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 p-adic analogue of the very weak language (<) on the real numbers to define a notion of minimality on the p-adics. Finally we give a universal-existential 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: p-adic numbers, quantifier elimination, cell decomposition, P- minimality, o-minimality, Hilbert's Tenth Problem, definability.

Full text: pdf.