Solvability of systems of polynomial congruences modulo a large prime
We consider the following polynomial congruences problem: given a prime p, and a set of polynomials f(1),..., f(m) is an element of F-p[x(1),..., x(n)] of total degree at most d, solve the system f(1) = ... = f(m) = 0 for solution(s) in F-p(n). 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 F-p.
polynomial congruence algorithm algebraic set finite field