7.5.5 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 representation 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:

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