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DEFINABLE HENSELIAN VALUATION RINGS

Published online by Cambridge University Press:  22 December 2015

ALEXANDER PRESTEL
ALEXANDER PRESTEL*
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF KONSTANZ KONSTANZ, GERMANYE-mail: alex.prestel@uni-konstanz.de
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Abstract

We give model theoretic criteria for the existence of ∃∀ and ∀∃- formulas in the ring language to define uniformly the valuation rings ${\cal O}$ of models $\left( {K,\,{\cal O}} \right)$ of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings ${\cal O}$ of valued henselian fields $\left( {K,\,{\cal O}} \right)$ whose residue class field k is finite, pseudofinite, or hilbertian. We also obtain ∀∃-formulas φ2 and φ4 such that φ2 defines uniformly k[[t]] in k(t) whenever k is finite or the function field of a real or complex curve, and φ4 replaces φ2 if k is any number field.

Keywords

definabilityvaluation rings
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1260 - 1267
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

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