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

Let $M \subseteq \H^{>0}$ be a pure scale on standard power domains. In this post, I gave the base step of the construction of a qaa field $(\F,L,T)$ as claimed here. The goal of this post is to finish this construction. Step 0.5: apply a $\log$-shift to the qaa field $(\F_0,L_0,T_0)$, that is, set \F’_1…

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…