Submission date: 6 November 2012.

Abstract:

We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z_p inside Q_p uniformly for all p. For any fixed finite extension of Q_p, we give an existential formula and a universal formula in the ring language which define the valuation ring.

This paper will appear in Annals of Pure and Applied Logic.

Mathematics Subject Classification: Primary 11D88, 11U09; Secondary 11U05

Keywords and phrases: Definability, Diophantine sets, Hilbert's Tenth Problem

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