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7.5.20 dmodloc_lib

Status: experimental
Localization of algebraic D-modules and applications
Daniel Andres, daniel.andres@math.rwth-aachen.de

Support: DFG Graduiertenkolleg 1632 `Experimentelle und konstruktive Algebra'

Let I be a left ideal in the n-th polynomial Weyl algebra D=K[x]<d> and let f be a polynomial in K[x].

If D/I is a holonomic module over D, it is known that the localization of D/I at f is also holonomic. The procedure Dlocalization computes an ideal J in D such that this localization is isomorphic to D/J as D-modules.

If one regards I as an ideal in the rational Weyl algebra as above, K(x)<d>*I, and intersects with K[x]<d>, the result is called the Weyl closure of I. The procedures WeylClosure (if I has finite holonomic rank) and WeylClosure1 (if I is in the first Weyl algebra) can be used for computations.

As an application of the Weyl closure, the procedure annRatSyz computes a holonomic part of the annihilator of a rational function by computing certain syzygies. The full annihilator can be obtained by taking the Weyl closure of the result.

If one regards the left ideal I as system of linear PDEs, one can find its polynomial solutions with polSol (if I is holonomic) or polSolFiniteRank (if I is of finite holonomic rank). Rational solutions can be obtained with ratSol.

The procedure bfctBound computes a possible multiple of the b-function for f^s*u at a generic root of f. Here, u stands for [1] in D/I.

This library also offers the procedures holonomicRank and DsingularLocus to compute the holonomic rank and the singular locus of the D-module D/I.

(OT) T. Oaku, N. Takayama: `Algorithms for D-modules', Journal of Pure and Applied Algebra, 1998.
(OTT) T. Oaku, N. Takayama, H. Tsai: `Polynomial and rational solutions of holonomic systems', Journal of Pure and Applied Algebra, 2001.
(OTW) T. Oaku, N. Takayama, U. Walther: `A Localization Algorithm for D-modules', Journal of Symbolic Computation, 2000.
(Tsa) H. Tsai: `Algorithms for algebraic analysis', PhD thesis, 2000.

Procedures: Dlocalization  computes the localization of a D-module WeylClosure  computes the Weyl closure of an ideal in the Weyl algebra WeylClosure1  computes the Weyl closure of operator in first Weyl algebra holonomicRank  computes the holonomic rank of I DsingularLocus  computes the singular locus of a D-module polSol  computes basis of polynomial solutions to the given system polSolFiniteRank  computes basis of polynomial solutions to given system ratSol  computes basis of rational solutions to the given system bfctBound  computes multiple of b-function for f^s*u annRatSyz  computes part of annihilator of rational function g/f dmodGeneralAssumptionCheck  check general assumptions extendWeyl  extends basering (Weyl algebra) by given vars polyVars  checks whether f contains only variables indexed by v monomialInIdeal  computes all monomials appearing in generators of ideal vars2pars  converts variables specified by v into parameters minIntRoot2  finds minimal integer root in a list of roots maxIntRoot  finds maximal integer root in a list of roots dmodAction  computes the natural action of a D-module on K[x] dmodActionRat  computes the natural action of a D-module on K(x) simplifyRat  simplifies rational function addRat  adds rational functions multRat  multiplies rational functions diffRat  derives rational function commRing  deletes non-commutative relations from ring rightNFWeyl  computes right NF wrt right ideal (var(k)) in Weyl algebra
See also: bfun_lib; dmod_lib; dmodapp_lib; dmodvar_lib; gmssing_lib.