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Proceedings of the American Mathematical Society

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


Authors: Ioulia N. Baoulina, Anurag Bishnoi and Pete L. Clark
Journal: Proc. Amer. Math. Soc. 147 (2019), 4107-4122
MSC (2010): Primary 11T06; Secondary 11D79, 11G25
DOI: https://doi.org/10.1090/proc/14181
Published electronically: July 8, 2019
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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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Additional Information

Ioulia N. Baoulina
Affiliation: Department of Mathematics, Moscow State Pedagogical University, Krasnoprudnaya str. 14, Moscow 107140, Russia
Email: jbaulina@mail.ru

Anurag Bishnoi
Affiliation: Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
Email: anurag.2357@gmail.com

Pete L. Clark
Affiliation: Department of Mathematics, University of Georgia, 1023 D. W. Brooks Drive, Athens, Georgia 30605
Email: plclark@gmail.com

DOI: https://doi.org/10.1090/proc/14181
Received by editor(s): September 17, 2017
Received by editor(s) in revised form: December 20, 2017
Published electronically: July 8, 2019
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2019 American Mathematical Society