Algebra & Number Theory

The existential theory of equicharacteristic henselian valued fields

Sylvy Anscombe and Arno Fehm

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

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.

Export citation


  • S. Anscombe and A. Fehm, “Characterizing diophantine henselian valuation rings and valuation ideals”, preprint, 2016.
  • W. Anscombe and J. Koenigsmann, “An existential ${\emptyset}$-definition of $\mathbb{F}\sb q[\![t]\!]$ in $\mathbb{F}\sb q(\!(t)\!)$”, J. Symb. Log. 79:4 (2014), 1336–1343.
  • S. Anscombe and F.-V. Kuhlmann, “Notes on extremal and tame valued fields”, 2016, hook \posturlhook. To appear in J. Sym. Logic.
  • R. Cluckers, J. Derakhshan, E. Leenknegt, and A. Macintyre, “Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields”, Ann. Pure Appl. Logic 164:12 (2013), 1236–1246.
  • J. Denef and H. Schoutens, “On the decidability of the existential theory of $\mathbb{F}\sb p[\![t]\!]$”, pp. 43–60 in Valuation theory and its applications (Saskatoon, 1999), vol. II, edited by F.-V. Kuhlmann et al., Fields Inst. Commun. 33, Amer. Math. Soc., Providence, RI, 2003.
  • I. Efrat, Valuations, orderings, and Milnor $K$-theory, Mathematical Surveys and Monographs 124, Amer. Math. Soc., Providence, RI, 2006.
  • A. J. Engler and A. Prestel, Valued fields, Springer, Berlin, 2005.
  • A. Fehm, “Existential $\varnothing$-definability of Henselian valuation rings”, J. Symb. Log. 80:1 (2015), 301–307.
  • Y. S. Gurevich and A. I. Kokorin, “Universal equivalence of ordered Abelian groups”, Algebra i Logika Sem. 2:1 (1963), 37–39. In Russian.
  • J. Koenigsmann, “Undecidability in number theory”, pp. 159–195 in Model theory in algebra, analysis and arithmetic, edited by H. D. Macpherson and C. Toffalori, Lecture Notes in Math. 2111, Springer, Heidelberg, 2014.
  • F.-V. Kuhlmann, “Elementary properties of power series fields over finite fields”, J. Symbolic Logic 66:2 (2001), 771–791.
  • F.-V. Kuhlmann, “Value groups, residue fields, and bad places of rational function fields”, Trans. Amer. Math. Soc. 356:11 (2004), 4559–4600.
  • F.-V. Kuhlmann, “Valuation theory”, book in progress, 2011, hook \posturlhook.
  • F.-V. Kuhlmann, “The algebra and model theory of tame valued fields”, J. Reine Angew. Math. (online publication May 2014).
  • F.-V. Kuhlmann, M. Pank, and P. Roquette, “Immediate and purely wild extensions of valued fields”, Manuscripta Math. 55:1 (1986), 39–67.
  • D. Marker, Model theory: an introduction, Graduate Texts in Mathematics 217, Springer, New York, 2002.
  • A. Prestel, “Definable Henselian valuation rings”, J. Symb. Log. 80:4 (2015), 1260–1267.
  • A. Prestel and C. N. Delzell, Mathematical logic and model theory: a brief introduction, Springer, London, 2011.
  • J.-P. Serre, Local fields, Graduate Texts in Mathematics 67, Springer, New York, 1979.