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On Plane Polynomial Vector Fields and the Poincaré Problem

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons44379

El Kahoui,  M'hammed
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Citation

El Kahoui, M. (2002). On Plane Polynomial Vector Fields and the Poincaré Problem. Electronic journal of differential equations, 2002(37), 1-23.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0019-EBD5-C
Abstract
In this paper we address the Poincaré problem, on plane polynomial vector fields, under some conditions on the nature of the singularities of invariant curves. Our main idea consists in transforming a given vector field of degree m into another one of degree at most m+1 having its invariant curves in projective quasi-generic position. This allows us to give bounds on degree for some well known classes of curves such as the nonsingular ones and curves with ordinary nodes. We also give a bound on degree for any invariant curve in terms of the maximum Tjurina number of its singularities and the degree of the vector field.