Further Improvements to the Chevalley-Warning Theorems

TL;DR

This paper enhances lower bound estimates for solutions of systems over finite fields, refining Heath-Brown's improvements to the Chevalley-Warning Theorems and demonstrating sharpness in multiple cases.

Contribution

It provides improved bounds and demonstrates their sharpness, advancing the understanding of solution counts for systems over finite fields.

Findings

Improved lower bounds for solutions of systems over finite fields.

Exclusion of systems with solutions forming an affine space.

Sharpness of bounds demonstrated in several cases.

Abstract

We study lower bound estimates for the number of solutions of systems of equations over finite fields. Heath-Brown improved the lower bounds given by the classical \emph{Chevalley-Warning Theorems} by excluding systems of equations whose solutions form an affine space. We improve each of Heath-Brown's results and demonstrate sharpness in several cases.