Milnor number

Remarks

The quotient $\mathcal{O}_{n+1} / J(f)$ is called the local algebra or Milnor algebra of $f$. The number $\mu(f)$ is finite if and only if $f$ has at most an isolated singularity at $0$, and $\mu(f) = 0$ exactly when $0$ is not a critical point.

Example: $f(x, y) = x^p + y^q$ with integers $p, q \ge 2$. Then $J(f) = (x^{p-1}, y^{q-1})$, and the monomials $x^a y^b$ with $0 \le a \le p - 2$ and $0 \le b \le q - 2$ form a basis of the quotient. So $\mu(f) = (p - 1)(q - 1)$. For $f(z) = z_0^2 + \cdots + z_n^2$, $J(f)$ is the maximal ideal and $\mu(f) = 1$.

Non-example of a finite Milnor number: $f(x, y) = x^2 y$. Here $J(f) = (2xy, x^2)$, and the powers $y, y^2, y^3, \dots$ are linearly independent modulo $J(f)$, so $\mu(f) = \infty$. The singularity is not isolated: the gradient vanishes on the whole line $x = 0$.

Let $\nabla f = (\partial f / \partial z_0, \dots, \partial f / \partial z_n)$. For an isolated singularity, $\mu(f)$ is also the topological degree of the map $z \mapsto \nabla f(z) / \norm{\nabla f(z)}$ from a small sphere about $0$ to the unit sphere. It is also the rank of the middle homology of the Milnor fibre.