Lang-Weil Type Estimates in Finite Difference Fields

Abstract

We prove a uniform estimate of the number of points for difference algebraic varieties in finite difference fields in the spirit of Lang-Weil. More precisely, we give uniform lower and upper bounds for the number of rational points of a difference variety in terms of its transformal dimension. As a main technical ingredient, we prove an equidimensionality result for Frobenius reductions of difference varieties.