  Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order
$q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic
rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two
positive integers $m,n$, define a box $\mathcal B_m$ to be a subset of
$\mathbb{Z}_q^n$ with $q^{nm}$ elements such that $\mathcal B_m$ modulo $p^m$
is equal to $(\mathbb{Z}_q/p^m \mathbb{Z}_q)^n$. For a collection of
nonconstant polynomials $f_1,\dots,f_s\in \mathbb{Z}_q[x_1,\ldots,x_n]$ and
positive integers $m_1,\dots,m_s$, define the set of common zeros inside the
box $\mathcal B_m$ to be $$V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod
{p^{m_i}}\mbox{ for all } 1\leq i\leq s\}.$$ It is an interesting problem to
give the sharp estimates for the $p$-divisibility of $|V|$. This problem has
been partially solved for the three cases: (i) $m=m_1=\cdots=m_s=1$, which is
just the Ax-Katz theorem, (ii) $m=m_1=\cdots=m_s>1$, which was solved by Katz,
Marshal and Ramage, and (iii) $m=1$, and $ m_1,\dots,m_s\geq 1$, which was
recently solved by Cao, Wan and Grynkiewicz. Based on the multi-fold addition
and multiplication of the finite Witt rings over $\mathbb{F}_q$, we investigate
the remaining unconsidered case of $m>1$ and $m\neq m_j$ for some $1\leq j\leq
s$, and finally provide a complete answer to this problem.
