The existential theory of equicharacteristic henselian valued fields

Abstract

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 ${\mathbb{F}}_q((t))$.

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• Axiomatizing the existential theory of Fq((t))
• Characterizing diophantine henselian valuation rings and valuation ideals
• An undecidability result for the asymptotic theory of $p$-adic fields
• Ax-Kochen-Ershov principles for finitely ramified henselian fields
• Notes on extremal and tame valued fields
• Decidability via the tilting correspondence
• Universal-existential theories of fields
• Two examples concerning existential undecidability in fields
• Existential uniform $p$-adic integration and descent for integrability and largest poles