Analytic continuations of $\log$-$\exp$-analytic germs

Abstract:

We describe maximal, in a sense made precise, $\mathbb {L}$-analytic continuations of germs at $+\infty$ of unary functions definable in the o-minimal structure $\mathbb {R}_\textrm {an,exp}$ on the Riemann surface $\mathbb {L}$ 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 $\mathcal {R}_{\text {poly}}$ of the valuation ring of all polynomially bounded definable germs.