A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions

TL;DR

This paper extends classical theorems on solution counts of polynomial systems over finite fields by incorporating polynomial substitutions, leading to generalized divisibility results including the Ax-Katz theorem and p-weight degree considerations.

Contribution

It introduces a unified framework for divisibility of solutions in polynomial systems over finite fields using polynomial substitutions, generalizing Chevalley-Warning, Morlaye-Joly, and Ax-Katz theorems.

Findings

Generalized divisibility conditions for polynomial solutions over finite fields.

Recovered classical theorems as special cases.

Extended results to p-adic divisibilities and p-weight degrees.

Abstract

We give conditions under which the number of solutions of a system of polynomial equations over a finite field F_q of characteristic p is divisible by p. Our setup involves the substitution t_i |-> f_i(t_i) for auxiliary polynomials f_1,...,f_n in F_q[t]. We recover as special cases results of Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving a result that recovers the Ax-Katz Theorem. We also consider p-weight degrees, recovering work of Moreno-Moreno, Moreno-Castro and Castro-Castro-Velez.