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