Patrick Speissegger
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…
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…
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…
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…