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.