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7.5.5 dmodapp_lib

Applications of algebraic D-modules
Viktor Levandovskyy, levandov@math.rwth-aachen.de
Daniel Andres, daniel.andres@math.rwth-aachen.de

Support: DFG Graduiertenkolleg 1632 'Experimentelle und konstruktive Algebra'

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.


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