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

Tom 198 / 2008

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


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.


  • A. DolichDepartment of Mathematics
    Chicago State University
    Chicago, IL 60628, U.S.A.
  • P. SpeisseggerDepartment of Mathematics & Statistics
    McMaster University
    Hamilton, ON, Canada L8S 4K1

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