rationalFunctionSolutions -- rational solutions of a holonomic system

Synopsis

Usage:

rationalFunctionSolutions I

rationalFunctionSolutions(I,f)

rationalFunctionSolutions(I,f,w)

rationalFunctionSolutions(I,ff)

rationalFunctionSolutions(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 Gröbner deformations and works for ideals $I$ of PDE's - see the paper Polynomial and rational solutions of a holonomic system by Oaku, Takayama and Tsai (2000).

i1 : makeWA(QQ[x])

o1 = QQ[x, dx]

o1 : PolynomialRing, 1 differential variable(s)

i2 : I = ideal((x+1)*dx+5)

o2 = ideal(x*dx + dx + 5)

o2 : Ideal of QQ[x, dx]

i3 : rationalFunctionSolutions I

                     1
o3 = {-------------------------------}
       5     4      3      2
      x  + 5x  + 10x  + 10x  + 5x + 1

o3 : List

Caveat

The most efficient method to find rational solutions of a system of differential equations is to find the singular locus, then try to find its irreducible factors. With these, call rationalFunctionSolutions(I, ff, w), where w should be generic enough so that the polynomialSolutions routine will not complain of a non-generic weight vector.

Ways to use rationalFunctionSolutions :

rationalFunctionSolutions(Ideal)
rationalFunctionSolutions(Ideal,List)
rationalFunctionSolutions(Ideal,List,List)
rationalFunctionSolutions(Ideal,RingElement)
rationalFunctionSolutions(Ideal,RingElement,List)

For the programmer

The object rationalFunctionSolutions is a method function.