Let $\mathscr F$ be a singular foliation by curves on a complex manifold $M$, with only isolated singularities. The Milnor number $\mu_p(\mathscr F)$ at a point $p \in M$ is the topological index at $p$ of a local vector field $X$ that generates $\mathscr F$ in a neighborhood of $p$. If $X$ is given in local coordinates by
$$X = f_1 \, \dfrac \partial {\partial z_1} + \dots + f_n \, \dfrac \partial {\partial z_n},$$
then the Milnor number $\mu_p(\mathscr F)$ is given by
$$\mu_p(\mathscr F) = \dim_\mathbb C \dfrac {\mathbb C \{ z_1, \dots, z_n \}} {\langle f_1, \dots, f_n \rangle},$$
and some authors take this algebraic expression to be the definition. However, if we are doing differential geometry, then only the earlier topological definition provides an actual reason to care about $\mu_p(\mathscr F)$, IMO.
The algebraic multiplicity of $\mathscr F$ at $p$ is defined as
$$m_p(\mathscr F) = \min \{ \mathrm{ord}(f) : f \in \langle f_1, \dots, f_n \rangle \}.$$
What is the reason to care about this algebraic multiplicity? Does it have a geometric interpretation?