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Moduli spaces of a family of topologically non quasi-homogeneous functions. (English) Zbl 1416.32015

Summary: 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 [Mosc. Math. J. 11, No. 1, 41–72 (2011; Zbl 1222.32056)] and the quasi homogeneous case [J. Singul. 14, 3–33 (2016; Zbl 1338.32028)] treated by Y. Genzmer and E. 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.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32G13 Complex-analytic moduli problems
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