The local Poincar\'e problem for irreducible branches

TL;DR

This paper establishes a lower bound for the vanishing multiplicity of a holomorphic foliation at the origin based on the equisingularity class of an invariant irreducible curve, and characterizes cases with bounded multiplicity.

Contribution

It introduces a sharp lower bound for the multiplicity of holomorphic foliations in relation to invariant curves and characterizes dicritical singularities with explicit bounds.

Findings

Lower bound for multiplicity based on equisingularity class

Sharpness of the established lower bound

Explicit bounds for dicritical singularities

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.