## The Hardy field of $\Ranexp$

(Joint work with Tobias Kaiser; this is essentially a reposting of this post) The goal of this post is to describe the Hardy field $\H = \Hanexp$ of the expansion $\Ranexp$ of the real field by all restricted analytic functions and the exponential function, based on van den Dries, Macintyre and Marker’s papers on $\Ranexp$…

## Ilyashenko algebras based on log monomials

This post describes the construction of quasianalytic algebras of functions with simple logarithmic transseries as asymptotic expansions. (It is a slightly altered version of this post.) The resulting algebra of germs contains Ilyashenko’s set $\A$ of almost regular germs introduced previously. The set $\A$ is closed under composition, but it is not closed under addition…

## Almost regular germs

Let $\xi:S^2 = \RR^2 \cup \{P\} \into \RR^2$ be a real analytic vector field. Let $\C$ be a cell decomposition definable in $\Ran$ as obtained for $\xi\rest{\RR^2}$, and enlarge $\C$ by adding $\{P\}$. Exercise 1 Use o-minimality to show that all but fginitely many singularities of $\xi$ are removable. Based on Exercise 1, we assume…

## Back to reality

Let $\xi:\RR^2 \into \RR^2$ be a vector field of class $C^1$, and assume that the expansion $\RR_\xi$ of the real field by $\xi$ is o-minimal. In the the previous post, I found a $C^1$-cell decomposition $\C$, definable in $\RR_\xi$, and I introduced the forward progression map $f_\xi:B \into B$ associated to this cell decomposition. As…

## Dulac’s Problem

Let $\xi = (\xi_1,\xi_2):\RR^2 \into \RR^2$ be a vector field of class $C^1$. Recall that the singular set of $\xi$ is the set $$Z(\xi):= \set{z \in \RR^2:\ \xi(z) = 0}.$$ The general theory of ordinary differential equations shows that, for every connected open $U \subseteq \RR^2 \setminus Z(\xi)$ and every $z \in U$, there exists…