Algebra & Number Theory

The existential theory of equicharacteristic henselian valued fields

Sylvy Anscombe and Arno Fehm

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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)).

Article information

Algebra Number Theory, Volume 10, Number 3 (2016), 665-683.

Received: 18 September 2015
Revised: 9 February 2016
Accepted: 15 March 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]
Secondary: 12L12: Model theory [See also 03C60] 12J10: Valued fields 11U05: Decidability [See also 03B25] 12L05: Decidability [See also 03B25]

model theory henselian valued fields decidability diophantine equations


Anscombe, Sylvy; Fehm, Arno. The existential theory of equicharacteristic henselian valued fields. Algebra Number Theory 10 (2016), no. 3, 665--683. doi:10.2140/ant.2016.10.665.

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