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

A. Dolich¹ ; P. Speissegger²

¹Department of Mathematics Chicago State University Chicago, IL 60628, U.S.A.
²Department of Mathematics & Statistics McMaster University Hamilton, ON, Canada L8S 4K1

Fundamenta Mathematicae, Tome 198 (2008) no. 1, pp. 17-60

Résumé

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.

Détail

DOI : 10.4064/fm198-1-2

Keywords: vector field mathbb construct under certain assumptions ordered model theoretic structure associated flow set limit cycles nbsp represented definable set allows restatements dulacs problem question whether has finitely many limit cycles model theoretic terms involving recently developed notion rank other involving notion o minimality