Separatrices for real analytic vector fields in the plane

Abstract

Let X be a germ of real analytic vector field at (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 ν0(X) or the Milnor number µ0(X) is even, then X has a formal separatrix, that is, a formal invariant curve at 0 ∈ R 2 . This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

On Milnor and Tjurina numbers of foliations

“…Remark that ξ p (F ) ≥ 0 and, by definition, ξ p (F ) = 0 if and only if SN(F ) = ∅, that is, if and only if F is of second type. In several papers (see for example [10], [4]) the tangency excess of F is denoted by τ p (F ). In this paper, we denote it by ξ p (F ) since we keep the letter τ for the Tjurina number of a curve or a foliation.…”