Zeros of polynomials over finite Witt rings.

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 \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} 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 [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m=1, and m_1,\dots,m_s\geq 1, which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. 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. [ABSTRACT FROM AUTHOR]