# 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 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.