The existential theory of equicharacteristic henselian valued fields

Anscombe, Sylvy; Fehm, Arno

doi:10.2140/ant.2016.10.665

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The existential theory of equicharacteristic henselian valued fields

Sylvy Anscombe and Arno Fehm

Vol. 10 (2016), No. 3, 665–683

DOI: 10.2140/ant.2016.10.665

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 $F_{q} ((t))$ .

Keywords

model theory, henselian valued fields, decidability, diophantine equations

Mathematical Subject Classification 2010

Primary: 03C60

Secondary: 12L12, 12J10, 11U05, 12L05

Milestones

Received: 18 September 2015

Revised: 9 February 2016

Accepted: 15 March 2016

Published: 12 June 2016

Authors

Sylvy Anscombe
Jeremiah Horrocks Institute University of Central Lancashire Preston PR1 2HE United Kingdom

Arno Fehm
School of Mathematics University of Manchester Oxford Road Manchester M13 9PL United Kingdom

Vol. 10, No. 3, 2016