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  "body": "\n## Definition\n\nLet $n \\ge 0$, let $U$ be an open neighbourhood of $0$ in $\\C^{n+1}$ and let $f \\colon U \\to \\C$ be a [[holomorphic function|507C34W0ZT]] with $f(0) = 0$.\nThe [[germ|VZXF7RNFKS]] $f$ has an *isolated singularity* at $0$ if:\n\n1. $\\frac{\\partial f}{\\partial z_j}(0) = 0$ for $j = 0, \\dots, n$, so $0$ is a [[critical point|0YTCMFB4HC]] of $f$;\n2. there is a neighbourhood $U' \\subseteq U$ of $0$ in which $0$ is the only point where all the partial derivatives $\\partial f / \\partial z_j$ vanish.\n\nThe germ at $0$ of the hypersurface $V = f^{-1}(0)$ is then called an *isolated hypersurface singularity*.\n\n## Remarks\n\nExample: $f(z) = z_0^2 + z_1^2 + \\cdots + z_n^2$.\nIts gradient is $(2z_0, \\dots, 2z_n)$, which vanishes only at $0$, so both clauses hold.\n\nNon-example: $f(z_0, z_1, z_2) = z_0 z_1$ on $\\C^3$.\nIts gradient is $(z_1, z_0, 0)$, which vanishes on the whole $z_2$-axis, so clause 2 fails.\n\nLet $J$ be the ideal of the ring $\\C\\{z_0, \\dots, z_n\\}$ of convergent power series generated by the partial derivatives of $f$.\nBy the Rückert Nullstellensatz, clause 2 holds if and only if $\\C\\{z_0, \\dots, z_n\\} / J$ is finite-dimensional over $\\C$.\nIts dimension is the Milnor number of $f$.\nIf $f$ has no repeated factor, the points near $0$ where $V$ is singular are the points of $V$ where the gradient of $f$ vanishes.\n",
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