### 7.5.14 dmodloc_lib

Status: experimental

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:

> 7.5.14.0. Dlocalization |  | computes the localization of a D-module
> 7.5.14.0. WeylClosure |  | computes the Weyl closure of an ideal in the Weyl algebra
> 7.5.14.0. WeylClosure1 |  | computes the Weyl closure of operator in first Weyl algebra
> 7.5.14.0. holonomicRank |  | computes the holonomic rank of I
> 7.5.14.0. DsingularLocus |  | computes the singular locus of a D-module
> 7.5.14.0. polSol |  | computes basis of polynomial solutions to the given system
> 7.5.14.0. polSolFiniteRank |  | computes basis of polynomial solutions to given system
> 7.5.14.0. ratSol |  | computes basis of rational solutions to the given system
> 7.5.14.0. bfctBound |  | computes multiple of b-function for f^s*u
> 7.5.14.0. annRatSyz |  | computes part of annihilator of rational function g/f
> 7.5.14.0. dmodGeneralAssumptionCheck |  | check general assumptions
> 7.5.14.0. extendWeyl |  | extends basering (Weyl algebra) by given vars
> 7.5.14.0. polyVars |  | checks whether f contains only variables indexed by v
> 7.5.14.0. monomialInIdeal |  | computes all monomials appearing in generators of ideal
> 7.5.14.0. vars2pars |  | converts variables specified by v into parameters
> 7.5.14.0. minIntRoot2 |  | finds minimal integer root in a list of roots
> 7.5.14.0. maxIntRoot |  | finds maximal integer root in a list of roots
> 7.5.14.0. dmodAction |  | computes the natural action of a D-module on K[x]
> 7.5.14.0. dmodActionRat |  | computes the natural action of a D-module on K(x)
> 7.5.14.0. simplifyRat |  | simplifies rational function
> 7.5.14.0. addRat |  | adds rational functions
> 7.5.14.0. multRat |  | multiplies rational functions
> 7.5.14.0. diffRat |  | derives rational function
> 7.5.14.0. commRing |  | deletes non-commutative relations from ring
> 7.5.14.0. rightNFWeyl |  | computes right NF wrt right ideal (var(k)) in Weyl algebra

See also: bfun_lib; dmod_lib; dmodapp_lib; dmodvar_lib; gmssing_lib.

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