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# 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 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.

• polynomialSolutions -- polynomial solutions of a holonomic system
• rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
• DHom -- D-homomorphisms between holonomic D-modules

## 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 .