4*t_4|.|-------- │ │ │ │ +00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027bf0: 0a7c 2a74 5f32 2a74 5f35 2d31 302a 745f .|*t_2*t_5-10*t_ │ │ │ │ +00027c00: 332a 745f 352d 3333 2a74 5f34 2a74 5f35 3*t_5-33*t_4*t_5 │ │ │ │ +00027c10: 2b35 2a74 5f35 5e32 2b31 342a 745f 302a +5*t_5^2+14*t_0* │ │ │ │ +00027c20: 745f 362b 3134 2a74 5f31 2a74 5f36 2b35 t_6+14*t_1*t_6+5 │ │ │ │ +00027c30: 2a74 5f32 2a74 5f36 2b31 342a 745f 347c *t_2*t_6+14*t_4| │ │ │ │ +00027c40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c90: 2d2d 2d2d 2d7c 0a7c 2a74 5f36 2d33 312a -----|.|*t_6-31* │ │ │ │ -00027ca0: 745f 352a 745f 362d 3234 2a74 5f36 5e32 t_5*t_6-24*t_6^2 │ │ │ │ -00027cb0: 2b33 322a 745f 332a 745f 372d 3235 2a74 +32*t_3*t_7-25*t │ │ │ │ -00027cc0: 5f34 2a74 5f37 2d31 392a 745f 352a 745f _4*t_7-19*t_5*t_ │ │ │ │ -00027cd0: 372b 3437 2a74 5f36 2a74 5f37 2b34 362a 7+47*t_6*t_7+46* │ │ │ │ -00027ce0: 745f 332a 747c 0a7c 2d2d 2d2d 2d2d 2d2d t_3*t|.|-------- │ │ │ │ +00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027c90: 0a7c 2a74 5f36 2d33 312a 745f 352a 745f .|*t_6-31*t_5*t_ │ │ │ │ +00027ca0: 362d 3234 2a74 5f36 5e32 2b33 322a 745f 6-24*t_6^2+32*t_ │ │ │ │ +00027cb0: 332a 745f 372d 3235 2a74 5f34 2a74 5f37 3*t_7-25*t_4*t_7 │ │ │ │ +00027cc0: 2d31 392a 745f 352a 745f 372b 3437 2a74 -19*t_5*t_7+47*t │ │ │ │ +00027cd0: 5f36 2a74 5f37 2b34 362a 745f 332a 747c _6*t_7+46*t_3*t| │ │ │ │ +00027ce0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d30: 2d2d 2d2d 2d7c 0a7c 5f38 2d33 362a 745f -----|.|_8-36*t_ │ │ │ │ -00027d40: 342a 745f 382d 3335 2a74 5f35 2a74 5f38 4*t_8-35*t_5*t_8 │ │ │ │ -00027d50: 2d33 312a 745f 362a 745f 382c 745f 312a -31*t_6*t_8,t_1* │ │ │ │ -00027d60: 745f 332d 372a 745f 312a 745f 352b 745f t_3-7*t_1*t_5+t_ │ │ │ │ -00027d70: 312a 745f 362b 745f 342a 745f 362d 372a 1*t_6+t_4*t_6-7* │ │ │ │ -00027d80: 745f 352a 747c 0a7c 2d2d 2d2d 2d2d 2d2d t_5*t|.|-------- │ │ │ │ +00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027d30: 0a7c 5f38 2d33 362a 745f 342a 745f 382d .|_8-36*t_4*t_8- │ │ │ │ +00027d40: 3335 2a74 5f35 2a74 5f38 2d33 312a 745f 35*t_5*t_8-31*t_ │ │ │ │ +00027d50: 362a 745f 382c 745f 312a 745f 332d 372a 6*t_8,t_1*t_3-7* │ │ │ │ +00027d60: 745f 312a 745f 352b 745f 312a 745f 362b t_1*t_5+t_1*t_6+ │ │ │ │ +00027d70: 745f 342a 745f 362d 372a 745f 352a 747c t_4*t_6-7*t_5*t| │ │ │ │ +00027d80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027dd0: 2d2d 2d2d 2d7c 0a7c 5f36 2b32 2a74 5f36 -----|.|_6+2*t_6 │ │ │ │ -00027de0: 5e32 2d74 5f33 2a74 5f37 2c74 5f30 2a74 ^2-t_3*t_7,t_0*t │ │ │ │ -00027df0: 5f33 2d34 362a 745f 302a 745f 352d 3339 _3-46*t_0*t_5-39 │ │ │ │ -00027e00: 2a74 5f31 2a74 5f35 2d34 332a 745f 322a *t_1*t_5-43*t_2* │ │ │ │ -00027e10: 745f 352d 3431 2a74 5f33 2a74 5f35 2d32 t_5-41*t_3*t_5-2 │ │ │ │ -00027e20: 362a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d 6*t_4|.|-------- │ │ │ │ +00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027dd0: 0a7c 5f36 2b32 2a74 5f36 5e32 2d74 5f33 .|_6+2*t_6^2-t_3 │ │ │ │ +00027de0: 2a74 5f37 2c74 5f30 2a74 5f33 2d34 362a *t_7,t_0*t_3-46* │ │ │ │ +00027df0: 745f 302a 745f 352d 3339 2a74 5f31 2a74 t_0*t_5-39*t_1*t │ │ │ │ +00027e00: 5f35 2d34 332a 745f 322a 745f 352d 3431 _5-43*t_2*t_5-41 │ │ │ │ +00027e10: 2a74 5f33 2a74 5f35 2d32 362a 745f 347c *t_3*t_5-26*t_4| │ │ │ │ +00027e20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e70: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2d32 382a -----|.|*t_5-28* │ │ │ │ -00027e80: 745f 355e 322d 3335 2a74 5f30 2a74 5f36 t_5^2-35*t_0*t_6 │ │ │ │ -00027e90: 2d33 362a 745f 312a 745f 362b 3230 2a74 -36*t_1*t_6+20*t │ │ │ │ -00027ea0: 5f32 2a74 5f36 2d33 362a 745f 342a 745f _2*t_6-36*t_4*t_ │ │ │ │ -00027eb0: 362b 392a 745f 352a 745f 362b 3135 2a74 6+9*t_5*t_6+15*t │ │ │ │ -00027ec0: 5f36 5e32 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d _6^2+|.|-------- │ │ │ │ +00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027e70: 0a7c 2a74 5f35 2d32 382a 745f 355e 322d .|*t_5-28*t_5^2- │ │ │ │ +00027e80: 3335 2a74 5f30 2a74 5f36 2d33 362a 745f 35*t_0*t_6-36*t_ │ │ │ │ +00027e90: 312a 745f 362b 3230 2a74 5f32 2a74 5f36 1*t_6+20*t_2*t_6 │ │ │ │ +00027ea0: 2d33 362a 745f 342a 745f 362b 392a 745f -36*t_4*t_6+9*t_ │ │ │ │ +00027eb0: 352a 745f 362b 3135 2a74 5f36 5e32 2b7c 5*t_6+15*t_6^2+| │ │ │ │ +00027ec0: 0a7c 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 2d2d 2d2d ---------------- │ │ │ │ -00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027f10: 2d2d 2d2d 2d7c 0a7c 3236 2a74 5f33 2a74 -----|.|26*t_3*t │ │ │ │ -00027f20: 5f37 2d35 2a74 5f34 2a74 5f37 2b33 352a _7-5*t_4*t_7+35* │ │ │ │ -00027f30: 745f 352a 745f 372d 3130 2a74 5f36 2a74 t_5*t_7-10*t_6*t │ │ │ │ -00027f40: 5f37 2d31 302a 745f 332a 745f 382d 3436 _7-10*t_3*t_8-46 │ │ │ │ -00027f50: 2a74 5f34 2a74 5f38 2b34 372a 745f 352a *t_4*t_8+47*t_5* │ │ │ │ -00027f60: 745f 382d 207c 0a7c 2d2d 2d2d 2d2d 2d2d t_8- |.|-------- │ │ │ │ +00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027f10: 0a7c 3236 2a74 5f33 2a74 5f37 2d35 2a74 .|26*t_3*t_7-5*t │ │ │ │ +00027f20: 5f34 2a74 5f37 2b33 352a 745f 352a 745f _4*t_7+35*t_5*t_ │ │ │ │ +00027f30: 372d 3130 2a74 5f36 2a74 5f37 2d31 302a 7-10*t_6*t_7-10* │ │ │ │ +00027f40: 745f 332a 745f 382d 3436 2a74 5f34 2a74 t_3*t_8-46*t_4*t │ │ │ │ +00027f50: 5f38 2b34 372a 745f 352a 745f 382d 207c _8+47*t_5*t_8- | │ │ │ │ +00027f60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00027f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027fb0: 2d2d 2d2d 2d7c 0a7c 3235 2a74 5f36 2a74 -----|.|25*t_6*t │ │ │ │ -00027fc0: 5f38 2c74 5f32 5e32 2d34 362a 745f 312a _8,t_2^2-46*t_1* │ │ │ │ -00027fd0: 745f 342d 3333 2a74 5f30 2a74 5f35 2d34 t_4-33*t_0*t_5-4 │ │ │ │ -00027fe0: 352a 745f 312a 745f 352d 3339 2a74 5f32 5*t_1*t_5-39*t_2 │ │ │ │ -00027ff0: 2a74 5f35 2d33 392a 745f 332a 745f 352d *t_5-39*t_3*t_5- │ │ │ │ -00028000: 3436 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d 46*t_|.|-------- │ │ │ │ +00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00027fb0: 0a7c 3235 2a74 5f36 2a74 5f38 2c74 5f32 .|25*t_6*t_8,t_2 │ │ │ │ +00027fc0: 5e32 2d34 362a 745f 312a 745f 342d 3333 ^2-46*t_1*t_4-33 │ │ │ │ +00027fd0: 2a74 5f30 2a74 5f35 2d34 352a 745f 312a *t_0*t_5-45*t_1* │ │ │ │ +00027fe0: 745f 352d 3339 2a74 5f32 2a74 5f35 2d33 t_5-39*t_2*t_5-3 │ │ │ │ +00027ff0: 392a 745f 332a 745f 352d 3436 2a74 5f7c 9*t_3*t_5-46*t_| │ │ │ │ +00028000: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028050: 2d2d 2d2d 2d7c 0a7c 342a 745f 352d 3239 -----|.|4*t_5-29 │ │ │ │ -00028060: 2a74 5f35 5e32 2d34 382a 745f 302a 745f *t_5^2-48*t_0*t_ │ │ │ │ -00028070: 362d 3338 2a74 5f31 2a74 5f36 2d33 302a 6-38*t_1*t_6-30* │ │ │ │ -00028080: 745f 322a 745f 362b 3139 2a74 5f34 2a74 t_2*t_6+19*t_4*t │ │ │ │ -00028090: 5f36 2d34 342a 745f 352a 745f 362d 3437 _6-44*t_5*t_6-47 │ │ │ │ -000280a0: 2a74 5f36 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d *t_6^|.|-------- │ │ │ │ +00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028050: 0a7c 342a 745f 352d 3239 2a74 5f35 5e32 .|4*t_5-29*t_5^2 │ │ │ │ +00028060: 2d34 382a 745f 302a 745f 362d 3338 2a74 -48*t_0*t_6-38*t │ │ │ │ +00028070: 5f31 2a74 5f36 2d33 302a 745f 322a 745f _1*t_6-30*t_2*t_ │ │ │ │ +00028080: 362b 3139 2a74 5f34 2a74 5f36 2d34 342a 6+19*t_4*t_6-44* │ │ │ │ +00028090: 745f 352a 745f 362d 3437 2a74 5f36 5e7c t_5*t_6-47*t_6^| │ │ │ │ +000280a0: 0a7c 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280f0: 2d2d 2d2d 2d7c 0a7c 322d 3336 2a74 5f30 -----|.|2-36*t_0 │ │ │ │ -00028100: 2a74 5f37 2d34 362a 745f 312a 745f 372b *t_7-46*t_1*t_7+ │ │ │ │ -00028110: 745f 322a 745f 372d 3434 2a74 5f33 2a74 t_2*t_7-44*t_3*t │ │ │ │ -00028120: 5f37 2b34 382a 745f 342a 745f 372d 3134 _7+48*t_4*t_7-14 │ │ │ │ -00028130: 2a74 5f35 2a74 5f37 2b34 2a74 5f36 2a74 *t_5*t_7+4*t_6*t │ │ │ │ -00028140: 5f37 2d33 367c 0a7c 2d2d 2d2d 2d2d 2d2d _7-36|.|-------- │ │ │ │ +000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000280f0: 0a7c 322d 3336 2a74 5f30 2a74 5f37 2d34 .|2-36*t_0*t_7-4 │ │ │ │ +00028100: 362a 745f 312a 745f 372b 745f 322a 745f 6*t_1*t_7+t_2*t_ │ │ │ │ +00028110: 372d 3434 2a74 5f33 2a74 5f37 2b34 382a 7-44*t_3*t_7+48* │ │ │ │ +00028120: 745f 342a 745f 372d 3134 2a74 5f35 2a74 t_4*t_7-14*t_5*t │ │ │ │ +00028130: 5f37 2b34 2a74 5f36 2a74 5f37 2d33 367c _7+4*t_6*t_7-36| │ │ │ │ +00028140: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028190: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f38 -----|.|*t_0*t_8 │ │ │ │ -000281a0: 2d34 362a 745f 312a 745f 382b 3437 2a74 -46*t_1*t_8+47*t │ │ │ │ -000281b0: 5f32 2a74 5f38 2d33 342a 745f 332a 745f _2*t_8-34*t_3*t_ │ │ │ │ -000281c0: 382d 3234 2a74 5f34 2a74 5f38 2d31 322a 8-24*t_4*t_8-12* │ │ │ │ -000281d0: 745f 352a 745f 382d 3437 2a74 5f36 2a74 t_5*t_8-47*t_6*t │ │ │ │ -000281e0: 5f38 2b34 377c 0a7c 2d2d 2d2d 2d2d 2d2d _8+47|.|-------- │ │ │ │ +00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028190: 0a7c 2a74 5f30 2a74 5f38 2d34 362a 745f .|*t_0*t_8-46*t_ │ │ │ │ +000281a0: 312a 745f 382b 3437 2a74 5f32 2a74 5f38 1*t_8+47*t_2*t_8 │ │ │ │ +000281b0: 2d33 342a 745f 332a 745f 382d 3234 2a74 -34*t_3*t_8-24*t │ │ │ │ +000281c0: 5f34 2a74 5f38 2d31 322a 745f 352a 745f _4*t_8-12*t_5*t_ │ │ │ │ +000281d0: 382d 3437 2a74 5f36 2a74 5f38 2b34 377c 8-47*t_6*t_8+47| │ │ │ │ +000281e0: 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028230: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2a74 5f38 -----|.|*t_7*t_8 │ │ │ │ -00028240: 2c74 5f31 2a74 5f32 2b36 2a74 5f31 2a74 ,t_1*t_2+6*t_1*t │ │ │ │ -00028250: 5f35 2b35 2a74 5f30 2a74 5f36 2d32 2a74 _5+5*t_0*t_6-2*t │ │ │ │ -00028260: 5f31 2a74 5f36 2d74 5f34 2a74 5f36 2d74 _1*t_6-t_4*t_6-t │ │ │ │ -00028270: 5f35 2a74 5f36 2b35 2a74 5f30 2a74 5f37 _5*t_6+5*t_0*t_7 │ │ │ │ -00028280: 2b74 5f31 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d +t_1*|.|-------- │ │ │ │ +00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028230: 0a7c 2a74 5f37 2a74 5f38 2c74 5f31 2a74 .|*t_7*t_8,t_1*t │ │ │ │ +00028240: 5f32 2b36 2a74 5f31 2a74 5f35 2b35 2a74 _2+6*t_1*t_5+5*t │ │ │ │ +00028250: 5f30 2a74 5f36 2d32 2a74 5f31 2a74 5f36 _0*t_6-2*t_1*t_6 │ │ │ │ +00028260: 2d74 5f34 2a74 5f36 2d74 5f35 2a74 5f36 -t_4*t_6-t_5*t_6 │ │ │ │ +00028270: 2b35 2a74 5f30 2a74 5f37 2b74 5f31 2a7c +5*t_0*t_7+t_1*| │ │ │ │ +00028280: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000282d0: 2d2d 2d2d 2d7c 0a7c 745f 372d 322a 745f -----|.|t_7-2*t_ │ │ │ │ -000282e0: 322a 745f 372d 372a 745f 352a 745f 372b 2*t_7-7*t_5*t_7+ │ │ │ │ -000282f0: 322a 745f 362a 745f 372d 322a 745f 312a 2*t_6*t_7-2*t_1* │ │ │ │ -00028300: 745f 382b 332a 745f 372a 745f 382c 745f t_8+3*t_7*t_8,t_ │ │ │ │ -00028310: 302a 745f 322b 745f 312a 745f 342b 352a 0*t_2+t_1*t_4+5* │ │ │ │ -00028320: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d t_0*t|.|-------- │ │ │ │ +000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000282d0: 0a7c 745f 372d 322a 745f 322a 745f 372d .|t_7-2*t_2*t_7- │ │ │ │ +000282e0: 372a 745f 352a 745f 372b 322a 745f 362a 7*t_5*t_7+2*t_6* │ │ │ │ +000282f0: 745f 372d 322a 745f 312a 745f 382b 332a t_7-2*t_1*t_8+3* │ │ │ │ +00028300: 745f 372a 745f 382c 745f 302a 745f 322b t_7*t_8,t_0*t_2+ │ │ │ │ +00028310: 745f 312a 745f 342b 352a 745f 302a 747c t_1*t_4+5*t_0*t| │ │ │ │ +00028320: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028370: 2d2d 2d2d 2d7c 0a7c 5f35 2b33 322a 745f -----|.|_5+32*t_ │ │ │ │ -00028380: 312a 745f 352d 3230 2a74 5f32 2a74 5f35 1*t_5-20*t_2*t_5 │ │ │ │ -00028390: 2d34 372a 745f 332a 745f 352d 3337 2a74 -47*t_3*t_5-37*t │ │ │ │ -000283a0: 5f34 2a74 5f35 2d32 352a 745f 355e 322b _4*t_5-25*t_5^2+ │ │ │ │ -000283b0: 3139 2a74 5f30 2a74 5f36 2b32 322a 745f 19*t_0*t_6+22*t_ │ │ │ │ -000283c0: 312a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d 1*t_6|.|-------- │ │ │ │ +00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028370: 0a7c 5f35 2b33 322a 745f 312a 745f 352d .|_5+32*t_1*t_5- │ │ │ │ +00028380: 3230 2a74 5f32 2a74 5f35 2d34 372a 745f 20*t_2*t_5-47*t_ │ │ │ │ +00028390: 332a 745f 352d 3337 2a74 5f34 2a74 5f35 3*t_5-37*t_4*t_5 │ │ │ │ +000283a0: 2d32 352a 745f 355e 322b 3139 2a74 5f30 -25*t_5^2+19*t_0 │ │ │ │ +000283b0: 2a74 5f36 2b32 322a 745f 312a 745f 367c *t_6+22*t_1*t_6| │ │ │ │ +000283c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000283d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000283e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000283f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028410: 2d2d 2d2d 2d7c 0a7c 2d32 352a 745f 322a -----|.|-25*t_2* │ │ │ │ -00028420: 745f 362b 3235 2a74 5f34 2a74 5f36 2d35 t_6+25*t_4*t_6-5 │ │ │ │ -00028430: 2a74 5f35 2a74 5f36 2b31 332a 745f 365e *t_5*t_6+13*t_6^ │ │ │ │ -00028440: 322b 352a 745f 302a 745f 372b 745f 312a 2+5*t_0*t_7+t_1* │ │ │ │ -00028450: 745f 372b 3339 2a74 5f33 2a74 5f37 2b32 t_7+39*t_3*t_7+2 │ │ │ │ -00028460: 382a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d 8*t_4|.|-------- │ │ │ │ +00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028410: 0a7c 2d32 352a 745f 322a 745f 362b 3235 .|-25*t_2*t_6+25 │ │ │ │ +00028420: 2a74 5f34 2a74 5f36 2d35 2a74 5f35 2a74 *t_4*t_6-5*t_5*t │ │ │ │ +00028430: 5f36 2b31 332a 745f 365e 322b 352a 745f _6+13*t_6^2+5*t_ │ │ │ │ +00028440: 302a 745f 372b 745f 312a 745f 372b 3339 0*t_7+t_1*t_7+39 │ │ │ │ +00028450: 2a74 5f33 2a74 5f37 2b32 382a 745f 347c *t_3*t_7+28*t_4| │ │ │ │ +00028460: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000284b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2d39 2a74 -----|.|*t_7-9*t │ │ │ │ -000284c0: 5f35 2a74 5f37 2d33 392a 745f 362a 745f _5*t_7-39*t_6*t_ │ │ │ │ -000284d0: 372b 342a 745f 302a 745f 382b 745f 312a 7+4*t_0*t_8+t_1* │ │ │ │ -000284e0: 745f 382d 3336 2a74 5f33 2a74 5f38 2d31 t_8-36*t_3*t_8-1 │ │ │ │ -000284f0: 342a 745f 342a 745f 382d 3236 2a74 5f35 4*t_4*t_8-26*t_5 │ │ │ │ -00028500: 2a74 5f38 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d *t_8-|.|-------- │ │ │ │ +000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000284b0: 0a7c 2a74 5f37 2d39 2a74 5f35 2a74 5f37 .|*t_7-9*t_5*t_7 │ │ │ │ +000284c0: 2d33 392a 745f 362a 745f 372b 342a 745f -39*t_6*t_7+4*t_ │ │ │ │ +000284d0: 302a 745f 382b 745f 312a 745f 382d 3336 0*t_8+t_1*t_8-36 │ │ │ │ +000284e0: 2a74 5f33 2a74 5f38 2d31 342a 745f 342a *t_3*t_8-14*t_4* │ │ │ │ +000284f0: 745f 382d 3236 2a74 5f35 2a74 5f38 2d7c t_8-26*t_5*t_8-| │ │ │ │ +00028500: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028550: 2d2d 2d2d 2d7c 0a7c 3337 2a74 5f36 2a74 -----|.|37*t_6*t │ │ │ │ -00028560: 5f38 2c74 5f30 2a74 5f31 2d33 392a 745f _8,t_0*t_1-39*t_ │ │ │ │ -00028570: 312a 745f 342b 3430 2a74 5f31 2a74 5f35 1*t_4+40*t_1*t_5 │ │ │ │ -00028580: 2d33 372a 745f 302a 745f 362d 3339 2a74 -37*t_0*t_6-39*t │ │ │ │ -00028590: 5f31 2a74 5f36 2b31 392a 745f 342a 745f _1*t_6+19*t_4*t_ │ │ │ │ -000285a0: 362d 3339 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d 6-39*|.|-------- │ │ │ │ +00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028550: 0a7c 3337 2a74 5f36 2a74 5f38 2c74 5f30 .|37*t_6*t_8,t_0 │ │ │ │ +00028560: 2a74 5f31 2d33 392a 745f 312a 745f 342b *t_1-39*t_1*t_4+ │ │ │ │ +00028570: 3430 2a74 5f31 2a74 5f35 2d33 372a 745f 40*t_1*t_5-37*t_ │ │ │ │ +00028580: 302a 745f 362d 3339 2a74 5f31 2a74 5f36 0*t_6-39*t_1*t_6 │ │ │ │ +00028590: 2b31 392a 745f 342a 745f 362d 3339 2a7c +19*t_4*t_6-39*| │ │ │ │ +000285a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000285b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000285c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000285d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000285f0: 2d2d 2d2d 2d7c 0a7c 745f 352a 745f 362d -----|.|t_5*t_6- │ │ │ │ -00028600: 3338 2a74 5f30 2a74 5f37 2b33 392a 745f 38*t_0*t_7+39*t_ │ │ │ │ -00028610: 312a 745f 372b 3139 2a74 5f32 2a74 5f37 1*t_7+19*t_2*t_7 │ │ │ │ -00028620: 2b31 382a 745f 352a 745f 372d 3139 2a74 +18*t_5*t_7-19*t │ │ │ │ -00028630: 5f36 2a74 5f37 2b31 392a 745f 312a 745f _6*t_7+19*t_1*t_ │ │ │ │ -00028640: 382b 3230 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d 8+20*|.|-------- │ │ │ │ +000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000285f0: 0a7c 745f 352a 745f 362d 3338 2a74 5f30 .|t_5*t_6-38*t_0 │ │ │ │ +00028600: 2a74 5f37 2b33 392a 745f 312a 745f 372b *t_7+39*t_1*t_7+ │ │ │ │ +00028610: 3139 2a74 5f32 2a74 5f37 2b31 382a 745f 19*t_2*t_7+18*t_ │ │ │ │ +00028620: 352a 745f 372d 3139 2a74 5f36 2a74 5f37 5*t_7-19*t_6*t_7 │ │ │ │ +00028630: 2b31 392a 745f 312a 745f 382b 3230 2a7c +19*t_1*t_8+20*| │ │ │ │ +00028640: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028690: 2d2d 2d2d 2d7c 0a7c 745f 372a 745f 382c -----|.|t_7*t_8, │ │ │ │ -000286a0: 745f 305e 322b 3132 2a74 5f31 2a74 5f34 t_0^2+12*t_1*t_4 │ │ │ │ -000286b0: 2b32 302a 745f 302a 745f 352b 3237 2a74 +20*t_0*t_5+27*t │ │ │ │ -000286c0: 5f31 2a74 5f35 2d38 2a74 5f32 2a74 5f35 _1*t_5-8*t_2*t_5 │ │ │ │ -000286d0: 2b33 372a 745f 332a 745f 352b 3238 2a74 +37*t_3*t_5+28*t │ │ │ │ -000286e0: 5f34 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d _4*t_|.|-------- │ │ │ │ +00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028690: 0a7c 745f 372a 745f 382c 745f 305e 322b .|t_7*t_8,t_0^2+ │ │ │ │ +000286a0: 3132 2a74 5f31 2a74 5f34 2b32 302a 745f 12*t_1*t_4+20*t_ │ │ │ │ +000286b0: 302a 745f 352b 3237 2a74 5f31 2a74 5f35 0*t_5+27*t_1*t_5 │ │ │ │ +000286c0: 2d38 2a74 5f32 2a74 5f35 2b33 372a 745f -8*t_2*t_5+37*t_ │ │ │ │ +000286d0: 332a 745f 352b 3238 2a74 5f34 2a74 5f7c 3*t_5+28*t_4*t_| │ │ │ │ +000286e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028730: 2d2d 2d2d 2d7c 0a7c 352b 3330 2a74 5f35 -----|.|5+30*t_5 │ │ │ │ -00028740: 5e32 2d34 362a 745f 302a 745f 362b 3234 ^2-46*t_0*t_6+24 │ │ │ │ -00028750: 2a74 5f31 2a74 5f36 2d34 302a 745f 322a *t_1*t_6-40*t_2* │ │ │ │ -00028760: 745f 362b 3235 2a74 5f34 2a74 5f36 2b31 t_6+25*t_4*t_6+1 │ │ │ │ -00028770: 362a 745f 352a 745f 362d 3335 2a74 5f36 6*t_5*t_6-35*t_6 │ │ │ │ -00028780: 5e32 2b32 397c 0a7c 2d2d 2d2d 2d2d 2d2d ^2+29|.|-------- │ │ │ │ +00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028730: 0a7c 352b 3330 2a74 5f35 5e32 2d34 362a .|5+30*t_5^2-46* │ │ │ │ +00028740: 745f 302a 745f 362b 3234 2a74 5f31 2a74 t_0*t_6+24*t_1*t │ │ │ │ +00028750: 5f36 2d34 302a 745f 322a 745f 362b 3235 _6-40*t_2*t_6+25 │ │ │ │ +00028760: 2a74 5f34 2a74 5f36 2b31 362a 745f 352a *t_4*t_6+16*t_5* │ │ │ │ +00028770: 745f 362d 3335 2a74 5f36 5e32 2b32 397c t_6-35*t_6^2+29| │ │ │ │ +00028780: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000287a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000287b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000287d0: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f37 -----|.|*t_0*t_7 │ │ │ │ -000287e0: 2b31 322a 745f 312a 745f 372d 3335 2a74 +12*t_1*t_7-35*t │ │ │ │ -000287f0: 5f32 2a74 5f37 2d38 2a74 5f33 2a74 5f37 _2*t_7-8*t_3*t_7 │ │ │ │ -00028800: 2d31 382a 745f 342a 745f 372b 3432 2a74 -18*t_4*t_7+42*t │ │ │ │ -00028810: 5f35 2a74 5f37 2d31 322a 745f 362a 745f _5*t_7-12*t_6*t_ │ │ │ │ -00028820: 372d 362a 747c 0a7c 2d2d 2d2d 2d2d 2d2d 7-6*t|.|-------- │ │ │ │ +000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000287d0: 0a7c 2a74 5f30 2a74 5f37 2b31 322a 745f .|*t_0*t_7+12*t_ │ │ │ │ +000287e0: 312a 745f 372d 3335 2a74 5f32 2a74 5f37 1*t_7-35*t_2*t_7 │ │ │ │ +000287f0: 2d38 2a74 5f33 2a74 5f37 2d31 382a 745f -8*t_3*t_7-18*t_ │ │ │ │ +00028800: 342a 745f 372b 3432 2a74 5f35 2a74 5f37 4*t_7+42*t_5*t_7 │ │ │ │ +00028810: 2d31 322a 745f 362a 745f 372d 362a 747c -12*t_6*t_7-6*t| │ │ │ │ +00028820: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00028830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028870: 2d2d 2d2d 2d7c 0a7c 5f30 2a74 5f38 2b31 -----|.|_0*t_8+1 │ │ │ │ -00028880: 322a 745f 312a 745f 382d 3135 2a74 5f33 2*t_1*t_8-15*t_3 │ │ │ │ -00028890: 2a74 5f38 2b39 2a74 5f34 2a74 5f38 2b32 *t_8+9*t_4*t_8+2 │ │ │ │ -000288a0: 302a 745f 352a 745f 382d 3330 2a74 5f36 0*t_5*t_8-30*t_6 │ │ │ │ -000288b0: 2a74 5f38 2b34 2a74 5f37 2a74 5f38 7d29 *t_8+4*t_7*t_8}) │ │ │ │ -000288c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00028870: 0a7c 5f30 2a74 5f38 2b31 322a 745f 312a .|_0*t_8+12*t_1* │ │ │ │ +00028880: 745f 382d 3135 2a74 5f33 2a74 5f38 2b39 t_8-15*t_3*t_8+9 │ │ │ │ +00028890: 2a74 5f34 2a74 5f38 2b32 302a 745f 352a *t_4*t_8+20*t_5* │ │ │ │ +000288a0: 745f 382d 3330 2a74 5f36 2a74 5f38 2b34 t_8-30*t_6*t_8+4 │ │ │ │ +000288b0: 2a74 5f37 2a74 5f38 7d29 2020 2020 207c *t_7*t_8}) | │ │ │ │ +000288c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000288e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028910: 2d2d 2d2d 2d2b 0a7c 6938 203a 202d 2d20 -----+.|i8 : -- │ │ │ │ -00028920: 7769 7468 6f75 7420 7468 6520 6f70 7469 without the opti │ │ │ │ -00028930: 6f6e 2027 436f 6469 6d42 7349 6e76 3d3e on 'CodimBsInv=> │ │ │ │ -00028940: 3427 2c20 6974 2074 616b 6573 2061 626f 4', it takes abo │ │ │ │ -00028950: 7574 2020 2020 2020 2020 2020 2020 2020 ut │ │ │ │ -00028960: 2020 2020 207c 0a7c 2020 2020 2074 696d |.| tim │ │ │ │ -00028970: 6520 7073 693d 6170 7072 6f78 696d 6174 e psi=approximat │ │ │ │ -00028980: 6549 6e76 6572 7365 4d61 7028 7068 692c eInverseMap(phi, │ │ │ │ -00028990: 436f 6469 6d42 7349 6e76 3d3e 3429 2020 CodimBsInv=>4) │ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000289c0: 2032 2e34 3832 3439 7320 2863 7075 293b 2.48249s (cpu); │ │ │ │ -000289d0: 2031 2e38 3130 3537 7320 2874 6872 6561 1.81057s (threa │ │ │ │ -000289e0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ -000289f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a00: 2020 2020 207c 0a7c 2d2d 2061 7070 726f |.|-- appro │ │ │ │ -00028a10: 7869 6d61 7465 496e 7665 7273 654d 6170 ximateInverseMap │ │ │ │ -00028a20: 3a20 7374 6570 2031 206f 6620 3320 2020 : step 1 of 3 │ │ │ │ +00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00028910: 0a7c 6938 203a 202d 2d20 7769 7468 6f75 .|i8 : -- withou │ │ │ │ +00028920: 7420 7468 6520 6f70 7469 6f6e 2027 436f t the option 'Co │ │ │ │ +00028930: 6469 6d42 7349 6e76 3d3e 3427 2c20 6974 dimBsInv=>4', it │ │ │ │ +00028940: 2074 616b 6573 2061 626f 7574 2020 2020 takes about │ │ │ │ +00028950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028960: 0a7c 2020 2020 2074 696d 6520 7073 693d .| time psi= │ │ │ │ +00028970: 6170 7072 6f78 696d 6174 6549 6e76 6572 approximateInver │ │ │ │ +00028980: 7365 4d61 7028 7068 692c 436f 6469 6d42 seMap(phi,CodimB │ │ │ │ +00028990: 7349 6e76 3d3e 3429 2020 2020 2020 2020 sInv=>4) │ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000289b0: 0a7c 202d 2d20 7573 6564 2031 2e39 3733 .| -- used 1.973 │ │ │ │ +000289c0: 3436 7320 2863 7075 293b 2031 2e37 3232 46s (cpu); 1.722 │ │ │ │ +000289d0: 3539 7320 2874 6872 6561 6429 3b20 3073 59s (thread); 0s │ │ │ │ +000289e0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000289f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028a00: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465 .|-- approximate │ │ │ │ +00028a10: 496e 7665 7273 654d 6170 3a20 7374 6570 InverseMap: step │ │ │ │ +00028a20: 2031 206f 6620 3320 2020 2020 2020 2020 1 of 3 │ │ │ │ 00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 2032 206f 6620 3320 2020 : step 2 of 3 │ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028a50: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465 .|-- approximate │ │ │ │ +00028a60: 496e 7665 7273 654d 6170 3a20 7374 6570 InverseMap: step │ │ │ │ +00028a70: 2032 206f 6620 3320 2020 2020 2020 2020 2 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 2d2d 2061 7070 726f |.|-- appro │ │ │ │ -00028ab0: 7869 6d61 7465 496e 7665 7273 654d 6170 ximateInverseMap │ │ │ │ -00028ac0: 3a20 7374 6570 2033 206f 6620 3320 2020 : step 3 of 3 │ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028aa0: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465 .|-- approximate │ │ │ │ +00028ab0: 496e 7665 7273 654d 6170 3a20 7374 6570 InverseMap: step │ │ │ │ +00028ac0: 2033 206f 6620 3320 2020 2020 2020 2020 3 of 3 │ │ │ │ 00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00028ae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028af0: 0a7c 2020 2020 2020 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 = -- │ │ │ │ -00028b50: 7261 7469 6f6e 616c 206d 6170 202d 2d20 rational map -- │ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028b40: 0a7c 6f38 203d 202d 2d20 7261 7469 6f6e .|o8 = -- ration │ │ │ │ +00028b50: 616c 206d 6170 202d 2d20 2020 2020 2020 al map -- │ │ │ │ 00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028b90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bb0: 2020 2020 2020 2020 5a5a 2020 2020 2020 ZZ │ │ │ │ +00028bb0: 2020 5a5a 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ 00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028be0: 2020 2020 207c 0a7c 2020 2020 2073 6f75 |.| sou │ │ │ │ -00028bf0: 7263 653a 2073 7562 7661 7269 6574 7920 rce: subvariety │ │ │ │ -00028c00: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078 of Proj(--[x , x │ │ │ │ -00028c10: 202c 2078 202c 2078 202c 2078 202c 2078 , x , x , x , x │ │ │ │ -00028c20: 202c 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ -00028c30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00028bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028be0: 0a7c 2020 2020 2073 6f75 7263 653a 2073 .| source: s │ │ │ │ +00028bf0: 7562 7661 7269 6574 7920 6f66 2050 726f ubvariety of Pro │ │ │ │ +00028c00: 6a28 2d2d 5b78 202c 2078 202c 2078 202c j(--[x , x , x , │ │ │ │ +00028c10: 2078 202c 2078 202c 2078 202c 2020 2020 x , x , x , │ │ │ │ +00028c20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c50: 2020 2020 2020 2020 3937 2020 3020 2020 97 0 │ │ │ │ -00028c60: 3120 2020 3220 2020 3320 2020 3420 2020 1 2 3 4 │ │ │ │ -00028c70: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00028c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028c90: 2020 2020 207b 2020 2020 2020 2020 2020 { │ │ │ │ +00028c50: 2020 3937 2020 3020 2020 3120 2020 3220 97 0 1 2 │ │ │ │ +00028c60: 2020 3320 2020 3420 2020 3520 2020 2020 3 4 5 │ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028c80: 0a7c 2020 2020 2020 2020 2020 2020 207b .| { │ │ │ │ +00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00028cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028cd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cf0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00028cf0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00028d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028d30: 2020 2020 2020 7820 7820 202d 2038 7820 x x - 8x │ │ │ │ -00028d40: 7820 202b 2032 3578 2020 2d20 3235 7820 x + 25x - 25x │ │ │ │ -00028d50: 7820 202d 2032 3278 2078 2020 2b20 7820 x - 22x x + x │ │ │ │ -00028d60: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ -00028d70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028d80: 2020 2020 2020 2031 2033 2020 2020 2032 1 3 2 │ │ │ │ -00028d90: 2033 2020 2020 2020 3320 2020 2020 2032 3 3 2 │ │ │ │ -00028da0: 2034 2020 2020 2020 3320 3420 2020 2030 4 3 4 0 │ │ │ │ -00028db0: 2035 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00028dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028d20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028d30: 7820 7820 202d 2038 7820 7820 202b 2032 x x - 8x x + 2 │ │ │ │ +00028d40: 3578 2020 2d20 3235 7820 7820 202d 2032 5x - 25x x - 2 │ │ │ │ +00028d50: 3278 2078 2020 2b20 7820 7820 2020 2020 2x x + x x │ │ │ │ +00028d60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028d70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028d80: 2031 2033 2020 2020 2032 2033 2020 2020 1 3 2 3 │ │ │ │ +00028d90: 2020 3320 2020 2020 2032 2034 2020 2020 3 2 4 │ │ │ │ +00028da0: 2020 3320 3420 2020 2030 2035 2020 2020 3 4 0 5 │ │ │ │ +00028db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028e20: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00028e30: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028e10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028e20: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00028e30: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00028e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028e70: 2020 2020 2020 7820 202b 2031 3778 2078 x + 17x x │ │ │ │ -00028e80: 2020 2d20 3134 7820 202d 2031 3378 2078 - 14x - 13x x │ │ │ │ -00028e90: 2020 2b20 3334 7820 7820 202b 2034 3478 + 34x x + 44x │ │ │ │ -00028ea0: 2078 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ -00028eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028ec0: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ -00028ed0: 3320 2020 2020 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