Milnor number

Summary

The Milnor number $\mu$ of a holomorphic germ $f$ at a critical point is the dimension of its local algebra, see the definition. The local algebra is the quotient of the power series ring by the partial derivatives of $f$. It is finite exactly when $f$ has an isolated singularity. It measures how many simple critical points are merged at the point: a small generic perturbation of $f$ has exactly $\mu$ nondegenerate critical points nearby. Topologically, it is the number of spheres in the Milnor fibre of $f$.