  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.
