The local Poincaré problem for irreducible branches

  • José Maria Cano

    Universidad de Valladolid, Spain
  • Pedro Fortuny Ayuso

    Universidad de Oviedo, Gijón, Spain
  • Javier Ribón

    Universidade Federal Fluminense, Niterói, Brazil
The local Poincaré problem for irreducible branches cover
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Abstract

Let be a germ of holomorphic foliation defined in a neighborhood of the origin of that has a germ of irreducible holomorphic invariant curve . We provide a lower bound for the vanishing multiplicity of at the origin in terms of the equisingularity class of . Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of can be bounded in terms of that of , and provide an explicit bound in this case.

Cite this article

José Maria Cano, Pedro Fortuny Ayuso, Javier Ribón, The local Poincaré problem for irreducible branches. Rev. Mat. Iberoam. 37 (2021), no. 6, pp. 2229–2244

DOI 10.4171/RMI/1260