Abstract.
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.
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Marín, D. Sur les notions de quasi-homogénéité de feuilletages holomorphes endimension deux. Bull Braz Math Soc, New Series 36, 177–185 (2005). https://doi.org/10.1007/s00574-005-0034-2
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DOI: https://doi.org/10.1007/s00574-005-0034-2