Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-30T04:15:19.745Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative decidability and definability in henselian valued fields

Published online by Cambridge University Press:  12 March 2014

Joseph Flenner
Joseph Flenner*
Affiliation:
University of Notre Dame, Department of Mathematics, 255 Hurley Hall, Notre Dame, IN 46556, USA, E-mail: jflenner@nd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))?

Two applications are given: first, a constructive quantifier elimination relative to the leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation.

Keywords

Henselian valued fielddecidabilityquantifier eliminationelimination of imaginaries
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1240 - 1260
Copyright
Copyright © Association for Symbolic Logic 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ax, James and Kochen, Simon, Diophantine problems over local fields. I, American Journal of Mathematics, vol. 87 (1965), pp. 605–630.CrossRefGoogle Scholar
[2]Ax, James and Kochen, Simon, Diophantine problems over local fields. II. A complete set of axioms for p-adic number theory, American Journal of Mathematics, vol. 87 (1965), pp. 631–648.CrossRefGoogle Scholar
[3]Ax, James and Kochen, Simon, Diophantine problems over local fields. III. Decidable fields, Annals of Mathematics (2), vol. 83 (1966), pp. 437–456.CrossRefGoogle Scholar
[4]Cluckers, Raf and Loeser, François, b-minimality, Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195–227.CrossRefGoogle Scholar
[5]Cohen, Paul J., Decision procedures for real and p-adic fields, Communications on Pure and Applied Mathematics, vol. 22 (1969), pp. 131–151.CrossRefGoogle Scholar
[6]Eršov, Ju. L., On the elementary theory of maximal normed fields, Doklady Akademii Nauk SSSR, vol. 165 (1965), pp. 21–23 (Russian).Google Scholar
[7]Haskell, Deirdre, Hrushovski, Ehud, and Macpherson, Dugald, Definable sets in algebraically closed valued fields: elimination of imaginarles, Journal für die reine und angewandte Mathematik, vol. 597 (2006), pp. 175–236.Google Scholar
[8]Haskell, Deirdre, Hrushovski, Ehud, and Macpherson, Dugald, Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, vol. 30, Association for Symbolic Logic, 2008.Google Scholar
[9]Holly, Jan E., Canonical forms for definable subsets of algebraically closed and real closed valued fields, this Journal, vol. 60 (1995), no. 3, pp. 843–860.Google Scholar
[10]Holly, Jan E., Prototypes for definable subsets of algebraically closed valued fields, this Journal, vol. 62 (1997), no. 4, pp. 1093–1141.Google Scholar
[11]Hrushovski, Ehud and Kazhdan, David, Integration in valued fields, Algebraic geometry and number theory, Progress in Mathematics, vol. 253, Birkhäuser, Boston, MA, 2006, pp. 261–405.CrossRefGoogle Scholar
[12]Hrushovski, Ehud and Martin, Ben, Zeta functions from definable equivalence relations, 2006, Preprint http://arxiv.org/abs/matli/0701011.Google Scholar
[13]Kuhlmann, Franz-Viktor, Quantifier elimination for henselian fields relative to additive and multiplicative congruences, Israel Journal of Mathematics, vol. 85 (1994), no. 1–3, pp. 277–306.CrossRefGoogle Scholar
[14]Macintyre, Angus, On definable subsets of p-adic fields, this Journal, vol. 41 (1976), no. 3, pp. 605–610.Google Scholar
[15]Mellor, T., Imaginaries in real closed valued fields, Annals of Pure and Applied Logic, vol. 139 (2006), no. 1–3, pp. 230–279.CrossRefGoogle Scholar
[16]Ribenboim, Paulo, Equivalent forms of Hensel's lemma, Expositiones Mathematicae, vol. 3 (1985), no. 1, pp. 3–24.Google Scholar
[17]Robinson, Abraham, Complete theories, North-Holland Publishing Co., 1956.Google Scholar
[18]Yin, Yimu, Henselianity and the Denef-Pas language, this Journal, vol. 74 (2009), pp. 655–664.Google Scholar