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arXiv · 1709.05699

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

Ioulia N. Baoulina·Anurag Bishnoi·Pete L. Clark
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Abstract

We give conditions under which the number of solutions of a system of polynomial equations over a finite field F_q of characteristic p is divisible by p. Our setup involves the substitution t_i |-> f_i(t_i) for auxiliary polynomials f_1,...,f_n in 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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Ioulia N. Baoulina, Anurag Bishnoi, Pete L. Clark. 2017-09-17. A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions. https://doi.org/10.1090/proc/14181

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