Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 37, pp. 1-23.
Title: On plane polynomial vector fields and the Poincare problem
Author: M'hammed El Kahoui (Cadi Ayyad Univ., Marrakech, Morocco)
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.
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.