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  "body": "\n## Definition\n\nLet $\\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\\R^n \\to \\R$, and let $\\mathfrak{m}$ be its maximal [[ideal|096Z9MP2ME]] of germs vanishing at $0$.\nLet $f \\in \\mathcal{E}_n$ have a [[critical point|0YTCMFB4HC]] at $0$, and let $J(f) \\subseteq \\mathfrak{m}$ be the ideal generated by $\\partial f / \\partial x_1, \\ldots, \\partial f / \\partial x_n$.\nThe *codimension* of $f$ is\n$$\\operatorname{codim}(f) = \\dim_\\R \\mathfrak{m} / J(f),$$\nwhich may be infinite.\n\n## Remarks\n\nExample: for $f = x^4$ on $\\R$, $J(f) = (x^3)$ and $\\mathfrak{m} / J(f)$ has basis the classes of $x$ and $x^2$, so $\\operatorname{codim}(f) = 2$.\nFor a nondegenerate critical point $J(f) = \\mathfrak{m}$, so the codimension is $0$.\nNon-example: the definition needs a critical point at $0$; for $f = x$ one has $J(f) = \\mathcal{E}_1 \\not\\subseteq \\mathfrak{m}$, and a regular point is not given a codimension in this sense.\nWhen it is finite, $\\operatorname{codim}(f) = \\dim_\\R \\mathcal{E}_n / J(f) - 1$.\nFor a holomorphic germ, the same quotient of the ring of convergent power series has dimension the Milnor number $\\mu(f)$.\nIt equals the least number of parameters of a [[versal unfolding|KSAFWZVB8K]] of $f$, not counting an additive constant.\nConventions vary between authors; some count the additive constant as well and use $\\dim_\\R \\mathcal{E}_n / J(f)$.\n",
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