Liouville closure?

Since $\K$ is not closed under exponentiation, it is fair to ask whether the Liouville closure of $\K$ is again an Ilyashenko field. This appears to be the case; indeed, working with Margaret Thomas, we conjecture that if $(\la, \M, T)$ is any Ilyashenko field with $\M$ the set of (not necessarily convergent) LE-monomials, then…

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…

Ilyashenko algebras: putting it all together

Let $f = (f_0, \dots, f_k)$ be such that each $f_i \in \H$ is infinitely increasing and $f_0 \gt \cdots \gt f_k$. To see what it takes to generalize our construction of the Ilyashenko algebra $(\F,L,T)$ to more general monomials $f$, recall the construction in the following schematic:  \begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix}…