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  "body": "\n## Statement\n\nLet $F(t, x)$ be a real-valued [[smooth function|0V33BNKZ9S]] on a neighbourhood of $(0, 0)$ in $\\R \\times \\R^n$.\nLet $k \\ge 1$, and assume that\n$$F(0, 0) = \\partial_t F(0, 0) = \\cdots = \\partial_t^{k-1} F(0, 0) = 0, \\qquad \\partial_t^k F(0, 0) \\ne 0.$$\nThen there are smooth functions $q(t, x)$ near $(0, 0)$ with $q(0, 0) \\ne 0$, and $a_0(x), \\ldots, a_{k-1}(x)$ near $0 \\in \\R^n$ with $a_i(0) = 0$, such that near $(0, 0)$\n$$q(t, x) \\, F(t, x) = t^k + a_{k-1}(x) \\, t^{k-1} + \\cdots + a_1(x) \\, t + a_0(x).$$\n\n## Remarks\n\nThis is the smooth analogue of the [[Weierstrass preparation theorem|49JSD49P7X]] for holomorphic functions.\nThe hypothesis says that $t \\mapsto F(t, 0)$ vanishes to order exactly $k$ at $0$.\nIt is necessary: since $q(0, 0) \\ne 0$ and $a_i(0) = 0$, the function $F(t, 0)$ vanishes at $t = 0$ to the same order as $t^k$.\nIn the smooth case the proof is much harder than in the holomorphic case, because there is no Cauchy integral formula.\nUnlike the holomorphic case, the factorisation need not be unique.\nA closely related form is the division theorem: every smooth $G$ near $(0, 0)$ can be written as $G = Q F + \\sum_{i \u003c k} r_i(x) t^i$ with smooth $Q$ and $r_i$.\nMather gave a form for modules of [[germs|7SWSRGDT4A]].\nLet $\\varphi \\colon (\\R^n, 0) \\to (\\R^p, 0)$ be a smooth map germ, and let $A$ be a finitely generated module over the ring $\\mathcal{E}_n$ of smooth germs on $\\R^n$.\nLet $\\mathfrak{m}_p$ be the maximal ideal of the ring $\\mathcal{E}_p$ of smooth germs on $\\R^p$.\nThen $A$ is finitely generated over $\\mathcal{E}_p$ through $\\varphi$ if and only if $A / \\varphi^*(\\mathfrak{m}_p) A$ is a finite-dimensional real vector space.\nExample: for $F(t, x) = t^2 - x$ the theorem holds with $q = 1$, $a_1 = 0$ and $a_0(x) = -x$.\n\n## Explanations\n\n### L1 human\n\nNear a point where a smooth function of $t$ vanishes to order $k$, the theorem says that the function behaves like a polynomial of degree $k$ in $t$.\nThe coefficients of this polynomial depend smoothly on the remaining variables.\nSo questions about the zeros of a smooth function reduce to questions about the roots of a polynomial.\n\n### L2 human\n\nThe preparation theorem is the main algebraic tool of the local theory of smooth maps.\nIn Mather's module form it turns infinitesimal conditions into finite-dimensional linear algebra.\nIt is used to prove that [[infinitesimal stability implies stability|T6BK8Q8PK8]], and results on finite determinacy and versal unfoldings.\n",
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