Dmodules -- algorithms for D-modules

Description

How to make Weyl algebras:

WeylAlgebra -- make or retrieve a monoid

Basic commands:

gbw -- Groebner basis 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
makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
stafford -- computes 2 generators for a given ideal in the Weyl algebra

Some examples of D-modules:

gkz -- GKZ A-hypergeometric ideal
AppellF1 -- Appell F1 system of PDE's
PolyAnn -- annihilator of a polynomial in Weyl algebra
RatAnn -- annihilator of a rational function in Weyl algebra

Basic invariants of D-modules:

Ddim -- dimension of a D-module
holonomicRank -- rank of a D-module
charIdeal -- characteristic ideal of a D-module
singLocus -- singular locus of a D-module

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
paramBpoly -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
factorBFunction -- factorization of a b-function
getIntRoots -- get integer roots of a b-function
AnnFs -- annihilator ideal of f^s
AnnIFs -- annihilator ideal of f^s 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)

Applications:

localCohom -- local cohomology
deRham -- deRham cohomology groups for the complement of a hypersurface
PolySols -- polynomial solutions of a holonomic system
RatSols -- rational solutions of a holonomic system
diffOps -- differential operators of up to the given order for a quotient polynomial ring
populateCechComplexCC, pruneCechComplexCC -- characteristic cycles of local cohomology

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 <leykin@math.ima.edu>
Harrison Tsai

Exports

Functions
- AnnFs, see AnnFs(RingElement) -- the annihilating ideal of f^s
- AnnIFs, see AnnIFs(Ideal,RingElement) -- the annihilating ideal of f^s for an arbitrary D-module
- AppellF1, see AppellF1(List) -- Appell F1 system of PDE's
- bFunction -- 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
- 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
- Drank -- an old name of holonomicRank
- 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
- ExternalProduct -- external product of modules or complexes
- factorBFunction, see factorBFunction(RingElement) -- factorization of a b-function
- Fourier -- Fourier transform for Weyl algebra
- FourierInverse -- Inverse Fourier map (D-modules)
- gbw -- Groebner basis w.r.t. a weight
- getDtrace -- (internal) -- get the INFOLEVEL switch
- getHomSwitch -- (internal) -- get the HOMOGENIZATION switch
- getIntRoots, see getIntRoots(RingElement) -- get integer roots of a b-function
- gkz -- GKZ A-hypergeometric ideal
- 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
- 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
- localCohom -- local cohomology
- makeCyclic, see makeCyclic(Matrix) -- finds a cyclic generator of a D-module
- makeWA, see makeWA(PolynomialRing) -- an abreviation for makeWeylalgebra
- makeWeylAlgebra, see makeWeylAlgebra(PolynomialRing) -- Weyl algebra corresponding to a polynomial ring
- 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, see PolyAnn(RingElement) -- annihilator of a polynomial in 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
- setHomSwitch, see setHomSwitch(Boolean) -- toggles the use of homogeneous Weyl algebra
- singLocus -- singular locus of a D-module
- stafford, see stafford(Ideal) -- computes 2 generators for a given ideal in the Weyl algebra
- WeylClosure -- Weyl closure of an ideal
Symbols
- Alg
- annFS -- a key created by Dlocalize
- AnnG -- a key created by makeCyclic
- BasisElts -- a key of the hashtable generated by diffOps
- Bfunction -- a key created by Dlocalize
- Boperator -- a key attached by globalB and Dlocalize
- Boundaries -- a key in the hashtable created by Drestriction/Dintegration
- Bpolynomial -- a key attached by globalB
- CohomologyGroups -- a key in the hashtable created by deRham
- Cycles -- a key in the hashtable created by Drestriction/Dintegration
- dpairInds -- a key attached by createDpairs
- dpairVars -- a key attached by createDpairs
- Duality -- an option for PolySols=>Alg
- Explicit -- a key in the hashtable created by Drestriction/Dintegration
- Exponents -- a key in the hashtable created by Drestriction/Dintegration
- GD -- an option for PolySols=>Alg
- GenCycles -- a key in the hashtable created by Drestriction/Dintegration
- Generator -- a key created by makeCyclic
- GeneratorPower -- a key created by Dlocalize
- GroundField
- HomologyModules -- a key in a hashtable; an option of DExt
- Info
- IntegrateBfunction -- a key created by Dlocalize
- IntegrateComplex -- a key in the hashtable created by Dintegration
- IntRing -- a strategy option for b-functions
- LocMap -- a key created by Dlocalize
- LocModule -- a key created by Dlocalize
- LocStrategy
- None -- an option for DExt=>Special
- NonGeneric -- a strategy option for b-functions
- Oaku -- an option for Dlocalize=>Strategy
- OaTa -- an option for localCohom=>Strategy
- OaTaWa -- an option for localCohom => LocStrategy
- OmegaRes -- a key in the hashtable created by deRham
- optGB -- indicates whether Grobner basis should be computed
- OTW -- an option for Dlocalize=>Strategy
- Output
- PolyGens -- a key of the hashtable generated by diffOps
- PreCycles -- a key in the hashtable created by deRham
- projMap1 -- a key attached by ExternalProduct
- projMap2 -- a key attached by ExternalProduct
- ReducedB -- a strategy option for global b-functions
- Schreyer -- strategy for computing a resolution of a D-module
- Special
- TransferCycles -- a key in the hashtable created by deRham
- TryGeneric -- a strategy option for b-functions
- twistInvMap -- a key attached by ExternalProduct
- TwistMap -- indicates whether TwistMap should be computed
- twistMap -- a key attached by ExternalProduct
- Vars
- Vhomogenize -- strategy for computing a resolution of a D-module
- ViaAnnFs -- a strategy option for global b-functions
- VResolution -- a key in the hashtable created by Drestriction/Dintegration
- Walther -- an option for localCohom=>Strategy