Oberseminar Modelltheorie, Geometrie und Gruppentheorie, Münster, 1st December 2022

Title. Existential theories of classes of fields — henselian and otherwise

Abstract. I have spoken before about joint work with Fehm on the existential theory of power series fields $F(\!(t)\!)$, and more recently on work with Dittmann and Fehm on the analogous theories in a language with an additional parameter for the uniformizer. In the former case we found a transfer of decidability: the existential theory is decidable if and only if the existential theory of $F$, as a field, is decidable. In the latter case, we obtained the same transfer, but in positive characteristic this is conditional on a consequence of local uniformization, a major open conjecture. In an ongoing project (again with Fehm), we broaden these transfer results to deal with classes of residue fields. One surprising result gives Turing equivalences between the existential theory of $\mathbb{Q}$ and a number of existential theories of henselian and large fields.