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Canonical forms for definable subsets of algebraically closed and real closed valued fields

Published online by Cambridge University Press:  12 March 2014

Jan E. Holly
Jan E. Holly*
Affiliation:
Robert S. Dow Neurological Sciences Institute, Portland, Oregon 97209-1595, E-mail: [email protected]
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Abstract

We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming “Swiss cheeses” in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in “most” valued fields F, if f(x), g(x) ∈ F[x] and v is the valuation map, then the set {x: v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of “valued trees”, which we define formally.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 3 , September 1995 , pp. 843 - 860
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[BGR]Bosch, S., Güntzer, U., and Remmert, R., Non-Archimedean analysis, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
[CD]Cherlin, Gregory and Dickmann, Max A., Real closed rings. II: Model theory, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 213–231.CrossRefGoogle Scholar
[De]Delon, Françoise, Espaces ultramétriques this Journal, vol. 49 (1984), pp. 405–424.Google Scholar
[Di]Dickmann, M. A., Elimination of quantifiers for ordered valuation rings this Journal, vol. 52 (1987), pp. 116–128.Google Scholar
[FP]Fresnel, Jean and van der Put, Marius, Géométrie analytique rigide et applications, Birkhäuser, Boston, Massachusetts, 1981.Google Scholar
[HM]Haskell, Deirdre and Macpherson, Dugald, Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113–162.CrossRefGoogle Scholar
[Ho]Holly, Jan Élise, Definable equivalence relations and disc spaces of algebraically closed valued fields, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1992.Google Scholar
[MMD]Macintyre, Angus, McKenna, Kenneth, and van den Dries, Lou, Elimination of quantifiers in algebraic structures, Advances in Mathematics, vol. 47 (1983), pp. 74–87.CrossRefGoogle Scholar
[Po]Poizat, Bruno, Une théorie de Galois imaginaire this Journal, vol. 48 (1983), pp. 1151–1170.Google Scholar
[RTV]Rammal, R., Toulouse, G., and Virasoro, M. A., Ultrametricity for physicists, Reviews of Modern Physics, vol. 58 (1986), pp. 765–788.CrossRefGoogle Scholar
[Ro]Robinson, Abraham, Complete theories, North-Holland, Amsterdam, 1956.Google Scholar