| Résume | In joint work with Dittmann and Fehm (2023) we showed that decidability of the existential theory of (Fq((t)),t) follows from a property we called (R4), studied before by Kuhlmann in connection with problems of local uniformization. The property (R4) is: every large field k is existentially closed in every extension that admits a k-rational k-place. The result from 2023 was an improvement on a result of Denef and Schoutens (2003), who showed that Resolution of Singularities implies the same decidability problem, and related to a previous result from other work of Fehm and I (2016). Recently Dittmann has proved a general statement, also dependent on (R4), that extends the known results for existential theories of henselian valuation rings in a range of settings.
More recently, with Fehm (2026), we studied weakenings of (R4) to deal only with existential closedness (again of large k in extensions with k-rational k-places) restricted to existential formulas with at most a certain number of quantifiers. In turn this yields the decidability of the corresponding fragments of the existential theory of k((t)), relative to the corresponding fragment of the existential theory of k.
In this talk I will explain this newer work (previous GTM talks having already addressed the other results), and explore further potential extensions of this fragmented approach which seem to be related to Kuhlmann's work on valuation regular function fields over defectless fields, from the theory of tame valued fields. |
