P-Adic valuations associated to fibers and images of polynomial functions

Abstract

The Chevalley-Warning and the Ax-Katz theorems, which provide the p-adic valuations for the number of solutions of the system of polynomials in finite fields, are stated in terms of the degrees of the associated polynomials. Recently, Clark et al. reformulated the theorems in terms of the cardinalities of fibers and images of the associated polynomial functions, and raised a question in this setting. In this paper, we give a negative answer to that question and improve their results in general case using p-adic lifting via the Teichmüller character.