Model Theory of Differentially Closed Fields with a Generic Automorphism

TL;DR

This paper establishes the model companion of difference-differential fields, called DCFA, and explores its properties, including SU rank, Zilber's dichotomy, and Abelian definable groups, advancing the model theory of these fields.

Contribution

It proves the existence of the model companion DCFA and analyzes its key properties, including rank, dichotomy, and group structure, extending previous results in difference-differential fields.

Findings

DCFA is the model companion of difference-differential fields.

SU rank relates to transcendence degree in DCFA.

A version of Zilber's dichotomy is established for DCFA.

Abstract

E. Hrushovski proved tha the theory of difference-differential fields has a model companion. We prove this result and other maind properties of this theory that we call DCFA. We describe the SU rank a its relation with transcendence degree. We prove a version of Zilber's dichotomy and we give a description of Abelian definable groups.