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

Our preprint on the analytic continuation of germs at $+\infty$ of unary functions definable in $\Ranexp$ is now on the ArXiv. Here is its introduction:

The o-minimal structure $\Ranexp$, see van den Dries and Miller or van den Dries, Macintyre and Marker, is one of the most important regarding applications, because it defines all elementary functions (with the necessary restriction on periodic ones such as $\sin$ or $\cos$). Holomorphic functions definable in $\Ranexp$ have turned out to be crucial in applications to diophantine geometry, see for instance Pila and Peterzil and Starchenko.

It is known from the first two papers above that every function definable in the o-minimal structure $\Ranexp$ is piecewise analytic. This implies that, if $f$ is the germ at $+\infty$ of a one-variable function definable in $\Ranexp$, also called a $\log$-$\exp$-analytic germ here, there is an open domain $U \subseteq \CC$ and a complex analytic continuation $\ff:U \into \CC$ of $\f$, or an open domain $\frU subseteq \LL$ and an $\LL$-analytic continuation $\f:\frU \into \LL$, where $\LL$ is the Riemann surface of the logarithm. Concerning complex analytic continuations $\ff:U \into \CC$ of $f$, it is shown by Kaiser that $\ff$ can be chosen to be definable. Wilkie characterizes those $f$ for which $\ff$ extends definably on some right translate of a sector properly containing a right half-plane of $\CC$; he then applies this continuation result to a diophantine problem.

The aim of our paper is to describe $\LL$-analytic continuations $\f:\frU \into \LL$: we find a maximal $\frU$ (in a sense to be made precise) such that $\f$ is half-bounded, that is, either $\f$ or $1/\f$ is bounded. We obtain this statement from the more precise Continuation Theorem, which only applies to infinitely increasing $f$, that is, those $f$ for which $\lim_{x \to +\infty} f(x) = +\infty$ holds.

These $\LL$-analytic continuations of $f$ depend on two integer-valued quantities associated to $f$: the exponential height $\eh(f)$ of $f$ and the level $\level(f)$ of $f$. The former measures the logarithmic-exponential complexity of $f$; roughly speaking, if $f$ is unbounded, then $\eh(\exp \circ f) = \eh(f)+1$, while if $f$ is bounded, then $\eh(\exp \circ f) = \eh(f)$. The latter measures the exponential order of growth of the germ $f$; we refer the reader to Marker and Miller [Levelled o-minimal structures. (English summary)
Real algebraic and analytic geometry (Segovia, 1995).
Rev. Mat. Univ. Complut. Madrid 10 (1997), Special Issue, suppl., 241–249] for details. The level extends to all $\log$-$\exp$-analytic germs in an obvious manner.


We show that $\level(f) \le \eh(f)$, for all $\log$-$\exp$-analytic germs $f$. The two are not equal in general: we have $\level(x+e^{-x}) = 0 \ne 1 = \eh(x+e^{-x})$.

What we find in the Continuation Corollary is that, if $\f:\frU \into \LL$ is a maximal, half-bounded $\LL$-analytic continuation of $f$, then the size (in a sense to be made precise) of $\frU$ is determined by $\eh(f)$ and, conversely, that the size of $\frU$ determines an upper bound on $\eh(f)$. Moreover, if $f$ is infinitely increasing, we also find that $\f$ is injective and, in this case, $\level(f)$ determines the size of the image $\f(\frU)$; this is summarized in what we call the Simplified Continuation Theorem. The Continuation Theorem is more technical, but it is the central result of our paper, as our applications actually rely on these extra technicalities.

We include two applications of the Continuation Theorem and its corollaries. In the first application, we give an upper bound on $\eh(f^{-1})$, in terms of $\eh(f)$ and $\level(f)$, of an infinitely increasing $\log$-$\exp$-analytic germ $f$, where $f^{-1}$ denotes the compositional inverse of $f$. In the second application, we strengthen Wilkie’s Theorem 1.11 on definable complex analytic continuations of germs belonging to the residue field of the valuation ring of all polynomially bounded $\log$-$\exp$-analytic germs.

The main motivation for us to prove the Continuation Theorem, however, is to show that all principal monomials of $\H$ can be used in asymptotic expansions to obtain a quasianalytic Ilyashenko field $\K$ extending the Ilyashenko field $\F$ constructed in my paper. The details of this application, some of which are outlined here, its motivations and the construction of $\K$ are the subject of a forthcoming paper.

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