arXiv subjects
Publications and source records attributed to Piotr Błaszkiewicz.
We consider valued fields equipped with an action of a given finite group by automorphisms. We show that the theory of such structures admits a model companion which is axiomatized in an Ax-Kochen/Ershov fashion. We prove this model companion has $\operatorname{NTP}_2$ and that the valuation is definable in the pure field language.
We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. Our argument uses Chebotarev's density theorem. We also discuss some related questions.
Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann we introduce and study Caristi-Kirk and Oettli-Théra ball spaces. We show that if the underlying metric space is complete, then these have a very strong property: every ball contains a singleton ball. This fact provides quick proofs for several results which are equivalent to the Caristi-Kirk Fixed Point Theorem, namely Ekeland's Variational Principles, the Oettli-Théra Theorem, Takahashi's Theorem and the Flower Petal Theorem.