Zariski's multiplicity conjecture for holomorphic foliations

Let $\mathcal{F}$ and $\mathcal{G}$ be holomorphic foliations by curves at $(\mathbb{C}^{n},0)$, with algebraic multiplicities denoted by $\nu(\mathcal{F},0)$ and $\nu(\mathcal{G},0)$. Say that they are topologically equivalent if a homeomorphism of $(\mathbb{C}^{n},0)$ sends every integral curve of one foliation onto an integral curve of the other. Zariski's multiplicity conjecture for holomorphic foliations. If $\mathcal{F}$ and $\mathcal{G}$ are topologically equivalent, then

$$\nu(\mathcal{F},0)=\nu(\mathcal{G},0).$$

This is the foliation analogue of Zariski's multiplicity conjecture and concerns a basic proposed topological invariant of holomorphic foliations and ordinary differential equations. The paper describes it as a fundamental open problem and proves invariance in several classes and under additional hypotheses.