Real Algebraic and Analytic Geometry
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194. A. Dolich, Patrick Speissegger:
An ordered structure of rank two related to Dulac's Problem.

e-mail: ,
homepage: http://www.math.toronto.edu/~speisseg/

Submission: 2006, March 2.

Abstract:
For a vector field $\xi$ on $\RR^2$ 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 $\urank$-rank and the other involving the notion of o-minimality. .

Mathematics Subject Classification (2000): 37C27, 03C64.

Keywords and Phrases: Vector fields, limit cycles, model theory, ordered structures.

Full text, 42p.: dvi 210k, ps.gz 527k, pdf 788k.