Zilber's Dichotomy for Differentially Closed Fields with an Automorphism

TL;DR

This paper proves the full Zilber's dichotomy for difference-differential fields of characteristic zero using arc space techniques, advancing the understanding of their model theory and extending methods to partial differential fields with automorphisms.

Contribution

It establishes the full Zilber's dichotomy for DCFA by employing arc space techniques, removing previous limitations and generalizing to partial differential fields with automorphisms.

Findings

Full Zilber's dichotomy proved for DCFA

Arc space techniques effectively used in this context

Method generalizes to partial differential fields with automorphisms

Abstract

The theory of difference-differential fields of characteristic zero has a model-companion denoted by $\it DCFA$. Previously we proved a weak version of Zilber's dichotomy for $\it DCFA$. In this paper we use arc spaces techniques as developed by Moosa, Pillay and Scanlon to suppress the extra hypothesis needed before and prove the full Zilber's dichotomy for $\it DCFA$, we also state how these techniques generalise to partial differential fields with an automorphism.