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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
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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

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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