Zeros of polynomials over finite Witt rings

Abstract

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 $f_1,\dots ,f_s\in {\mathbb{Z}}_q[x_1,\dots ,x_n]$ 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 $m_1,\dots ,m_s\ge 1$, 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 $m\ne m_j$ for some $1\le j\le s$, and finally provide a complete answer to this problem.