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 G 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).

中文翻译：

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具有有限群作用的微分场模型理论

让 G是一个有限群。我们探讨了特征零微分场类的模型理论性质 米通勤派生配备 G-微分域自同构的作用。在语言中 G-微分环（即带有用于推导和自同构的符号的环语言），我们证明这个类有一个模型伴侣——表示 G -DCF0,米. 然后我们部署在第一作者的论文 [DM Hoffmann, Model theoretic dynamics in a Galois fashion,安。纯应用。逻辑 170(7) (2019) 755–804] 表明任何模型 G -DCF0,米是超级简单的（但不稳定时 G是非平凡的），一个 PAC 微分场（因此在第二作者和 Tressl 的意义上差异很大 [Differentially large fields, preprint (2020), arXiv:2005.00888，可在https://arxiv.org/abs/2005.00888])，并承认在添加参数元组后消除想象。我们还解决了有界 PAC 微分场理论的模型完整性和超简单性（扩展了 Chatzidakis 和 Pillay [通用结构和简单理论，安。纯应用。逻辑 95(1998) 71-92] 关于有界 PAC 场）。