Moduli spaces of a family of topologically non quasi-homogeneous functions

Abstract

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element $f$ in this class induces a germ of foliation ($df = 0$). Proceeding similarly to the homogeneous case [2] and the quasi homogeneous case [3] treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.