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  "body": "\n## Statement\n\nLet $n \\ge 1$, write $z' = (z_1, \\dots, z_{n-1})$, and let $\\mathcal{O}_n = \\C\\{z_1, \\dots, z_n\\}$ and $\\mathcal{O}_{n-1} = \\C\\{z'\\}$ be [[rings of convergent power series|EVPZ7ZG1HB]].\nLet $f \\in \\mathcal{O}_n$ and $d \\ge 0$ be such that $f(0, \\dots, 0, z_n) = z_n^d\\, v(z_n)$ with $v(0) \\ne 0$; that is, $f$ is regular of order $d$ in $z_n$.\nThen for every $g \\in \\mathcal{O}_n$ there are unique $q \\in \\mathcal{O}_n$ and $r \\in \\mathcal{O}_{n-1}[z_n]$ such that\n$$g = q f + r \\quad \\text{and} \\quad \\deg_{z_n} r \u003c d.$$\n\n## Remarks\n\nThe same statement holds for [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0$, since these form a ring isomorphic to $\\mathcal{O}_n$.\nOne proof first treats a divisor $f = P$ that is a Weierstrass polynomial of degree $d$.\nOn a small polydisc $\\abs{z'} \u003c \\varepsilon$, $\\abs{z_n} \u003c \\delta$, the polynomial $P(z', \\cdot)$ has its $d$ roots inside $\\abs{z_n} \u003c \\delta$ for each $z'$.\nThe Cauchy integral\n$$q(z', z_n) = \\frac{1}{2\\pi i} \\oint_{\\abs{w} = \\delta} \\frac{g(z', w)}{P(z', w)\\,(w - z_n)}\\, dw$$\ndefines a holomorphic $q$, and the Cauchy formula for $g$ gives\n$$r = g - q P = \\frac{1}{2\\pi i} \\oint_{\\abs{w} = \\delta} \\frac{g(z', w)}{P(z', w)} \\cdot \\frac{P(z', w) - P(z', z_n)}{w - z_n}\\, dw.$$\nThe last quotient is a polynomial in $z_n$ of degree $d - 1$, so $r$ is a polynomial in $z_n$ of degree less than $d$.\nA general $f$ regular of order $d$ is a unit times a Weierstrass polynomial.\nThis can be shown directly through the symmetric functions of the roots of $f(z', \\cdot)$, and it reduces the general case to the first one.\nExample: for $n = 1$ and $f = z^d$, the theorem says $g = q z^d + r$, where $r$ is the Taylor polynomial of $g$ of degree $d - 1$.\nThe regularity hypothesis is needed: for $f = z_1$ in $\\mathcal{O}_2$ and any $d$, the series $g = z_2^d$ is not of the form $q z_1 + r$ with $\\deg_{z_2} r \u003c d$, since setting $z_1 = 0$ would give $z_2^d = r(0, z_2)$.\nConversely, applying the theorem to $g = z_n^d$ gives the [[Weierstrass preparation theorem|49JSD49P7X]]: $f$ is a unit times the [[Weierstrass polynomial|SDCKNWSWNM]] $z_n^d - r$.\nA linear change of coordinates makes any nonzero $f \\in \\mathcal{O}_n$ regular in $z_n$ of some order, so the hypothesis is mild.\n\n## Explanations\n\n### L1 human\n\nThe theorem is a division with remainder, like the division of integers or of polynomials.\nOne divides a power series $g$ by a power series $f$ that looks, along one axis, like $z_n^d$.\nThe remainder is then a polynomial in $z_n$ of degree less than $d$, with coefficients that are power series in the other variables.\n\n### L2 human\n\nDivision is the engine behind the algebra of the ring $\\mathcal{O}_n$.\nIt implies the preparation theorem, and with it one proves by induction on $n$ that $\\mathcal{O}_n$ is Noetherian and factorial.\nIt also shows that $\\mathcal{O}_n/(f)$ is a free module of rank $d$ over $\\mathcal{O}_{n-1}$ when $f$ is regular of order $d$.\nThis is the algebraic form of the local parametrization of hypersurfaces.\n",
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