The local Poincaré problem for irreducible branches
- José Maria Cano 1
- Pedro Fortuny Ayuso 2
- Javier Ribón 3
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1
Universidad de Valladolid
info
- 2 Universidad de Oviedo, Gijón
- 3 Universidade Federal Fluminense, Niterói
ISSN: 0213-2230
Year of publication: 2021
Volume: 37
Issue: 6
Pages: 2229-2244
Type: Article
More publications in: Revista matemática iberoamericana
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Abstract
Let F be a germ of holomorphic foliation defined in a neighborhood of the origin of C2 that has a germ of irreducible holomorphic invariant curve γ. We provide a lower bound for the vanishing multiplicity of F at the origin in terms of the equisingularity class of γ. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of F can be bounded in terms of that of γ, and provide an explicit bound in this case.