Submission date: 20 January 2014.

Abstract:

We give model theoretic criteria for \exists\forall and \forall\exists-formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an \exists \forall-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudo-finite, or hilbertian. We also obtain \forall \exists-formulas φ_2 and φ_4 such that φ_2 defines uniformly k[[t]] in k((t)) whenever k is finite or the function field of a real or complex curve, and φ_4 does the job if k is any number field.

Mathematics Subject Classification: 03C60, 12L12, 12J10

Keywords and phrases:

Full text arXiv 1401.4813: pdf, ps.