DEFINABLE HENSELIAN VALUATIONS

Definable Henselian Valuation Rings

We give model theoretic criteria for the existence of∃∀ and∀∃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 ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...

Henselian Implies Large

In this note we show that the quotient field of a domain which is Henselian with respect to a non-trivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

Henselian Function Fields

Definable Representations of Definable C Groups

Let G be a compact affine definable C group and let r be∞ or ω. We prove that the representative definable C functions on G is dense in the space of continuous functions on G. Moreover we compare the category of compact affine definable C groups D with that of compact real algebraic groups A.

Existential ∅-Definability of Henselian Valuation Rings

In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F [[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. §