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Axiomatizing the existential theory of $\mathbb{F}_q((t))$

Sylvy Anscombe, Philip Dittmann and Arno Fehm

Vol. 17 (2023), No. 11, 2013–2032
Abstract

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.

Keywords
local fields, positive characteristic, henselian valued field, existential theory, decision algorithm, resolution of singularities, local uniformization
Mathematical Subject Classification
Primary: 03C60, 11D88, 11G25, 12L05
Milestones
Received: 2 June 2022
Revised: 15 January 2023
Accepted: 6 March 2023
Published: 3 October 2023
Authors
Sylvy Anscombe
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Université Paris Cité
Paris
France
Philip Dittmann
Institut fĂĽr Algebra
Technische Universität Dresden
Dresden
Germany
Arno Fehm
Institut fĂĽr Algebra
Technische Universität Dresden
Dresden
Germany

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