A generalization of the theorems of Chevalley-warning and ax-Katz via polynomial substitutions

Ioulia N. Baoulina, Anurag Bishnoi, Pete L. Clark

Research output: Contribution to journalArticle

Abstract

We give conditions under which the number of solutions of a system of polynomial equations over a finite field Fq of characteristic p is divisible by p. Our setup involves the substitution ti → f(ti) for auxiliary polynomials f1, . . ., fn ∈ Fq[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.

Original languageEnglish
Pages (from-to)4107-4122
Number of pages16
JournalProceedings of the American Mathematical Society
Volume147
Issue number10
DOIs
Publication statusPublished - 1 Jan 2019
Externally publishedYes

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