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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
References
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Additional Information
  • Ioulia N. Baoulina
  • Affiliation: Department of Mathematics, Moscow State Pedagogical University, Krasnoprudnaya str. 14, Moscow 107140, Russia
  • MR Author ID: 332448
  • ORCID: 0000-0003-3743-8690
  • Email: jbaulina@mail.ru
  • Anurag Bishnoi
  • Affiliation: Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
  • MR Author ID: 1165212
  • Email: anurag.2357@gmail.com
  • Pete L. Clark
  • Affiliation: Department of Mathematics, University of Georgia, 1023 D. W. Brooks Drive, Athens, Georgia 30605
  • MR Author ID: 767639
  • Email: plclark@gmail.com
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4107-4122
  • MSC (2010): Primary 11T06; Secondary 11D79, 11G25
  • DOI: https://doi.org/10.1090/proc/14181
  • MathSciNet review: 4002529