Functional degrees and arithmetic applications II: The Group-Theoretic Prime Ax-Katz Theorem

TL;DR

This paper extends the Ax-Katz $p$-adic congruences and Moreno-Moreno's $p$-weight refinement to finite commutative rings of prime characteristic using a group-theoretic approach, providing new lower bounds on zero counts.

Contribution

It introduces a purely group-theoretic framework to generalize Ax-Katz and Moreno-Moreno results over finite rings, unifying and extending previous $p$-adic divisibility bounds.

Findings

Provides a lower bound on the $p$-adic divisibility of zero counts

Generalizes Ax-Katz congruences to finite rings of prime characteristic

Combines group-theoretic methods with functional calculus for proofs

Abstract

We give a version of Ax-Katz's $p$-adic congruences and Moreno-Moreno's $p$-weight refinement that holds over any finite commutative ring of prime characteristic. We deduce this from a purely group-theoretic result that gives a lower bound on the $p$-adic divisibility of the number of simultaneous zeros of a system of maps $f_j: A\to B_j$ from a fixed ``source'' finite commutative group $A$ of exponent $p$ to varying ``target'' finite commutative $p$-groups $B_j$. Our proof combines Wilson's proof of Ax-Katz over $\mathbb{F}_p$ with the functional calculus of Aichinger-Moosbauer.