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.