Analytic continuations of log-exp-analytic germs

TL;DR

This paper studies the maximal analytic continuations of certain definable functions at infinity within the o-minimal structure R_an,exp, providing bounds on their complexity and extending Wilkie's theorem on complex continuations.

Contribution

It introduces a precise notion of maximal analytic continuation for germs definable in R_an,exp and establishes bounds on the complexity of their inverses and extensions.

Findings

Provides an upper bound on the logarithmic-exponential complexity of inverses.

Strengthens Wilkie's theorem on complex analytic continuations.

Describes maximal analytic continuations of definable germs at infinity.

Abstract

We describe maximal, in a sense made precise, analytic continuations of germs at infinity of unary functions definable in the o-minimal structure R_an,exp on the Riemann surface of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie's theorem on definable complex analytic continuations of germs belonging to the residue field of the valuation ring of all polynomially bounded definable germs.