Divisibility on point counting over finite Witt rings.

Let F q denote the finite field of q elements with characteristic p. Let Z q denote the unramified extension of the p -adic integers Z p with residue field 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 Z q / p m Z q , where the n variables of the polynomials are restricted to run through a box lifting F q n. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q -divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field F q (the case m = 1). Taking q = p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case. [ABSTRACT FROM AUTHOR]