On the computation of rational solutions of underdetermined systems over a finite field

Abstract

We design and analyze an algoritm for computing solutions with coefficients in a finite field of underdetermined systems defined over . The algoritm is based on reductions to zero-dimensional searches. The searches are performed on “vertical strips”, namely parallel linear spaces of suitable dimension in a given direction. Our results show that, on average, less than three searches suffice to obtain a solution of the original system, with a probability of success which grows exponentially with the number of searches. The analysis of our algorithm relies on results on the probability that the solution set (over the algebraic closure of ) of a random system with coefficients in satisfies certain geometric and algebraic properties which is of independent interest.