The theory DCF$_p$A exists for $p>0$

TL;DR

This paper proves the existence of a model companion for differential-difference fields in positive characteristic, extending the theory of differentially closed fields with automorphisms to characteristic p>0.

Contribution

It establishes the model companion DCF$_p$A for differential-difference fields in characteristic p>0, and provides alternative axiomatizations for DCF and DCF$_0$A.

Findings

Existence of DCF$_p$A for p>0

New axiomatizations for DCF and DCF$_0$A

Extension of differential field theory to positive characteristic

Abstract

We prove that the (elementary) class of differential-difference fields in characteristic $p>0$ admits a model-companion. In the terminology of Chatzidakis-Pillay, this says that the class of differentially closed fields of characteristic $p$ equipped with a generic differential-automorphism is elementary; i.e., DCF$_p$A exists. Along the way, we provide alternative first-order axiomatisations for DCF (differentially closed fields) and also for DCF$_0$A.