Lang-Weil Type Estimates in Finite Difference Fields

TL;DR

This paper establishes uniform bounds on the number of rational points of difference algebraic varieties over finite difference fields, extending Lang-Weil estimates to this setting.

Contribution

It introduces a Lang-Weil type estimate for difference algebraic varieties, including bounds based on transformal dimension and an equidimensionality result for Frobenius reductions.

Findings

Derived uniform bounds for rational points

Proved equidimensionality of Frobenius reductions

Extended Lang-Weil estimates to difference algebraic varieties

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.