The existential theory of equicharacteristic henselian valued fields

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of Fq((t))

Definable valuations on ordered fields

We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings $\mathcal{L}_{\mathrm{r}}$ and in the richer language of ordered rings $\mathcal{L}_{\mathrm{or}}$. We analyse and compare definability in both languages and show the following contrary results: while there are convex valuations that are definable in the language $\mathcal{L}_{\mathrm{or}}$ but not in the language $\mathcal{L}_{\mathrm{r}}$, any $\mathcal{L}_{\mathrm{or}}$-definable henselian valuation is already $\mathcal{L}_{\mathrm{r}}$-definable. To prove the latter, we show that the value group and the ordered residue field of an ordered henselian valued field are stably embedded (as an ordered abelian group, respectively as an ordered field). Moreover, we show that in almost real closed fields any $\mathcal{L}_{\mathrm{or}}$-definable valuation is henselian.Comment: 17 page

Panorama of p-adic model theory

ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud par une revue de la bibliographie

An example of a PP-minimal structure without definable Skolem functions

We show there are intermediate $P$-minimal structures between the semi-algebraic and sub-analytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are $P$-minimal structures which do not admit classical cell decomposition.Comment: 9 pages, (added missing grant acknowledgement

Axiomatizing the existential theory of Fq((t))

We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of - and therefore an algorithm to decide - the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true.Comment: New Remark 4.18 and expanded Remark 2.

Diophantine problems over tamely ramified fields

Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen-Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic $0$ and unramified mixed characteristic. It also encompasses the conditional existential decidability results known for $\mathbb{F}_p((t))$ and its finite extensions, due to Denef-Schoutens. On the other hand, it also applies to the setting of infinite ramification, providing us with an abundance of infinitely ramified extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$ that are existentially decidable.Comment: Local corrections, improved exposition, new Section

Characterizing Diophantine Henselian valuation rings and valuation ideals

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal