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.