The local Poincaré problem for irreducible branches

Abstract

Let F be a germ of holomorphic foliation defined in a neigh- borhood 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.