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…

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}…

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…

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