Axiomatizing the existential theory of 𝔽q((t))

Anscombe, Sylvy; Dittmann, Philip; Fehm, Arno

doi:10.2140/ant.2023.17.2013

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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

DOI: 10.2140/ant.2023.17.2013

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 für Algebra Technische Universität Dresden Dresden Germany

Arno Fehm
Institut für Algebra Technische Universität Dresden Dresden Germany

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Vol. 17, No. 11, 2023