### 7.7.5 dmodapp_lib

Library:
  dmodapp.lib
Purpose:
  Applications of algebraic D-modules
Authors:
  Viktor Levandovskyy, levandov@math.rwth-aachen.de
  Daniel Andres, daniel.andres@math.rwth-aachen.de

  Support: DFG Graduiertenkolleg 1632 'Experimentelle und konstruktive Algebra'

Overview:
  Let K be a field of characteristic 0, R = K[x1,...,xN] and D be the Weyl algebra in variables x1,...,xN,d1,...,dN. In this library there are the following procedures for algebraic D-modules:

  - given a cyclic representation D/I of a holonomic module and a polynomial F in R, it is proved that the localization of D/I with respect to the mult. closed set of all powers of F is a holonomic D-module. Thus we aim to compute its cyclic representaion D/L for an ideal L in D. The procedures for the localization are DLoc, SDLoc and DLoc0.

  - annihilator in D of a given polynomial F from R as well as of a given rational function G/F from Quot(R). These can be computed via procedures annPoly resp. annRat.

  - Groebner bases with respect to weights (according to (SST), given an arbitrary integer vector containing weights for variables, one computes the homogenization of a given ideal relative to this vector, then one computes a Groebner basis and returns the dehomogenization of the result), initial forms and initial ideals in Weyl algebras with respect to a given weight vector can be computed with GBWeight, inForm, initialMalgrange and initialIdealW.

  - restriction and integration of a holonomic module D/I. Suppose I annihilates a function F(x1,...,xn). Our aim is to compute an ideal J directly from I, which annihilates
  - F(0,...,0,xk,...,xn) in case of restriction or
  - the integral of F with respect to x1,...,xm in case of integration. The corresponding procedures are restrictionModule, restrictionIdeal, integralModule and integralIdeal.

  - characteristic varieties defined by ideals in Weyl algebras can be computed with charVariety and charInfo.

  - appelF1, appelF2 and appelF4 return ideals in parametric Weyl algebras, which annihilate corresponding Appel hypergeometric functions.

References:
  (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric Differential Equations', Springer, 2000
  (OTW) Oaku, Takayama, Walther 'A Localization Algorithm for D-modules', Journal of Symbolic Computation, 2000
  (OT) Oaku, Takayama 'Algorithms for D-modules', Journal of Pure and Applied Algebra, 1998

Procedures:

> 7.7.5.0. annPoly |  | computes annihilator of a polynomial f in the corr. Weyl algebra
> 7.7.5.0. annRat |  | computes annihilator of rational function f/g in corr. Weyl algebra
> 7.7.5.0. DLoc |  | computes presentation of localization of D/I wrt symbolic power f^s
> 7.7.5.0. SDLoc |  | computes generic presentation of the localization of D/I wrt f^s
> 7.7.5.0. DLoc0 |  | computes presentation of localization of D/I wrt f^s based on SDLoc
> 7.7.5.0. GBWeight |  | computes Groebner basis of I wrt a weight vector
> 7.7.5.0. initialMalgrange |  | computes Groebner basis of initial Malgrange ideal
> 7.7.5.0. initialIdealW |  | computes initial ideal of wrt a given weight
> 7.7.5.0. inForm |  | computes initial form of poly/ideal wrt a weight
> 7.7.5.0. restrictionIdeal |  | computes restriction ideal of I wrt w
> 7.7.5.0. restrictionModule |  | computes restriction module of I wrt w
> 7.7.5.0. integralIdeal |  | computes integral ideal of I wrt w
> 7.7.5.0. integralModule |  | computes integral module of I wrt w
> 7.7.5.0. deRhamCohom |  | computes basis of n-th de Rham cohom. group
> 7.7.5.0. deRhamCohomIdeal |  | computes basis of n-th de Rham cohom. group
> 7.7.5.0. charVariety |  | computes characteristic variety of the ideal I
> 7.7.5.0. charInfo |  | computes char. variety, singular locus and primary decomp.
> 7.7.5.0. isFsat |  | checks whether the ideal I is F-saturated
> 7.7.5.0. appelF1 |  | creates an ideal annihilating Appel F1 function
> 7.7.5.0. appelF2 |  | creates an ideal annihilating Appel F2 function
> 7.7.5.0. appelF4 |  | creates an ideal annihilating Appel F4 function
> 7.7.5.0. fourier |  | applies Fourier automorphism to ideal
> 7.7.5.0. inverseFourier |  | applies inverse Fourier automorphism to ideal
> 7.7.5.0. bFactor |  | computes the roots of irreducible factors of an univariate poly
> 7.7.5.0. intRoots |  | dismisses non-integer roots from list in bFactor format
> 7.7.5.0. poly2list |  | decomposes the polynomial f into a list of terms and exponents
> 7.7.5.0. fl2poly |  | reconstructs a monic univariate polynomial from its factorization
> 7.7.5.0. insertGenerator |  | inserts an element into an ideal/module
> 7.7.5.0. deleteGenerator |  | deletes the k-th element from an ideal/module
> 7.7.5.0. engine |  | computes a Groebner basis with the algorithm specified by i
> 7.7.5.0. isInt |  | checks whether number n is actually an int
> 7.7.5.0. sortIntvec |  | sorts intvec

D-module See also: bfun_lib; dmod_lib; dmodvar_lib; gmssing_lib.

User manual for Singular version 4-0-3, 2016, generated by texi2html.