The first-order theories of local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are far less well understood than their characteristic zero analogues: the fields of real, complex and p-adic numbers. On the other hand, the existential theory of an equicharacteristic henselian valued field in the language of valued fields is controlled by the existential theory of its residue field. One is decidable if and only if the other is decidable. When we add a parameter to the language, things get more complicated. Denef and Schoutens gave an algorithm, assuming resolution of singularities, to decide the existential theory of rings like Fp[[t]], with the parameter t in the language. I will discuss their algorithm and present a new result (from ongoing work, with Dittmann and Fehm) that weakens the hypothesis to a form of local uniformization, and which works in greater generality.
- Théorie des Modèles et Groupes