Polynomial and rational solutions of holonomic systems

In this article, we give two new algorithms to find the polynomial and rati onal function solutions of a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k <x(1),..,x(n),parti al derivative (1),...,partial derivative (n)>, where k is a computable subf ield of the complex numbers. Both algorithms are based on the theory of D-m odules - the first algorithm obtains degree bounds on the solutions through Grobner deformations and b-functions while the second algorithm evaluates the dimension of the solutions through duality and restriction. (C) 2001 El sevier Science B.V All rights reserved.