Sur les notions de quasi-homogénéité de feuilletages holomorphes en dimension deux

In this note, we recall the different notions of quasi-homogeneity for singular germs of holomorphic foliations in the plane presented in [6]. The classical notion of quasi-homogenity allude to those functions which belong to its own jacobian ideal. Given a foliation in the plane, asking that the equation of the separatrix set is a classical quasi-homogeneous function we obtain a natural generalization in the context of foliations. On the other hand, topological quasi-homogeneity is characterized by the fact that every topologically trivial deformation whose sepatrix family is analytically trivial is an analytically trivial deformation. We give an explicit example of a topological quasi-homogeneous foliation which is not quasi-homogeneous in the sense given above. © 2005, Sociedade Brasileira de Matemática.