# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## An ordered structure of rank two related to Dulac's Problem

### Tom 198 / 2008

Fundamenta Mathematicae 198 (2008), 17-60 MSC: 37C27, 03C64. DOI: 10.4064/fm198-1-2

#### Streszczenie

For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a way that the set of all limit cycles of $\xi$ is represented by a definable set. This allows us to give two restatements of Dulac's Problem for $\xi$—that is, the question whether $\xi$ has finitely many limit cycles—in model-theoretic terms, one involving the recently developed notion of ${\rm U}^{\rm l}\!\!\!\!\rm^{^o}$-rank and the other involving the notion of o-minimality.

#### Autorzy

• A. DolichDepartment of Mathematics
Chicago State University
Chicago, IL 60628, U.S.A.
e-mail
• P. SpeisseggerDepartment of Mathematics & Statistics
McMaster University