A Note on the Chevalley--Warning Theorems

TL;DR

This paper refines the Chevalley--Warning theorems over finite fields, extending congruence results and sharpening lower bounds on the number of solutions under certain conditions.

Contribution

It extends Warning's congruence result from modulo p to modulo q and improves the lower bound on the number of solutions when Z is not a linear subspace.

Findings

Proves congruence of solution counts modulo q for parallel hyperplanes.

Sharpens the lower bound on the number of solutions under non-linear conditions.

Provides new insights into the structure of solution sets of polynomial systems over finite fields.

Abstract

Let $f_1,\...,f_r$ be polynomials in $n$ variables over a finite field $F$ of cardinality $q$ and characteristic $p$. Let $f_i$ have total degree $d_i$ and define $d=d_1+\...+d_r$. Write $Z$ for the set of common zeros of the $f_i$, over the field $F$. Warning showed that $#(Z\cap H_1)\equiv#(Z\cap H_2)\mod{p}$ for any two parallel affine hyperplanes $H_1,H_2$ in $F^n$. We prove that the same congruence holds to modulus $q$. Warning also proved that $# Z\ge q^{n-d}$ providing that $Z$ is non-empty. We sharpen this inequality in various ways, assuming that $Z$ is not a linear subspace of $F^n$.