Separatrices for real analytic vector fields in the plane

Abstract

Let $X$ be a germ of real analytic vector field at $({\mathbb{R}}^2,0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order ${\nu }_0(X)$ or the Milnor number ${\mu }_0(X)$ is even, then $X$ has a formal separatrix, that is, a formal invariant curve at $0\in {\mathbb{R}}^2$. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

16 pages