Patrick Speissegger
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…
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…
(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…