## 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…

## Ilyashenko algebras based on definable monomials: the construction (base step)

Let $\H$ be the Hardy field of $\Ranexp$, and let $M$ be a multiplicative $\RR$-subvector space of $\H^{>0}$; I continue to assume in this post that $M$ is a pure scale. A germ $h \in \H^{>0}$ is small if $h(x) \to 0$ as $x \to +\infty$. The construction discussed here works for the following type…

## Ilyashenko algebras based on definable monomials, revisited

In order to address some of the questions raised in this post, I introduce here some relevant definitions and recast the construction of Ilyashenko algebras based on $\log$ monomials using these new notions. As before, let $\H$ be the Hardy field of germs at $+\infty$ of all $h:\RR \into \RR$ definable in $\Ranexp$, and denote…

## Quasianalytic Ilyashenko algebras

The first version of my preprint on quasianalytic Ilyashenko algebras is available on arXiv. Feel free to leave comments here! Here is the abstract: I construct a quasianalytic field $\F$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\F$ is closed under differentiation and log-composition; in…

## Ilyashenko algebras based on definable monomials

This post generalizes the construction of quasianalytic Ilyashenko algebras based on log monomials to certain other definable monomials. This construction is joint work with my student Zeinab Galal. Recall that $\H = \Hanexp$ is the Hardy field of germs at $+\infty$ of univariate functions definable in $\Ranexp$, and that $\I$ is the set of all…

## Ilyashenko algebras based on log monomials

This post describes the construction of quasianalytic algebras of functions with simple logarithmic transseries as asymptotic expansions. The construction is based on Ilyashenko’s class of almost regular functions, as introduced in his book on Dulac’s problem. This class forms a group under composition, but it is not closed under addition or multiplication; to obtain a…

## Holomorphic extensions of definable germs

(Joint work with Tobias Kaiser) Recall from this post that not all germs in $\H$ have a holomorphic extension that maps definable real domains to definable real domains. In fact, the extension $\t_a$ of the translation $t_a$, for $a\gt 0$, does not even map real domains to real domains. So, in order to describe the…

## Angular level

(Joint work with Tobias Kaiser) The goal of this post is to introduce a rough measure of size for a real domain $U$, based on the level of its boundary function $f_U$. As before, “definable” means “definable in $\Ranexp$”. Let $\I$ be the set of all infinitely increasing $\,f \in \H$ and set \$\bo:= \H_{\gt…