a list, a basis of the rational solutions of I with poles along f or along the polynomials in ff using w for Groebner deformations
Description
The rational solutions of a holonomic system form a finite-dimensional vector space. The only possibilities for the poles of a rational solution are the codimension one components of the singular locus. An algorithm to compute rational solutions is based on Groebner deformations and works for ideals I of PDE's -- see the paper 'Polynomial and rational solutions of a holonomic system' by Oaku-Takayama-Tsai (2000).
i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]
o1 = W
o1 : PolynomialRing
i2 : I = ideal((x+1)*D+5)
o2 = ideal(x*D + D + 5)
o2 : Ideal of W
i3 : RatSols I
-1
o3 = {-------------------------------}
5 4 3 2
x + 5x + 10x + 10x + 5x + 1
o3 : List
Caveat
The most efficient method to find rational solutions is to find the singular locus, then try to find its irreducible factors. With these, call RatSols(I, ff, w), where w should be generic enough so that the PolySols routine will not complain of a non-generic weight vector.
See also
PolySols -- polynomial solutions of a holonomic system
RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)
DHom -- D-homomorphisms between holonomic D-modules