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- Title
DEFINABLE HENSELIAN VALUATION RINGS.
- Authors
PRESTEL, ALEXANDER
- 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.
- Subjects
HENSELIAN rings; VALUATION theory; DEFINABILITY theory (Mathematical logic); ALGEBRAIC curves; ALGEBRAIC field theory
- Publication
Journal of Symbolic Logic, 2015, Vol 80, Issue 4, p1260
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2014.52