We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e., they are never contained in non-trivial analytic families of equisingular invariant curves. In this case, we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.
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Pedro Fortuny Ayuso, Javier Ribón, The Poincaré problem for reducible curves. Rev. Mat. Iberoam. 40 (2024), no. 1, pp. 251–276DOI 10.4171/RMI/1451