Solvability of systems of polynomial congruences modulo a large prime

Abstract

We consider the following polynomial congruences problem: given a prime p, and a set of polynomials \( f_1,\ldots,f_m \in {\Bbb F}_p[x_1,\ldots,x_n] \) of total degree at most d, solve the system \( f_1 = \cdots = f_m = 0 \) for solution(s) in \( {\Bbb F}^n_p \). We give a randomized algorithm for the decision version of this problem. For a fixed number of variables, the sequential version of the algorithm has expected time complexity polynomial in d, m and log p; the parallel version has expected time complexity polylogarithmic in d, m and p, using a number of processors, polynomial in d, m and log p. The only point which prevents the algorithm from being deterministic is the lack of a deterministic polynomial time algorithm for factoring univariate polynomials over \( {\Bbb F}_p \).