Zeros of polynomials over finite Witt rings (2310.15637v1)

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 $f_1,\dots,f_s\in \mathbb{Z}_q[x_1,\ldots,x_n]$ 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.