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  "body": "\n## Statement\n\nLet $\\mathcal{O}_n = \\C\\{z_1, \\dots, z_n\\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n$ variables.\nLet $I$ be an [[ideal|096Z9MP2ME]] of $\\mathcal{O}_n$.\nIts *integral closure* $\\overline{I}$ is the set of $h \\in \\mathcal{O}_n$ that satisfy an equation\n$$h^m + a_1 h^{m-1} + \\cdots + a_m = 0 \\quad \\text{with } a_j \\in I^j \\text{ for } j = 1, \\dots, m,$$\nfor some $m \\ge 1$; it is an ideal containing $I$.\nThen, for every ideal $I$ of $\\mathcal{O}_n$ and every integer $k \\ge 1$,\n$$\\overline{I^{\\,n + k - 1}} \\subseteq I^k.$$\nIn particular $\\overline{I^n} \\subseteq I$.\n\n## Remarks\n\nJoël Briançon and Henri Skoda proved the theorem in 1974, using Skoda's $L^2$ estimates for the division problem.\nA modern proof uses [[multiplier ideals|BKM0TQSCD9]]: one has $\\overline{I^{\\,l}} \\subseteq \\mathcal{J}(I^{\\,l})$ for all $l$, and Skoda's theorem gives $\\mathcal{J}(I^{\\,l}) \\subseteq I \\cdot \\mathcal{J}(I^{\\,l-1})$ for $l \\ge n$, so induction gives the inclusion.\nLipman and Teissier, and Lipman and Sathaye, extended the result to regular local rings.\n\nA corollary answers a question of Mather.\nIf $f \\in \\mathcal{O}_n$ and $f(0) = 0$, then $f$ lies in the integral closure of its Jacobian ideal $J(f) = (\\partial f / \\partial z_1, \\dots, \\partial f / \\partial z_n)$, and hence $f^n \\in J(f)$.\n\nThe exponent cannot be lowered to $1$ in general.\nFor $n = 2$ and $I = (x^2, y^2)$, the element $xy$ satisfies $(xy)^2 - x^2 y^2 = 0$ with $x^2 y^2 \\in I^2$, so $xy \\in \\overline{I}$, but $xy \\notin I$.\nThe theorem gives instead $\\overline{I^2} \\subseteq I$.\n\n## Explanations\n\n### L1 human\n\nThe integral closure of an ideal adds the elements that are \"as small as\" elements of the ideal.\nThis means that they satisfy a polynomial equation with coefficients from powers of the ideal.\nThis can be strictly bigger than the ideal.\nThe theorem says that taking enough powers makes up for the difference, and the number of powers needed depends only on the number of variables.\n\n### L2 human\n\nIn analytic terms, $h \\in \\overline{I}$ when $\\abs{h} \\le C (\\abs{g_1} + \\cdots + \\abs{g_r})$ near $0$, where $g_1, \\dots, g_r$ generate $I$.\nThe theorem turns such a size bound into an algebraic membership in a power of $I$.\nThis is how it is used in the study of the Jacobian ideal and of the [[Milnor number|1ZE51P7GRQ]].\nIt is a model for many effective results in commutative algebra and for applications of $L^2$ methods in algebraic geometry.\n",
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