# Dmodules -- algorithms for D-modules

## Description

To begin, read the D-modules tutorial.

### Basic commands:

• gbw -- Groebner bases w.r.t. a weight
• inw -- initial form/ideal w.r.t. a weight
• Fourier -- Fourier transform for Weyl algebra
• Dtransposition -- standard transposition for Weyl algebra
• makeCyclic -- finds a cyclic generator of a D-module
• stafford -- computes 2 generators for a given ideal in the Weyl algebra

### Some examples of D-modules:

• gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)
• AppellF1 -- Appell F1 system of PDE's
• PolyAnn -- annihilator of a polynomial in the Weyl algebra
• RatAnn -- annihilator of a rational function in Weyl algebra

### B-functions:

• bFunction -- b-function
• globalBFunction -- global b-function (else known as the Bernstein-Sato polynomial)
• globalB -- compute global b-function and b-operator for a D-module and a polynomial
• globalBoperator -- compute a b-operator of a polynomial
• generalB -- global generalized Bernstein-Sato polynomial
• localBFunction -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
• paramBpoly -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
• factorBFunction -- factorization of a b-function
• bFunctionRoots -- get roots of a b-function
• getIntRoots -- get integer roots of a b-function
• AnnFs -- annihilator ideal of fs
• AnnIFs -- annihilator ideal of fs for an arbitrary D-module

### Resolutions and Functors:

• Dresolution -- resolution of a D-module
• Dlocalize -- localization of a D-module
• WeylClosure -- Weyl closure of an ideal
• Ddual -- holonomic dual of a D-module
• Drestriction -- restriction modules of a D-module
• Dintegration -- integration modules of a D-module
• DHom -- D-homomorphisms between holonomic D-modules
• DExt -- Ext groups between holonomic modules
• PolyExt -- Ext groups between a holonomic module and a polynomial ring
• RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)

### Programming aids:

• createDpairs -- pairs up the variables in Weyl algebra
• Dtrace -- set the depth of comments made by D-module routines
• setHomSwitch -- toggles the use of homogeneous Weyl algebra

## Authors

• Anton Leykin
• Harrison Tsai

## Version

This documentation describes version 1.4.0.1 of Dmodules.

## Source code

The source code from which this documentation is derived is in the file Dmodules.m2. The auxiliary files accompanying it are in the directory Dmodules/.

## Exports

• Functions and commands
• AnnFs -- the annihilating ideal of f^s
• "AnnIFs" -- see AnnIFs(Ideal,RingElement) -- the annihilating ideal of f^s for an arbitrary D-module
• AppellF1 -- Appell F1 system of PDE's
• bFunction -- b-function
• "bFunctionRoots" -- see bFunctionRoots(RingElement) -- get roots of a b-function
• BMM -- the characteristic cycle of the localized $D$-module
• charIdeal -- characteristic ideal of a D-module
• "createDpairs" -- see createDpairs(PolynomialRing) -- pairs up the variables in Weyl algebra
• cssExpts -- the exponents of the canonical series solutions of I in the direction of a weight vector
• cssExptsMult -- the exponents (and multiplicities) of the canonical series solutions of I in the direction of a weight vector
• Ddim -- dimension of a D-module
• Ddual -- holonomic dual of a D-module
• deRham -- deRham cohomology groups for the complement of a hypersurface
• "deRhamAll" -- see deRhamAll(RingElement) -- deRham complex for the complement of a hypersurface
• DExt -- Ext groups between holonomic modules
• DHom -- D-homomorphisms between holonomic D-modules
• diffOps -- differential operators of up to the given order for a quotient polynomial ring
• Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateAll" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateClasses" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateComplex" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateIdeal" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• Dintegration -- integration modules of a D-module
• DintegrationAll -- integration modules of a D-module (extended version)
• DintegrationClasses -- integration classes of a D-module
• DintegrationComplex -- derived integration complex of a D-module
• DintegrationIdeal -- integration ideal of a D-module
• Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationAll" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationMap" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• Dlocalize -- localization of a D-module
• DlocalizeAll -- localization of a D-module (extended version)
• DlocalizeMap -- localization map from a D-module to its localization
• Dprune -- prunes a matrix over a Weyl algebra
• Dres -- abbreviation for Dresolution
• Dresolution -- resolution of a D-module
• Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictAll" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictClasses" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictComplex" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictIdeal" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• Drestriction -- restriction modules of a D-module
• DrestrictionAll -- restriction modules of a D-module (extended version)
• DrestrictionClasses -- restriction classes of a D-module
• DrestrictionComplex -- derived restriction complex of a D-module
• DrestrictionIdeal -- restriction ideal of a D-module
• "Dtrace" -- see Dtrace(ZZ) -- set the depth of comments made by D-module routines
• Dtransposition -- standard transposition for Weyl algebra
• eulerOperators (missing documentation)
• ExternalProduct -- external product of modules or complexes
• extractDiffsAlgebra -- underlying polynomial ring in the differentials of a Weyl algebra
• extractVarsAlgebra -- underlying polynomial ring in the ordinary variables of a Weyl algebra
• "factorBFunction" -- see factorBFunction(RingElement) -- factorization of a b-function
• Fourier -- Fourier transform for Weyl algebra
• FourierInverse -- Inverse Fourier map (D-modules)
• gbw -- Groebner bases w.r.t. a weight
• "generalB" -- see generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
• genToDistractionGens -- the image in the thetaRing of a torus-fixed element in a Weyl algebra
• getDtrace -- (internal) -- get the INFOLEVEL switch
• getHomSwitch -- (internal) -- get the HOMOGENIZATION switch
• "getIntRoots" -- see getIntRoots(RingElement) -- get integer roots of a b-function
• gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)
• "globalB" -- see globalB(Ideal,RingElement) -- compute global b-function and b-operator for a D-module and a polynomial
• "globalBFunction" -- see globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)
• "globalBoperator" -- see globalBoperator(RingElement) -- compute a b-operator of a polynomial
• "hasRationalSing" -- see hasRationalSing(List) -- check if a complete intersection has at most rational singularities
• holonomicRank -- rank of a D-module
• inw -- initial form/ideal w.r.t. a weight
• isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
• "isInMultiplierIdeal" -- see isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
• isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
• "jumpingCoefficients" -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• kappaAnnF1PlanarCurve (missing documentation)
• kDiffFs (missing documentation)
• kOrderAnnFa (missing documentation)
• kOrderAnnFs (missing documentation)
• "lct" -- see lct(Ideal) -- compute the log canonical threshold for an ideal
• "localBFunction" -- see localBFunction(RingElement,Ideal) -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
• localCohom -- local cohomology
• "logCohomology" -- see logCohomology(RingElement) -- logarithmic cohomology groups in two variables
• makeCyclic -- finds a cyclic generator of a D-module
• makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
• "multiplierIdeal" -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
• "paramBpoly" -- see paramBpoly(RingElement,String) -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
• pInfo -- prints tracing info
• PolyAnn -- annihilator of a polynomial in the Weyl algebra
• PolyExt -- Ext groups between a holonomic module and a polynomial ring
• PolySols -- polynomial solutions of a holonomic system
• "populateCechComplexCC" -- see populateCechComplexCC(Ideal,List) -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules
• "pruneCechComplexCC" -- see pruneCechComplexCC(MutableHashTable) -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
• "pruneLocalCohom" -- see pruneLocalCohom(HashTable) -- prunes local cohomology modules
• "putWeylAlgebra" -- see putWeylAlgebra(HashTable) -- transforms output of diffOps into elements of Weyl algebra
• RatAnn -- annihilator of a rational function in Weyl algebra
• RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• RatSols -- rational solutions of a holonomic system
• reiffen (missing documentation)
• "setHomSwitch" -- see setHomSwitch(Boolean) -- toggles the use of homogeneous Weyl algebra
• singLocus -- singular locus of a D-module
• stafford -- computes 2 generators for a given ideal in the Weyl algebra
• thetaIdeal -- the image in the thetaRing of a torus-fixed ideal in a Weyl algebra
• toricIdealPartials (missing documentation)
• WeylClosure -- Weyl closure of an ideal
• Methods
• AnnFs(List) -- the annihilating ideal of f_1^{s_1}...f_r^{s_r}
• AnnFs(RingElement) -- the annihilating ideal of f^s
• AnnIFs(Ideal,RingElement) -- the annihilating ideal of f^s for an arbitrary D-module
• "AppellF1(List)" -- see AppellF1 -- Appell F1 system of PDE's
• bFunction(Ideal,List) -- b-function of an ideal
• bFunction(Module,List,List) -- b-function of a holonomic D-module
• bFunctionRoots(RingElement) -- get roots of a b-function
• "BMM(Ideal,RingElement)" -- see BMM -- the characteristic cycle of the localized $D$-module
• "BMM(List,RingElement)" -- see BMM -- the characteristic cycle of the localized $D$-module
• "charIdeal(Ideal)" -- see charIdeal -- characteristic ideal of a D-module
• "charIdeal(Module)" -- see charIdeal -- characteristic ideal of a D-module
• createDpairs(PolynomialRing) -- pairs up the variables in Weyl algebra
• "cssExpts(Ideal,List)" -- see cssExpts -- the exponents of the canonical series solutions of I in the direction of a weight vector
• "cssExptsMult(Ideal,List)" -- see cssExptsMult -- the exponents (and multiplicities) of the canonical series solutions of I in the direction of a weight vector
• "Ddim(Ideal)" -- see Ddim -- dimension of a D-module
• "Ddim(Module)" -- see Ddim -- dimension of a D-module
• "Ddual(Ideal)" -- see Ddual -- holonomic dual of a D-module
• "Ddual(Module)" -- see Ddual -- holonomic dual of a D-module
• "deRham(RingElement)" -- see deRham -- deRham cohomology groups for the complement of a hypersurface
• "deRham(ZZ,RingElement)" -- see deRham -- deRham cohomology groups for the complement of a hypersurface
• deRhamAll(RingElement) -- deRham complex for the complement of a hypersurface
• "DExt(Module,Module)" -- see DExt -- Ext groups between holonomic modules
• "DExt(Module,Module,List)" -- see DExt -- Ext groups between holonomic modules
• "DHom(Ideal,Ideal)" -- see DHom -- D-homomorphisms between holonomic D-modules
• "DHom(Module,Module)" -- see DHom -- D-homomorphisms between holonomic D-modules
• "DHom(Module,Module,List)" -- see DHom -- D-homomorphisms between holonomic D-modules
• "diffOps(Ideal,ZZ)" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
• "diffOps(RingElement,ZZ)" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
• "Dintegrate(Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "Dintegrate(Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "Dintegrate(ZZ,Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "Dintegrate(ZZ,Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateAll(Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateAll(Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateClasses(Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateClasses(Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateClasses(ZZ,Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateClasses(ZZ,Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateComplex(Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateComplex(Module,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "DintegrateIdeal(Ideal,List)" -- see Dintegrate -- Dintegrate* is an (OBSOLETE) abbreviation for Dintegration*
• "Dintegration(Ideal,List)" -- see Dintegration -- integration modules of a D-module
• "Dintegration(Module,List)" -- see Dintegration -- integration modules of a D-module
• "Dintegration(ZZ,Ideal,List)" -- see Dintegration -- integration modules of a D-module
• "Dintegration(ZZ,Module,List)" -- see Dintegration -- integration modules of a D-module
• "DintegrationAll(Ideal,List)" -- see DintegrationAll -- integration modules of a D-module (extended version)
• "DintegrationAll(Module,List)" -- see DintegrationAll -- integration modules of a D-module (extended version)
• "DintegrationClasses(Ideal,List)" -- see DintegrationClasses -- integration classes of a D-module
• "DintegrationClasses(Module,List)" -- see DintegrationClasses -- integration classes of a D-module
• "DintegrationClasses(ZZ,Ideal,List)" -- see DintegrationClasses -- integration classes of a D-module
• "DintegrationClasses(ZZ,Module,List)" -- see DintegrationClasses -- integration classes of a D-module
• "DintegrationComplex(Ideal,List)" -- see DintegrationComplex -- derived integration complex of a D-module
• "DintegrationComplex(Module,List)" -- see DintegrationComplex -- derived integration complex of a D-module
• "DintegrationIdeal(Ideal,List)" -- see DintegrationIdeal -- integration ideal of a D-module
• "Dlocalization(Ideal,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "Dlocalization(Module,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationAll(Ideal,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationAll(Module,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationMap(Ideal,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "DlocalizationMap(Module,RingElement)" -- see Dlocalization -- Dlocalization* is an OBSOLETE name for Dlocalize*
• "Dlocalize(Ideal,RingElement)" -- see Dlocalize -- localization of a D-module
• "Dlocalize(Module,RingElement)" -- see Dlocalize -- localization of a D-module
• "DlocalizeAll(Ideal,RingElement)" -- see DlocalizeAll -- localization of a D-module (extended version)
• "DlocalizeAll(Module,RingElement)" -- see DlocalizeAll -- localization of a D-module (extended version)
• "DlocalizeMap(Ideal,RingElement)" -- see DlocalizeMap -- localization map from a D-module to its localization
• "DlocalizeMap(Module,RingElement)" -- see DlocalizeMap -- localization map from a D-module to its localization
• "Dprune(Matrix)" -- see Dprune -- prunes a matrix over a Weyl algebra
• "Dprune(Module)" -- see Dprune -- prunes a matrix over a Weyl algebra
• "Dres(Ideal)" -- see Dres -- abbreviation for Dresolution
• "Dres(Ideal,List)" -- see Dres -- abbreviation for Dresolution
• "Dres(Module)" -- see Dres -- abbreviation for Dresolution
• "Dres(Module,List)" -- see Dres -- abbreviation for Dresolution
• "Dresolution(Ideal)" -- see Dresolution -- resolution of a D-module
• "Dresolution(Ideal,List)" -- see Dresolution -- resolution of a D-module
• "Dresolution(Module)" -- see Dresolution -- resolution of a D-module
• "Dresolution(Module,List)" -- see Dresolution -- resolution of a D-module
• "Drestrict(Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "Drestrict(Module,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "Drestrict(ZZ,Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "Drestrict(ZZ,Module,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictAll(Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictAll(Module,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictClasses(Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictClasses(Ideal,List,ZZ)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictClasses(Module,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictClasses(Module,List,ZZ)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictComplex(Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictComplex(Ideal,List,ZZ)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictComplex(Module,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictComplex(Module,List,ZZ)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "DrestrictIdeal(Ideal,List)" -- see Drestrict -- an (OBSOLETE) abbreviation for Drestriction
• "Drestriction(Ideal,List)" -- see Drestriction -- restriction modules of a D-module
• "Drestriction(Module,List)" -- see Drestriction -- restriction modules of a D-module
• "Drestriction(ZZ,Ideal,List)" -- see Drestriction -- restriction modules of a D-module
• "Drestriction(ZZ,Module,List)" -- see Drestriction -- restriction modules of a D-module
• "DrestrictionAll(Ideal,List)" -- see DrestrictionAll -- restriction modules of a D-module (extended version)
• "DrestrictionAll(Module,List)" -- see DrestrictionAll -- restriction modules of a D-module (extended version)
• "DrestrictionClasses(Ideal,List)" -- see DrestrictionClasses -- restriction classes of a D-module
• "DrestrictionClasses(Module,List)" -- see DrestrictionClasses -- restriction classes of a D-module
• "DrestrictionClasses(ZZ,Ideal,List)" -- see DrestrictionClasses -- restriction classes of a D-module
• "DrestrictionClasses(ZZ,Module,List)" -- see DrestrictionClasses -- restriction classes of a D-module
• "DrestrictionComplex(Ideal,List)" -- see DrestrictionComplex -- derived restriction complex of a D-module
• "DrestrictionComplex(Module,List)" -- see DrestrictionComplex -- derived restriction complex of a D-module
• "DrestrictionIdeal(Ideal,List)" -- see DrestrictionIdeal -- restriction ideal of a D-module
• Dtrace(ZZ) -- set the depth of comments made by D-module routines
• "Dtransposition(ChainComplex)" -- see Dtransposition -- standard transposition for Weyl algebra
• "Dtransposition(Ideal)" -- see Dtransposition -- standard transposition for Weyl algebra
• "Dtransposition(Matrix)" -- see Dtransposition -- standard transposition for Weyl algebra
• "Dtransposition(RingElement)" -- see Dtransposition -- standard transposition for Weyl algebra
• "ExternalProduct(ChainComplex,ChainComplex)" -- see ExternalProduct -- external product of modules or complexes
• "ExternalProduct(Module,Module)" -- see ExternalProduct -- external product of modules or complexes
• "extractDiffsAlgebra(PolynomialRing)" -- see extractDiffsAlgebra -- underlying polynomial ring in the differentials of a Weyl algebra
• "extractVarsAlgebra(PolynomialRing)" -- see extractVarsAlgebra -- underlying polynomial ring in the ordinary variables of a Weyl algebra
• factorBFunction(RingElement) -- factorization of a b-function
• "Fourier(Ideal)" -- see Fourier -- Fourier transform for Weyl algebra
• "Fourier(Matrix)" -- see Fourier -- Fourier transform for Weyl algebra
• "Fourier(RingElement)" -- see Fourier -- Fourier transform for Weyl algebra
• "FourierInverse(ChainComplex)" -- see FourierInverse -- Inverse Fourier map (D-modules)
• "FourierInverse(Ideal)" -- see FourierInverse -- Inverse Fourier map (D-modules)
• "FourierInverse(Matrix)" -- see FourierInverse -- Inverse Fourier map (D-modules)
• "FourierInverse(Module)" -- see FourierInverse -- Inverse Fourier map (D-modules)
• "FourierInverse(RingElement)" -- see FourierInverse -- Inverse Fourier map (D-modules)
• "gbw(Ideal,List)" -- see gbw -- Groebner bases w.r.t. a weight
• "gbw(Matrix,List)" -- see gbw -- Groebner bases w.r.t. a weight
• "generalB(List)" -- see generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
• generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
• "genToDistractionGens(RingElement,Ring)" -- see genToDistractionGens -- the image in the thetaRing of a torus-fixed element in a Weyl algebra
• getIntRoots(RingElement) -- get integer roots of a b-function
• "gkz(Matrix,List)" -- see gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)
• "gkz(Matrix,List,PolynomialRing)" -- see gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)
• globalB(Ideal,RingElement) -- compute global b-function and b-operator for a D-module and a polynomial
• globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)
• globalBoperator(RingElement) -- compute a b-operator of a polynomial
• hasRationalSing(List) -- check if a complete intersection has at most rational singularities
• "holonomicRank(Ideal)" -- see holonomicRank -- rank of a D-module
• "holonomicRank(Module)" -- see holonomicRank -- rank of a D-module
• "inw(Ideal,List)" -- see inw -- initial form/ideal w.r.t. a weight
• "inw(Matrix,List)" -- see inw -- initial form/ideal w.r.t. a weight
• "inw(RingElement,List)" -- see inw -- initial form/ideal w.r.t. a weight
• "isHolonomic(Ideal)" -- see isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
• "isHolonomic(Module)" -- see isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic
• isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
• "isTorusFixed(Ideal)" -- see isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
• jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• "jumpingCoefficients(Ideal,QQ,QQ)" -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• "jumpingCoefficients(Ideal,QQ,ZZ)" -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• "jumpingCoefficients(Ideal,ZZ,QQ)" -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• "jumpingCoefficients(Ideal,ZZ,ZZ)" -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
• lct(Ideal) -- compute the log canonical threshold for an ideal
• localBFunction(RingElement,Ideal) -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
• localCohom(Ideal) -- local cohomology of a polynomial ring
• localCohom(Ideal,Module) -- local cohomology of a D-module
• localCohom(List,Ideal) -- local cohomology of a polynomial ring
• localCohom(List,Ideal,Module) -- local cohomology of a D-module
• localCohom(ZZ,Ideal) -- local cohomology of a polynomial ring
• localCohom(ZZ,Ideal,Module) -- local cohomology of a D-module
• logCohomology(RingElement) -- logarithmic cohomology groups in two variables
• "makeCyclic(Matrix)" -- see makeCyclic -- finds a cyclic generator of a D-module
• "makeWeylAlgebra(PolynomialRing)" -- see makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
• "multiplierIdeal(Ideal,List)" -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
• multiplierIdeal(Ideal,QQ) -- multiplier ideal
• "multiplierIdeal(Ideal,ZZ)" -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
• paramBpoly(RingElement,String) -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
• "pInfo(ZZ,List)" -- see pInfo -- prints tracing info
• "pInfo(ZZ,Thing)" -- see pInfo -- prints tracing info
• "PolyAnn(RingElement)" -- see PolyAnn -- annihilator of a polynomial in the Weyl algebra
• "PolyExt(Ideal)" -- see PolyExt -- Ext groups between a holonomic module and a polynomial ring
• "PolyExt(Module)" -- see PolyExt -- Ext groups between a holonomic module and a polynomial ring
• "PolyExt(ZZ,Ideal)" -- see PolyExt -- Ext groups between a holonomic module and a polynomial ring
• "PolyExt(ZZ,Module)" -- see PolyExt -- Ext groups between a holonomic module and a polynomial ring
• "PolySols(Ideal)" -- see PolySols -- polynomial solutions of a holonomic system
• "PolySols(Ideal,List)" -- see PolySols -- polynomial solutions of a holonomic system
• "PolySols(Module)" -- see PolySols -- polynomial solutions of a holonomic system
• "PolySols(Module,List)" -- see PolySols -- polynomial solutions of a holonomic system
• populateCechComplexCC(Ideal,List) -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules
• pruneCechComplexCC(MutableHashTable) -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
• pruneLocalCohom(HashTable) -- prunes local cohomology modules
• putWeylAlgebra(HashTable) -- transforms output of diffOps into elements of Weyl algebra
• "RatAnn(RingElement)" -- see RatAnn -- annihilator of a rational function in Weyl algebra
• "RatAnn(RingElement,RingElement)" -- see RatAnn -- annihilator of a rational function in Weyl algebra
• "RatExt(Ideal)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(Ideal,RingElement)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(Module)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(Module,RingElement)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(ZZ,Ideal)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(ZZ,Ideal,RingElement)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(ZZ,Module)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatExt(ZZ,Module,RingElement)" -- see RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
• "RatSols(Ideal)" -- see RatSols -- rational solutions of a holonomic system
• "RatSols(Ideal,List)" -- see RatSols -- rational solutions of a holonomic system
• "RatSols(Ideal,List,List)" -- see RatSols -- rational solutions of a holonomic system
• "RatSols(Ideal,RingElement)" -- see RatSols -- rational solutions of a holonomic system
• "RatSols(Ideal,RingElement,List)" -- see RatSols -- rational solutions of a holonomic system
• setHomSwitch(Boolean) -- toggles the use of homogeneous Weyl algebra
• "singLocus(Ideal)" -- see singLocus -- singular locus of a D-module
• "singLocus(Module)" -- see singLocus -- singular locus of a D-module
• "stafford(Ideal)" -- see stafford -- computes 2 generators for a given ideal in the Weyl algebra
• "thetaIdeal(Ideal,Ring)" -- see thetaIdeal -- the image in the thetaRing of a torus-fixed ideal in a Weyl algebra
• "WeylClosure(Ideal)" -- see WeylClosure -- Weyl closure of an ideal
• "WeylClosure(Ideal,RingElement)" -- see WeylClosure -- Weyl closure of an ideal
• Symbols

## For the programmer

The object Dmodules is .