Restricted Variable Chevalley-Warning Theorems

TL;DR

This paper generalizes the Chevalley-Warning theorem to cases with variables restricted to specific subsets of finite fields, introducing new invariants and bounds for polynomial solutions.

Contribution

It introduces new restricted variable versions of the Chevalley-Warning theorem, including the invariant aromega(X) and bounds for solution sets over finite fields.

Findings

Established Chevalley-Warning type results for variables in Vandermonde subsets.

Defined the invariant aromega(X) to analyze solution bounds.

Explored classes of subsets with lower bounds on aromega(X).

Abstract

We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde subsets of $\F_q$ defined by G\'acs-Weiner and Sziklai-Tak\'ats. We then define an invariant $\uomega(X)$ of a nonempty subset of $\F_q^n$. Our second result involves $X$-restricted variables when the degrees of the polynomials are small compared to $\uomega(X)$. We end by exploring various classes of subsets for which $\uomega(X)$ can be bounded from below.