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  "body": "\n## Definition\n\nLet $f \\colon (\\C^n, 0) \\to (\\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with $f(0) = 0$.\nThe point $0$ is a *singular point* of $f$ if it is a [[critical point|0YTCMFB4HC]], that is, if $\\partial f/\\partial x_i (0) = 0$ for $i = 1, \\ldots, n$.\nTwo such germs $f$ and $g$ are *right equivalent* if $f = g \\circ \\varphi$ for a germ $\\varphi \\colon (\\C^n, 0) \\to (\\C^n, 0)$ of a biholomorphic map.\n*Singularity theory* studies singular germs of functions, maps and varieties up to such equivalences, and their invariants.\n\n## Remarks\n\nThe same definitions apply to real smooth germs, with diffeomorphisms in place of biholomorphic maps.\nExample: $f = x^2 + y^2$ has a singular point at $0$, since both partial derivatives vanish there.\nIt is right equivalent to $g = xy$ via $\\varphi(x, y) = (x + iy, x - iy)$, since $(x + iy)(x - iy) = x^2 + y^2$.\nNon-example: $f = x + y^2$ has $\\partial f/\\partial x = 1$ at $0$, so $0$ is not a singular point.\nBy the holomorphic Morse lemma, every germ with a nondegenerate critical point at $0$ is right equivalent to $x_1^2 + \\cdots + x_n^2$.\nAn isolated singularity has a finite Milnor number $\\mu(f) = \\dim_\\C \\C\\{x_1, \\ldots, x_n\\} / (\\partial f/\\partial x_1, \\ldots, \\partial f/\\partial x_n)$, which is invariant under right equivalence.\n",
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