On the decidability of the p-adic exponential ring.

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Abstract

Let Zp be the ring of p-adic integers and Ep be the map x-->exp(px) where exp denotes the exponential map determined by the usual power series. It defines an exponential ring (Zp, + , . , 0, 1, Ep). The goal of the thesis is to study the model theory of this structure. In particular, we are interested by the question of the decidability of this theory. The main theorem of the thesis is: Theorem: If the p-adic Schanuel's conjecture is true, then the theory of (Zp, + , . , 0, 1, Ep) is decidable. The proof involves: 1- A result of effective model-completeness (chapters 3 and 4): If F is a family of restricted analytic functions (i.e. power series with coefficients in the valuation ring and convergent on Zp) closed under decomposition functions and such that the set of terms in the language LF= (+, . , 0, 1, f; f in F) is closed under derivation, then we prove that the theory of Zp in the language LF is model-complete. And furthermore, if each term of LF has an effective Weierstrass bound, then the model-completeness is effective. 2- A resolution of the decision problem for existential formulas (assuming Schanuel's conjecture) in chapter 5. We also consider the problem of the decidability of the structure (Op, + , . , 0, 1, |, E_p) where Op denotes the valuation ring of Cp. We give a positive answer to this question assuming the p-adic Schanuel's conjecture.

Keyword(s)

decidability; model theory; p-adic exponential

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