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Preprint Number 281
281. Raf Cluckers and Eva Leenknegt A version of padic minimality Email: Submission date: 22 November 2010. Abstract: We introduce a very weak language L_p on Q_p, which is just rich enough to have the same definable subsets of the line Q_p than one has using the ring language. We prove that the only definable functions in the language L_p are trivial functions. We further give a definitional expansion L'_p of L_p in which Q_p has quantifier elimination and we obtain a weak cell decomposition. Our language L_p can serve as the padic analogue of the very weak language (<) on the real numbers to define a notion of minimality on the padics. Finally we give a universalexistential definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly in all p, and in all finite field extensions. Mathematics Subject Classification: Primary 03C10, 11U05, 03C07; Secondary 03C64. Keywords and phrases: padic numbers, quantifier elimination, cell decomposition, P minimality, ominimality, Hilbert's Tenth Problem, definability. Full text: pdf.

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