## Closure under composition?

The Ilyashenko field we construct in our most recent paper is not closed under composition, or even under $\log$-composition. How do we know this? By construction, every LE-series that is the asymptotic expansion of a germ $f$ in the Ilyashenko field $\K$ is an LE-series with convergent LE-monomials, but in general the series is divergent. Since the infinite…

## Ilyashenko algebras based on transserial asymptotic expansions

Our preprint extending my earlier construction of Ilyashenko algebras is now on the arXiv. The purpose of this paper is to extend Ilyashenko’s construction of the class of germs at $+\infty$ of almost regular functions to obtain a Hardy field containing them.  In addition, each germ in this Hardy field is uniquely characterized by an asymptotic…

## 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…

## 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…