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Characterization of second type plane foliations using Newton polygons
(Sciendo, 2022-05-01)
In this article we characterize the foliations that have the same Newton polygon that their union of formal separatrices, they are the foliations called of the second type. In the case of cuspidal foliations studied by Loray [ Lo ], we precise this characterization using the Poincaré-Hopf index. This index also characterizes the cuspidal foliations having the same process of singularity reduction that the union of its separatrices. Finally we give necessary and sufficient conditions when these cuspidal foliations are generalized curves, and a characterization when they have only one separatrix.
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Complete transversal and formal normal forms of germs of vector fields
(Sociedade Brasileira de Matemática, 2022-12-26)
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.
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Generalized Poincaré-Dulac singularities of holomorphic foliations
(Springer, 2024-12-01)
In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in (C2,0), those exhibiting a Poincaré-Dulac type singularity in their reduction process. This analytic classification is based in the holonomy of a certain component of the exceptional divisor. Finally, as a consequence, we show that these singularities exhibit a formal analytic rigidity.
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On the Approximate Polar Curves of Foliations
(MDPI, 2023-02-01)
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.
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On the Saito basis for plane curves
(Springer Nature, 2026-03-01)
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them.Using this algorithm, we compute analytic invariants for plane curves with multiplicity less than or equal to three.
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