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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: hollyj@ohsu.edu
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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

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