Difference Galois theory and dynamics

TL;DR

This paper develops a Galois theory for difference rings, connecting algebraic structures with dynamical systems, and applies it to prove properties of difference zeta functions over finite fields.

Contribution

It introduces a difference Galois theory inspired by categorical approaches and classifies difference ring extensions via a difference profinite Galois groupoid.

Findings

Classifies difference ring extensions using a difference profinite Galois groupoid.

Establishes a connection between difference algebra and symbolic dynamics.

Proves near-rationality of a difference zeta function over finite fields.

Abstract

We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference ring extensions split by a chosen Galois difference ring extension is classified by actions of the associated difference profinite Galois groupoid. In particular, difference locally \'etale extensions of a difference ring are classified by its difference profinite fundamental groupoid. The emergence of difference profinite spaces, viewed as discrete dynamical systems in the realm of topological dynamics, leads us to investigate the interaction of difference algebra and symbolic dynamics. As an application of this interaction, we prove the near-rationality of a certain difference zeta function counting solutions of systems of difference algebraic equations over algebraic closures of finite fields with Frobenius.