Polynomial and Rational Solutions of Holonomic Systems

Polynomial and rational solutions for linear ordinary differential equations can be obtained by algorithmic methods. For instance, the maple package DEtools provides efficient functions polysols and ratsols to find polynomial and rational solutions for a given linear ordinary differential equation with rational function coefficients. A natural analogue of the notion of linear ordinary differential equation in the several variable case is the notion of holonomic system. A holonomic system is a system of linear partial differential equations whose characteristic variety is middle dimensional. Chyzak [4] gave an algorithm to find the rational solutions of holonomic systems by using elimination in the ring of differential operators with rational function coefficients combined with Abramov’s algorithm for rational solutions of ordinary differential equations with parameters. To the authors, solving holonomic systems is analogous to solving systems of algebraic equations of zero-dimensional ideals. Under this analogy, the method of Chyzak corresponds to the elimination method for solving systems of algebraic equations. The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra