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