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Preprint Number 531
531. Raf Cluckers, Jamshid Derakhshan, Eva Leenknegt, and Angus Macintyre Uniformly defining valuation rings in Henselian valued fields with finite or pseudofinite residue fields Email: , , , 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 existentialuniversal 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 Full text: pdf.

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