Submission date: 4 August 2014.

Abstract:

In this paper, we combine classical techniques of model theory of p-adic subanalytic sets with results of tropical analytic geometry to obtain a result of effective model-completeness. We consider languages L_F=( +,.,0,1, P_n, f; for all n natural number and f in F) where F is a family of restricted analytic functions. Definable sets in this language are a collection of p-adic subanalytic sets. The main result of the paper gives conditions on F so that the structure with underlying set Z_p (the ring of p-adic integers) in this language is effectively model-complete. An interesting example of language satisfying our hypotheses is the case where F={exp(px)}. This example gives a structure of exponential ring to Z_p which is a natural p-adic equivalent to the (restricted) real exponential field.

Mathematics Subject Classification: 03C10

Keywords and phrases:

Full text arXiv 1408.0610: pdf, ps.