Henselian valued fields and inp-minimality

We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic $0$ in the RV language.Comment: v.2: 15 pages, minor corrections and presentation improvements; accepted to the Journal of Symbolic Logi

Finite Undecidability in Fields II: PAC, PRC and PpC Fields

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if \mbox{Cons}(\Sigma) is undecidable for every nonempty finite \Sigma \subseteq \mbox{Th}(K; \mathcal{L}). We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields), Haran (for PRC fields), and Efrat (for PpC fields) to prove all PAC, PRC, and (bounded) PpC fields are finitely undecidable. This work is drawn from the author's PhD thesis and is a sequel to arXiv:2210.12729.Comment: 24 page

NIP\rm NIP, and NTP2{\rm NTP}_2 division rings of prime characteristic

Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ division ring of characteristic $p$ and show that an $\rm NIP$ division ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ division rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\rm NIP$ or simple difference fields