Solving a Multivariable Congruence by Change of Term Order

We consider the congruence a P s i=1 b i h i mod I where h 1 ; : : : ; hs are given modulo a zero dimensional ideal I. We give two polynomial time algorithms for determining a Grr obner basis, relative to an arbitrary term order, of the module M of solutions of the congruence, and, in particular, for nding its minimal element. These are based on a generalization of an algorithm of Faug ere et al. and extend the 1{variable solution techniques that use the Euclidean algorithm and the Berlekamp{Massey algorithm.