Abstract. We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation F of (C 2 , 0) in terms of the Milnor number of F . This result gives a bound for the degree of an algebraic invariant curve C ⊂ P 2 C of a foliation G of P 2 C in terms of the degree of G, provided that all the branches of C are isolated separatrices.
In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.
In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. We will also give a sufficient condition, in the quasihomogeneous, three-dimensional case, for these foliations to be of generalized surface type.
In this paper, we study a class of singularities of codimension 1 holomorphic germs of foliations in (C 3 , 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.
We study in this paper several properties concerning singularities of foliations in (C 3 , 0) that are pull-back of dicritical foliations in (C 2 , 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 (C 2 , 0), the adaptations are not straightforward.
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