Vol. 10, No. 3, 2016

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The existential theory of equicharacteristic henselian valued fields

Sylvy Anscombe and Arno Fehm

Vol. 10 (2016), No. 3, 665–683
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 Fq((t)).

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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