Ax's lemma and Ax's theorem in the Aichinger-Moosbauer calculus

Abstract

In 1964, J. Ax gave a ten line proof of the Chevalley–Warning Theorem. The novelty was an innocuous result about summing a polynomial f∈Fq[t1,…,tN] over all x∈FqN. Recent work of Aichinger–Moosbauer provides a context in which we can seek to generalize Ax's Lemma to maps f:A→B between any two finite commutative groups, and indeed Aichinger–Moosbauer proved such a result when the target group B has prime exponent. In this paper we define a summation invariantσ(A,B) that fits naturally into the Aichinger–Moosbauer calculus. We give upper and lower bounds on σ(A,B) and its exact value in many (but not all) cases. We also give Diophantine applications, including a qualitative Ax–Katz Theorem for polynomials over any finite rng. The bounds that we get turn out to be closely related to Ax's part of the Ax–Katz Theorem.