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

Abstract

We design and analyze an algorithm for computing solutions with coefficients in a finite field Fq of underdetermined systems defined over Fq. The algorithm 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 Fq) of a random system with coefficients in Fq satisfies certain geometric and algebraic properties which is of independent interest.