Model theory of differential fields with finite group actions

Abstract

Let [Formula: see text] be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in [Formula: see text] commuting derivations equipped with a [Formula: see text]-action by differential field automorphisms. In the language of [Formula: see text]-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 [Formula: see text]. 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 [Formula: see text] is supersimple (but unstable when [Formula: see text] 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 bounded 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).