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  "body": "\n## Statement\n\nLet $\\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\\R^n \\to \\R$.\nLet $\\mathfrak{m} \\subset \\mathcal{E}_n$ be its maximal [[ideal|096Z9MP2ME]], the germs that vanish at $0$.\nFor $f \\in \\mathcal{E}_n$, let $J(f)$ be the Jacobian ideal, generated by $\\partial f / \\partial x_1, \\ldots, \\partial f / \\partial x_n$.\nA germ $g$ is *right equivalent* to $f$ if $g = f \\circ \\varphi$ for a germ of [[diffeomorphism|0KZZ70ZHXP]] $\\varphi \\colon (\\R^n, 0) \\to (\\R^n, 0)$.\nCall $f$ *$k$-determined* if every $g \\in \\mathcal{E}_n$ with the same [[$k$-jet|1QE0Z0756D]] at $0$ as $f$ is right equivalent to $f$.\n\n1. If $\\mathfrak{m}^{k+1} \\subseteq \\mathfrak{m}^2 J(f)$, then $f$ is $k$-determined.\n2. If $f$ is $k$-determined, then $\\mathfrak{m}^{k+1} \\subseteq \\mathfrak{m} J(f)$.\n3. Hence $f$ is $k$-determined for some $k$ if and only if $\\mathfrak{m}^r \\subseteq J(f)$ for some $r$.\n\n## Remarks\n\nThe proof of the first part uses the homotopy method: join $f$ to $g$ by $f_t = f + t(g - f)$ and look for diffeomorphisms $\\varphi_t$ with $f_t \\circ \\varphi_t = f$.\nDifferentiating in $t$ leads to an equation $g - f = -\\sum_i \\xi_i \\, \\partial f_t / \\partial x_i$ for a time-dependent vector field $\\xi$ vanishing at $0$, which can be solved by Nakayama's lemma because $g - f \\in \\mathfrak{m}^{k+1}$.\nExample: if $f$ has a nondegenerate [[critical point|0YTCMFB4HC]] at $0$, then $J(f) = \\mathfrak{m}$, so $\\mathfrak{m}^3 = \\mathfrak{m}^2 J(f)$ and $f$ is $2$-determined.\nWith the normal form of a quadratic form this gives the [[Morse lemma|ERSD8D4HWX]].\nFor $f = x^3$ on $\\R$, $J(f) = (x^2)$ and $\\mathfrak{m}^2 J(f) = (x^4) = \\mathfrak{m}^4$, so $x^3$ is $3$-determined.\nFor $f = x^2 y$ on $\\R^2$, the ideal $J(f) = (xy, x^2)$ contains no power of $y$, so $f$ is not finitely determined.\nMather and Tougeron proved such criteria; similar results hold for other equivalence relations and for holomorphic germs.\n\n## Explanations\n\n### L1 human\n\nA smooth function near a point has a Taylor expansion with infinitely many terms.\nFinite determinacy says that, for most interesting critical points, only finitely many terms matter.\nAfter the first $k$ terms, any change of the higher terms can be absorbed by a change of coordinates.\nFor example, near a nondegenerate minimum, the quadratic terms already decide the shape completely.\n\n### L2 human\n\nThe theorem is what makes the classification of singularities a finite problem.\nA $k$-determined germ is fixed by a polynomial, its $k$-jet, so the classification takes place in a finite-dimensional jet space under the action of a finite-dimensional group.\nThe condition $\\mathfrak{m}^r \\subseteq J(f)$ says that the local algebra $\\mathcal{E}_n / J(f)$ is finite-dimensional; for holomorphic germs its dimension is the Milnor number.\nThis underlies [[Arnold's classification of simple critical points|M5437FP73G]] and the theory of [[versal unfoldings|KSAFWZVB8K]].\n",
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