Restricted Variable Chevalley-Warning Theorems

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 Fq defined by Gács-Weiner and Sziklai-Takáts. We then define an invariant ω(X) of a nonempty subset of F n q . Our second result involves X-restricted variables when the degrees of the polynomials are small compared to ω(X). We end by exploring various classes of subsets for which ω(X) can be bounded from below.1 Some further information about the history of these results can be found in [CGS21, §1].