Geodesic
On plane polynomial vector fields and the Poincaré problem
Electronic journal of differential equations, Tome 2002 (2002)
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.
Classification : 34C05, 34A34, 34C14
Keywords: polynomial vector fields, invariant algebraic curves, intersection numbers, tjurina number, Bezout theorem 
@article{EJDE_2002__2002__a144,
     author = {El Kahoui,  M'hammed},
     title = {On plane polynomial vector fields and the {Poincar\'e} problem},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {1068.34506},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a144/}
}
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%A El Kahoui,  M'hammed
%T On plane polynomial vector fields and the Poincaré problem
%J Electronic journal of differential equations
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%F EJDE_2002__2002__a144
El Kahoui,  M'hammed. On plane polynomial vector fields and the Poincaré problem. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a144/