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  "body": "\n## Definition\n\nLet $f \\colon (\\R^n, 0) \\to \\R$ be a [[germ|7SWSRGDT4A]] of a [[smooth function|0V33BNKZ9S]].\nAn *$r$-parameter unfolding* of $f$ is a smooth germ $F \\colon (\\R^n \\times \\R^r, 0) \\to \\R$ with $F(x, 0) = f(x)$.\nAn $s$-parameter unfolding $G$ of $f$ is *induced* from $F$ if there are smooth germs $\\Phi \\colon (\\R^n \\times \\R^s, 0) \\to (\\R^n, 0)$, $\\alpha \\colon (\\R^s, 0) \\to (\\R^r, 0)$ and $c \\colon (\\R^s, 0) \\to \\R$ with $\\Phi(x, 0) = x$ and\n$$G(x, v) = F(\\Phi(x, v), \\alpha(v)) + c(v).$$\nThe unfolding $F$ is *versal* if every unfolding of $f$, with any number of parameters, is induced from $F$.\nA versal unfolding with the least possible number $r$ of parameters is *universal*, or *miniversal*.\n\n## Remarks\n\nLet $\\mathcal{E}_n$ be the ring of smooth germs at $0$ on $\\R^n$ and $J(f)$ the ideal generated by the partial derivatives of $f$.\nThe versality theorem says that $F$ is versal if and only if\n$$\\mathcal{E}_n = J(f) + \\R \\cdot 1 + \\R \\cdot \\frac{\\partial F}{\\partial u_1}(\\cdot, 0) + \\cdots + \\R \\cdot \\frac{\\partial F}{\\partial u_r}(\\cdot, 0),$$\nwhere $u_1, \\ldots, u_r$ are the parameters.\nExample: for $f = x^3$, $J(f) = (x^2)$ and $\\mathcal{E}_1 / J(f)$ has basis the classes of $1$ and $x$; for $F(x, u) = x^3 + u x$ one has $\\partial F / \\partial u (x, 0) = x$, so $F$ is versal, and universal, since one parameter is needed.\nNon-example: $G(x, u) = x^3 + u x^2$ is an unfolding of $x^3$, but $\\partial G / \\partial u (x, 0) = x^2$ lies in $J(f)$, so the class of $x$ is not reached and $G$ is not versal.\nIndeed, every function $G(\\cdot, u)$ has a critical point near $0$, while $x^3 + u x$ has none for $u \u003e 0$.\nA germ has a versal unfolding with finitely many parameters if and only if $\\mathcal{E}_n / J(f)$ is finite-dimensional.\nIf moreover $f$ has a critical point at $0$, a universal unfolding has $\\dim_\\R \\mathcal{E}_n / J(f) - 1$ parameters.\n",
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