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  "body": "\n## Definition\n\nLet $\\mathcal{O}_n = \\C\\{x_1, \\ldots, x_n\\}$ be the ring of [[germs|VZXF7RNFKS]] at $0$ of [[holomorphic functions|0NAHXR1FRH]] on $\\C^n$, that is, convergent power series.\nLet $f \\in \\mathcal{O}_n$ with $f(0) = 0$, and let $I$ be the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.\nThe *Tjurina number* of $f$ is\n$$\\tau(f) = \\dim_\\C \\mathcal{O}_n / I, \\qquad I = \\left( f, \\frac{\\partial f}{\\partial x_1}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right).$$\nIt may be infinite.\n\n## Remarks\n\nThe quotient is the Tjurina algebra of $f$.\nThe number $\\tau(f)$ is finite if and only if the hypersurface $f = 0$ has an isolated singular point at $0$ or is smooth there.\nExample: for $f = x^2 + y^3$ the ideal is $(x^2 + y^3, 2x, 3y^2) = (x, y^2)$, so $\\mathcal{O}_2 / (x, y^2)$ has basis $1, y$ and $\\tau(f) = 2$.\nNon-example: for $f = x^2$ on $\\C^2$ the ideal is $(x)$, and $\\mathcal{O}_2 / (x) \\cong \\C\\{y\\}$ is infinite-dimensional; the hypersurface $x^2 = 0$ is singular along the whole line $x = 0$, so the singularity is not isolated.\nThe Milnor number is $\\mu(f) = \\dim_\\C \\mathcal{O}_n / (\\partial f / \\partial x_1, \\ldots, \\partial f / \\partial x_n)$, so $\\tau(f) \\le \\mu(f)$.\nFor an isolated singularity, $\\tau(f) = \\mu(f)$ if and only if $f$ is quasi-homogeneous in suitable coordinates, by a theorem of K. Saito.\nThe Tjurina number depends only on the hypersurface germ, not on the choice of $f$: if $u$ is a unit of $\\mathcal{O}_n$, then $uf$ and $f$ give the same ideal.\n",
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    "name": "Tjurina number",
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  "msc": [
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  "name": "Tjurina number",
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