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

Baoulina, Ioulia; Bishnoi, Anurag; Clark, Pete

doi:10.1090/proc/14181

A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions
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by Ioulia N. Baoulina, Anurag Bishnoi and Pete L. Clark PDF

Proc. Amer. Math. Soc. 147 (2019), 4107-4122 Request permission

Abstract:

We give conditions under which the number of solutions of a system of polynomial equations over a finite field $\mathbb {F}_q$ of characteristic $p$ is divisible by $p$. Our setup involves the substitution $t_i \mapsto f(t_i)$ for auxiliary polynomials $f_1,\dots ,f_n \in \mathbb {F}_q[t]$. 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.

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