Divisibility on point counting over finite Witt rings

Abstract

Let Fq denote the finite field of q elements with characteristic p. Let Zq denote the unramified extension of the p-adic integers Zp with residue field Fq. 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 Zq/pmZq, where the n variables of the polynomials are restricted to run through a box lifting Fqn. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field Fq (the case m=1). Taking q=p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case.