RatSols -- rational solutions of a holonomic system

Synopsis

Usage:
RatSols I, RatSols(I,f), RatSols(I,f,w), RatSols(I,ff), RatSols(I,ff,w)
Inputs:
- I, an ideal, holonomic ideal in the Weyl algebra D
- f, a ring element, a polynomial
- ff, a list, a list of polynomials
- w, a list, a weight vector
Outputs:
- 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.

Ways to use `RatSols` :

RatSols(Ideal)
RatSols(Ideal,List)
RatSols(Ideal,List,List)
RatSols(Ideal,RingElement)
RatSols(Ideal,RingElement,List)

RatSols -- rational solutions of a holonomic system

Synopsis

Description

Caveat

See also

Ways to use RatSols :

Ways to use `RatSols` :