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

TL;DR

This paper presents an algorithm for efficiently computing rational solutions of underdetermined systems over finite fields, using reductions to zero-dimensional searches on linear spaces, with high success probability.

Contribution

The paper introduces a novel algorithm based on vertical strip searches for solving underdetermined systems over finite fields, with probabilistic analysis of its efficiency.

Findings

Less than three searches typically suffice to find a solution

Success probability increases exponentially with the number of searches

Analysis of solution set properties over algebraic closure

Abstract

We design and analyze an algorithm for computing solutions with coefficients in a finite field $\mathbb{F}_q$ of underdetermined systems defined over $\mathbb{F}_q$. 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 $\mathbb{F}_q$) of a random system with coefficients in $\mathbb{F}_q$ satisfies certain geometric and algebraic properties which is of independent interest.