Complex polynomial vector fields having a finitely curved orbit

Details Complex polynomial vector fields having a finitely curved orbit Complex polynomial vector fields having a finitely curved orbit

2006-12-05

arxiv.org/abs/math/0612122v1

Mathematics

Complex Variables

Scientific paper

In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positive answer to (i) under some non-degeneracy conditions on the singularities of the projective foliation induced by the vector field. For vector fields with a slightly more general class of singularities we prove a classification result that captures rational pull-backs of Poincar\'e-Dulac normal forms.

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