Dicritical nilpotent holomorphic foliations

We study in this paper several properties concerning singularities of foliations in $(\mathbb{C}^3,\mathbf{0})$ that are pull-back of dicritical foliations in $(\mathbb{C}^2,\mathbf{0})$. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in $(\mathbb{C}^2,\mathbf{0})$, the adaptations are not straightforward.Comment: 14 pages. Several mistakes corrected from the previous version. Several changes in the text, including a change in the titl

On the quasi-ordinary cuspidal foliations in (C3,0)

AbstractIn this paper, we study a class of singularities of codimension 1 holomorphic germs of foliations in (C3,0), namely those ones having only one separatrix, that is a quasi-ordinary surface, and whose reduction of singularities agrees with the combinatorial desingularization of the separatrix. We show that the analytic classification of these germs can be read in the holonomy of a certain component of the exceptional divisor of the desingularization

On the Approximate Polar Curves of Foliations

We present a decomposition theorem of the generic polar curves of a generalized curve foliation with only one separatrix and the Hamiltonian foliations defined by the approximate roots of the generatrix. This is a generalization to foliations of the decomposition theorem of approximate Jacobians given by García Barroso and Gwoździewicz for plane branches