Abstract:
Let $X$ be a germ of real analytic vector field at $(\mathbb{R}^{2},0)$ with an algebraically 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.
Key words and phrases:real analytic vector field, formal and analytic separatrix, reduction of singularities, index of vector fields, polar invariants, center-focus vector field.
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