Divisibility on point counting over finite Witt rings

TL;DR

This paper extends classical divisibility theorems for polynomial solutions over finite fields to solutions over finite Witt rings, incorporating combinatorial boxes and using Witt vector addition for a more transparent proof.

Contribution

It introduces a $q$-divisibility theorem for solutions over Witt rings with combinatorial boxes, generalizing and improving previous results like Ax-Katz and Grynkiewicz.

Findings

Proves $q$-divisibility for solutions in low algebraic complexity boxes.

Extends Ax-Katz theorem to Witt rings with combinatorial boxes.

Provides a new Witt vector addition-based proof approach.

Abstract

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate the $q$-divisibility for the number of solutions of a polynomial system in $n$ variables over the finite Witt ring $\mathbb{Z}_q/p^m\mathbb{Z}_q$, where the $n$ variables of the polynomials are restricted to run through a combinatorial box lifting $\mathbb{F}_q^n$. The introduction of the combinatorial box makes the problem much more complicated. We prove a $q$-divisibility theorem for any box of low algebraic complexity, including the simplest Teichm\"uller box.This extends the classical Ax-Katz theorem over finite field $\mathbb{F}_q$ (the case $m=1$). Taking $q=p$ to be a prime, our result extends and improves a recent combinatorial theorem of Grynkiewicz. Our different approach is based on the addition operation of Witt vectors and is conceptually much more transparent.