Abstract

I will discuss a characterization of the elements of the set $H$_n of degree $n$ homogeneous polynomial vector fields that are structurally stable with respect to perturbation in $H$_n, both on the plane and on the Poincaré sphere. This allows one to characterize elements of the set $W$_n of smooth vector fields on $R2$ beginning with terms of order $n$ at ($(0,0)$ that are structurally stable in a neighborhood of $(0,0)$ under perturbation in $W$_n. I will also determine the set of elements of $H$_n that are determining for topological equivalence at $(0,0)$, in the sense that the topological type of the singularity at $(0,0)$ is invariant under the addition of higher order terms. Some of these results hark back to earlier work of J. Llibre, J.S. Perez del Rio and J.A. Rodriguez.