Model theory of differential fields with finite group actions

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-DCF0,m. We then deploy the model-theoretic tools developed in the first author's paper [D. M. Hoffmann, Model theoretic dynamics in a Galois fashion, Ann. Pure Appl. Logic 170(7) (2019) 755-804] to show that any model of G-DCF0,m is supersimple (but unstable when C is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [Differentially large fields, preprint (2020), arXiv:2005.00888, available at https://arxiv.org/abs/2005.00888]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of hounded PAC-differential fields (extending the results of Chatzidakis and Pillay [Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998) 71-92] on bounded PAC-fields).