We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property (...) is first order. (shrink)

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We give geometric axioms for existentially closed Hasse fields. We prove a quantifier elimination result for existentially closed n-truncated Hasse fields and characterize them as reducts of existentially closed Hasse fields.

We state and prove a generalization of Ax’s theorem on the transcendence degree of solutions of the differential equation of the exponential map. We also discuss a positive characteristic analogue of this theorem.

Ethical principles concerning foresters were presented in this article in the light of the forest law passed in the interwar period for the whole country, as well as in the light of the documents issued by the local forest administration. For analysis were submitted especially archive documents on foresters’ responsibility for forest goods: wood and game. Ethical attitude of foresters translated, among other things, into efficient functioning of forest management all over Greater Poland.

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of the λ-functions. Along the way, we give new geometric axioms of DCFp.