Model theory of differential fields with finite group actions

TL;DR

This paper develops a model theory for differential fields with finite group automorphisms, establishing the existence of a model companion, G-DCF, and analyzing its properties such as supersimplicity and elimination of imaginaries.

Contribution

It introduces the theory G-DCF for G-differential fields, proving its model companion exists and analyzing its model-theoretic properties, extending previous work on PAC fields.

Findings

G-DCF exists as a model companion for G-differential fields.

Models of G-DCF are supersimple but unstable when G is nontrivial.

Theories of bounded PAC-differential fields are model-complete and supersimple.

Abstract

Let G be a finite group. We explore the model theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential field automorphisms. In the language of G-differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a model-companion - denoted G-DCF. We then deploy the model-theoretic tools developed in the first author's paper [11] to show that any model of G-DCF is supersimple (but unstable when G is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [30]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of bounded PAC-differential fields (extending the results of Chatzidakis-Pillay [5] on bounded PAC-fields).