On Plane Polynomial Vector Fields and the Poincare Problem

Date

2002-05-06

Authors

El Kahoui, M'hammed

Journal Title

Journal ISSN

Volume Title

Publisher

Southwest Texas State University, Department of Mathematics

Abstract

In this paper we address the Poincare 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.

Description

Keywords

Polynomial vector fields, Invariant algebraic curves, Intersection numbers, Tjurina number, Bezout theorem

Citation

El Kahoui, M. (2002). On plane polynomial vector fields and the Poincare problem. <i>Electronic Journal of Differential Equations, 2002</i>(37), pp. 1-23.

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Attribution 4.0 International

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