Electron. J. Diff. Eqns., Vol. 2002(2002), No. 37, pp. 1-23.

On plane polynomial vector fields and the Poincare problem

M'hammed El Kahoui

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.

Submitted November 20, 2001. Published May 6, 2002.
Math Subject Classifications: 34C05, 34A34, 34C14.
Key Words: Polynomial vector fields, Invariant algebraic curves, Intersection numbers, Tjurina number, Bezout theorem.

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M'hammed El Kahoui
Department of Mathematics
Faculty of Sciences Semlalia
Cadi Ayyad University
P.O Box 2390, Marrakech, Morocco
e-mail: elkahoui@ucam.ac.ma

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