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Merge pull request Macaulay2#3003 from mahrud/Dmodules
Dmodules update, new WeylAlgebras package
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,131 +1,157 @@ | ||
| -- -*- coding: utf-8 -*- | ||
| newPackage("Dmodules", | ||
| Version => "1.4.1.1", | ||
| Date => "February 2023", | ||
| Headline => "D-modules", | ||
| HomePage => "http://people.math.gatech.edu/~aleykin3/Dmodules", | ||
| AuxiliaryFiles => true, | ||
| Authors => { | ||
| {Name => "Anton Leykin", Email => "[email protected]"}, | ||
| {Name => "Harrison Tsai"} | ||
| }, | ||
| Keywords => {"D-modules"}, | ||
| DebuggingMode => false, | ||
| PackageImports => { | ||
| "PrimaryDecomposition", | ||
| "ReesAlgebra", | ||
| "Elimination", | ||
| "FourTiTwo" | ||
| } | ||
| ) | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
| -- Dmodules: the base package | ||
| -------------------------------------------------------------------------------- | ||
|
|
||
| -- Harry's basic files | ||
| load "./Dmodules/Dbasic.m2" | ||
| export { | ||
| "SetVariables", | ||
| "dpairInds", | ||
| "dpairVars", | ||
| "makeWeylAlgebra", | ||
| "makeWA" => "makeWeylAlgebra", | ||
| "createDpairs", | ||
| "extractVarsAlgebra", | ||
| "extractDiffsAlgebra", | ||
| -- createCommAlgebra | ||
| -- FourierLocal | ||
| "Fourier", | ||
| "FourierInverse", | ||
| "Dtransposition", | ||
| "Ddim", | ||
| "isHolonomic", | ||
| "holonomicRank", | ||
| -- GBprune | ||
| "Dprune", | ||
| -- reduceCompress | ||
| -- + a lot of tests | ||
| } | ||
| load "./Dmodules/Gbw.m2" | ||
| export { | ||
| "inw", | ||
| "gbw", | ||
| "characteristicIdeal", | ||
| "charIdeal" => "characteristicIdeal", | ||
| "DsingularLocus", | ||
| "singLocus" => "DsingularLocus", | ||
| "setHomSwitch", | ||
| "getHomSwitch", | ||
| } | ||
| -- Anton's basic files | ||
| load "./Dmodules/switch.m2" | ||
| export { | ||
| -- pInfo | ||
| "Dtrace", | ||
| } | ||
| load "./Dmodules/newRings.m2" | ||
| export { | ||
| -- createHomWeylAlgebra | ||
| -- createIntRing | ||
| -- createAssCommRing | ||
| -- TODO: export and document these | ||
| -- "ThetaRing", | ||
| -- "createThetaRing", | ||
| } | ||
|
|
||
| -- TODO | ||
| -- Anton's algorithms | ||
| load "./Dmodules/stafford.m2" -- TODO | ||
| export { | ||
| -- leadMonomial -- TODO: evict | ||
| "stafford", | ||
| } | ||
| load "./Dmodules/makeCyclic.m2" | ||
| export { | ||
| "AnnG", -- TODO: move to BernsteinSato? | ||
| "Generator", | ||
| "makeCyclic", | ||
| } | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
|
|
||
| scan({"Local", "Global"}, nm -> assert (isGlobalSymbol nm and value getGlobalSymbol nm === getGlobalSymbol nm)) | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
| -- computes the preimage of a submodule M of the target of f | ||
| -- (more precisely, M and <target f> should have the same ambient module) | ||
| -------------------------------------------------------------------------------- | ||
| preimage(Matrix, Module) := (f, M) -> ( | ||
| T := target f; | ||
| g := map(T/M, T); | ||
| kernel (g * f)) | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
| -- Tests | ||
| -------------------------------------------------------------------------------- | ||
|
|
||
| load "Dmodules/TST/Dbasic.m2" | ||
| load "Dmodules/TST/Gbw.m2" | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
| -- Documentation | ||
| -------------------------------------------------------------------------------- | ||
| Version => "1.4.1.1", | ||
| Date => "February 2023", | ||
| Headline => "D-modules", | ||
| HomePage => "http://people.math.gatech.edu/~aleykin3/Dmodules", | ||
| Authors => { | ||
| { Name => "Anton Leykin", Email => "[email protected]" }, | ||
| { Name => "Harrison Tsai" } | ||
| }, | ||
| Keywords => { "D-modules" }, | ||
| PackageExports => { | ||
| "WeylAlgebras", | ||
| "HolonomicSystems", | ||
| "BernsteinSato", | ||
| } | ||
| ) | ||
|
|
||
| beginDocumentation() | ||
|
|
||
| load "Dmodules/DOC/main.m2" | ||
| load "Dmodules/DOC/tutorial.m2" -- basic tutorial | ||
| load "Dmodules/DOC/basics.m2" -- basic commands | ||
| load "Dmodules/DOC/general.m2" | ||
| doc /// | ||
| Node | ||
| Key | ||
| Dmodules | ||
| Headline | ||
| D-modules package collection | ||
| Description | ||
| Text | ||
| To begin, read the @TO "D-modules tutorial"@. | ||
| Tree | ||
| :Packages included in this collection | ||
| @TOH WeylAlgebras@ | ||
| @TOH HolonomicSystems@ | ||
| @TOH BernsteinSato@ | ||
| Subnodes | ||
| "D-modules tutorial" | ||
|
|
||
| -------------------------------------------------------------------------------- | ||
| Node | ||
| Key | ||
| "D-modules tutorial" | ||
| Headline | ||
| Algebraic computations for linear differential equations | ||
| Description | ||
| Text | ||
| D-modules are modules over rings of differential operators over algebraic varieties. | ||
| This package is mostly concerned with computations in the {\em Weyl algebra}, | ||
| the ring of differential operators over affine space (over a field of characteristic zero). | ||
| Most algorithms in this package can be found in the book | ||
| {\em Gröbner deformations of Hypergeometric Differential Equations} by Saito, Sturmfels and Takayama, | ||
| hereafter referred to as [@HREF("https://mathscinet.ams.org/mathscinet/pdf/1734566.pdf","SST")@]. | ||
| This is also the best place to learn about computational D-module theory. The book | ||
| {\em Computational Algebraic Geometry with Macaulay2} has a chapter on D-modules and local cohomology. | ||
| A good introduction to D-module theory is {\em A primer of algebraic D-modules} by Coutinho. | ||
|
|
||
| The Weyl algebra $D_n$ is the free associative algebra in $2n$ variables | ||
| $x_1,\dots,x_n$, $\partial_1,\dots,\partial_n$, subject to the following relations: | ||
| the $x$'s commute with each other; the $\partial$'s commute with each other; $x_i$ commutes with $\partial_j$ if $i\neq j$; | ||
| and finally, $\partial_i x_i = x_i \partial_i +1$ (the Leibniz rule). | ||
|
|
||
|
|
||
| Example | ||
| D1 = QQ[z,dz, WeylAlgebra=>{z=>dz}] | ||
|
|
||
| Text | ||
| As a reality check, let us confirm that this is not a commutative ring. Here is the Leibniz rule. | ||
|
|
||
| Example | ||
| dz*z | ||
|
|
||
| Text | ||
| In order to type less, we can use the shortcuts makeWeylAlgebra or makeWA. | ||
|
|
||
| Example | ||
| R = QQ[x_1..x_4] | ||
| D4 = makeWA R | ||
| describe D4 | ||
|
|
||
| Text | ||
| Elements and ideals are handled in the usual Macaulay2 way. Let us look at the | ||
| @HREF("https://en.wikipedia.org/wiki/Hypergeometric_function#The_hypergeometric_differential_equation","Gauss hypergeometric equation")@ | ||
| for parameters $a=1, b=2, c=3$. | ||
|
|
||
| Example | ||
| use D1 | ||
| a = 1, b = 2, c = 3 | ||
| g = z*(1-z)*dz^2 + (c-(a+b+1)*z)*dz -a*b | ||
| I = ideal g | ||
|
|
||
| Text | ||
| The holonomicRank function computes the dimension of the solution space of a linear system of differential equations. | ||
|
|
||
| Example | ||
| holonomicRank I | ||
|
|
||
| Text | ||
| A-Hypergeometric systems, also known as GKZ systems (see [@HREF("https://mathscinet.ams.org/mathscinet/pdf/1734566.pdf","SST")@, Chapters 3 and 4]) are implemented. | ||
|
|
||
| Example | ||
| needsPackage "HolonomicSystems" | ||
| use D4 | ||
| A = matrix{{1,1,1,1},{0,1,3,4}} | ||
| b = {1,2} | ||
| H = gkz(A,b, D4) | ||
|
|
||
| Text | ||
| Holonomic D-ideals are analogous to zero-dimensional ideals in polynomial rings (see [@HREF("https://mathscinet.ams.org/mathscinet/pdf/1734566.pdf","SST")@, Section 1.4]). | ||
| Let us confirm that our GKZ systems are holonomic. | ||
|
|
||
| Example | ||
| isHolonomic H | ||
|
|
||
| Text | ||
| Once we know our ideal is holonomic, we can compute its holonomic rank. The above is a famous | ||
| GKZ example because the holonomic rank may change when the parameter vector $b$ is changed. | ||
|
|
||
| Example | ||
| holonomicRank H | ||
| holonomicRank sub(gkz(A,{1,0}), vars D4) | ||
|
|
||
| Text | ||
| We can compute the characteristic ideal and singular locus of a D-ideal [@HREF("https://mathscinet.ams.org/mathscinet/pdf/1734566.pdf","SST")@, Section 1.4]. Note that the output of charIdeal | ||
| belongs to a commutative ring, the associated graded ring of $D_n$ with respect to the order filtration. | ||
|
|
||
| Example | ||
| charIdeal H | ||
| singLocus H | ||
|
|
||
| Text | ||
| The singular locus of a GKZ system is the zero set of a polynomial called the Principal A-determinant, | ||
| which is a product of discriminants corresponding to faces of the matrix A | ||
| (see Chapters 8 and 9 of the book {\em Discriminants, Resultants and Multidimensional Determinants} | ||
| by Gelfand, Kapranov and Zelevinsky). Here is how to find the classic cubic discriminant. | ||
|
|
||
| Example | ||
| A1 = matrix{{1,1,1,1},{0,1,2,3}}, b1={0,0} | ||
| H1 = sub(gkz(A1,b1),vars D4) | ||
| factor (singLocus H1)_0 | ||
|
|
||
| SeeAlso | ||
| Dmodules | ||
| /// | ||
|
|
||
| end-- | ||
|
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| uninstallAllPackages() | ||
| restart | ||
| installPackage "WeylAlgebras" | ||
| installPackage "HolonomicSystems" | ||
| installPackage "BernsteinSato" | ||
|
|
||
| restart | ||
| uninstallPackage "Dmodules" | ||
| installPackage "Dmodules" | ||
| viewHelp Dmodules | ||
|
|
||
| restart | ||
| needsPackage "Dmodules" | ||
|
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||
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