Effective model-completeness for p-adic analytic structures

TL;DR

This paper extends the model-theoretic understanding of p-adic integers by constructing effective model-complete expansions of restricted analytic languages, with applications to p-adic exponential rings.

Contribution

It introduces a method to achieve model-completeness for p-adic analytic structures with effective bounds, generalizing previous results.

Findings

Constructed expansions of restricted analytic functions ensuring model-completeness.

Provided conditions for effective model-completeness in p-adic structures.

Applied results to p-adic exponential rings.

Abstract

In their paper 'p-adic and real subanalytic sets, J. Denef and L. van den Dries prove that the theory of the ring of p-adic integers admits the elimination of quantifiers in the language of p-adic restricted analytic functions expanded by a division symbol. In this paper, we are interested in restriction of this language: Let F be any family of restricted analytic functions, we construct an expansion of F so that the theory of the ring of p-adic integers is model-complete in the corresponding language. Next, we give conditions on F so that the model-completeness is effective. Finally, we apply our results in the context of p-adic exponential rings.