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  "body": "\n## Definition\n\nLet $\\mathcal{O}_{n+1} = \\C\\{z_0, \\dots, z_n\\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n + 1$ variables.\nLet $f \\in \\mathcal{O}_{n+1}$ with $f(0) = 0$.\nThe *Jacobian ideal* of $f$ is the [[ideal|096Z9MP2ME]] $J(f) = \\left( \\frac{\\partial f}{\\partial z_0}, \\dots, \\frac{\\partial f}{\\partial z_n} \\right)$ of $\\mathcal{O}_{n+1}$.\nThe *Milnor number* of $f$ at $0$ is\n$$\\mu(f) = \\dim_\\C \\mathcal{O}_{n+1} / J(f),$$\na non-negative integer or $\\infty$.\n\n## Remarks\n\nThe quotient $\\mathcal{O}_{n+1} / J(f)$ is called the *local algebra* or *Milnor algebra* of $f$.\nThe number $\\mu(f)$ is finite if and only if $f$ has at most an [[isolated singularity|VTGYEES3VS]] at $0$, and $\\mu(f) = 0$ exactly when $0$ is not a critical point.\n\nExample: $f(x, y) = x^p + y^q$ with integers $p, q \\ge 2$.\nThen $J(f) = (x^{p-1}, y^{q-1})$, 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.\nSo $\\mu(f) = (p - 1)(q - 1)$.\nFor $f(z) = z_0^2 + \\cdots + z_n^2$, $J(f)$ is the maximal ideal and $\\mu(f) = 1$.\n\nNon-example of a finite Milnor number: $f(x, y) = x^2 y$.\nHere $J(f) = (2xy, x^2)$, and the powers $y, y^2, y^3, \\dots$ are linearly independent modulo $J(f)$, so $\\mu(f) = \\infty$.\nThe singularity is not isolated: the gradient vanishes on the whole line $x = 0$.\n\nLet $\\nabla f = (\\partial f / \\partial z_0, \\dots, \\partial f / \\partial z_n)$.\nFor 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.\nIt is also the rank of the middle homology of the Milnor fibre.\n",
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