The local Poincaré problem for irreducible branches

Abstract

Let $\mathcal{F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\mathbb{C}}^2$ that has a germ of irreducible holomorphic invariant curve $\gamma$. We provide a lower bound for the vanishing multiplicity of $\mathcal{F}$ at the origin in terms of the equisingularity class of $\gamma$. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of $\mathcal{F}$ can be bounded in terms of that of $\gamma$, and provide an explicit bound in this case.