7.7.21 dmodloc_lib

Library:

dmodloc.lib

Purpose:

Localization of algebraic D-modules and applications

Author:

Daniel Andres, daniel.andres@math.rwth-aachen.de

Support: DFG Graduiertenkolleg 1632 `Experimentelle und konstruktive Algebra'

Overview:

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.

References:

(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:

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