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Dmodules :: Dmodules

Dmodules -- algorithms for D-modules

Description

To begin, read the D-modules tutorial.

How to make Weyl algebras:

Basic commands:

Some examples of D-modules:

Basic invariants of D-modules:

B-functions:

Resolutions and Functors:

Applications:

Programming aids:

Authors

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
  • Symbols