A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions
Ioulia N. Baoulina, Anurag Bishnoi, Pete L. Clark · Proceedings of the American Mathematical Society · 2019
We give conditions under which the number of solutions of a system of polynomial equations over a finite field F q \mathbb {F}_q of characteristic p p is divisible by p p . Our setup involves the substitution t i ↦ f ( t i ) t_i \mapsto f(t_i) for auxiliary polynomials f 1 , … , f n ∈ F q [ t ] 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 p -adic divisibilities, proving a result that recovers the Ax-Katz theorem. We also consider p p -weight degrees, recovering work of Moreno-Moreno, Moreno-Castro, and Castro-Castro-Velez.