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  "body": "\n## Statement\n\nLet $f \\colon (\\C^n, 0) \\to (\\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with a [[critical point|0YTCMFB4HC]] at $0$.\nCall $f$ *simple* if, in a jet space of high enough order, some neighbourhood of $f$ meets only finitely many right equivalence classes.\nEquivalently, $f$ has [[modality|8CFTJP1WN8]] $0$.\nThen $f$ is simple if and only if it has the following form in some holomorphic coordinates $(x, y, z_1, \\ldots, z_{n-2})$ centred at $0$.\nIt is one of the following germs plus $z_1^2 + \\cdots + z_{n-2}^2$, with the obvious changes when $n \\le 2$:\n\n1. $A_k$, $k \\ge 1$: $x^{k+1} + y^2$;\n2. $D_k$, $k \\ge 4$: $x^2 y + y^{k-1}$;\n3. $E_6$: $x^3 + y^4$;\n4. $E_7$: $x^3 + x y^3$;\n5. $E_8$: $x^3 + y^5$.\n\n## Remarks\n\nFor $n = 1$ only the germs $A_k$, written $x^{k+1}$, occur.\nIn each case the index is the Milnor number, $\\dim_\\C \\mathcal{O}_n / J(f)$, where $\\mathcal{O}_n$ is the ring of convergent power series and $J(f)$ is the ideal of partial derivatives.\nThe proof proceeds by the corank and the lowest-order terms of $f$, using [[finite determinacy|EGPP69XA4F]] to reduce to polynomials.\nIt shows that every germ outside the list deforms, by arbitrarily small perturbations, into members of a unimodal family.\nThese families are $x^3 + y^3 + z^3 + a xyz$, $x^4 + y^4 + a x^2 y^2$ and $x^3 + y^6 + a x^2 y^2$, with suitable conditions on $a$.\nExample: $x^2 y + y^3$ is a $D_4$ singularity; its cubic part factors into three distinct lines through $0$.\nOver $\\R$ the same list holds up to signs of the terms.\nFor $n = 3$ the zero sets of these germs are the [[ADE singularities|HQNXQBZ6C7]] of surfaces.\n\n## Explanations\n\n### L1 human\n\nCritical points of functions of several variables can be very complicated.\nArnold asked which of them are not part of a continuous family of different shapes, and found a short list.\nThe list is labelled by the same letters $A$, $D$, $E$ as the simply laced Dynkin diagrams of Lie theory.\nEverything else comes in families that depend on at least one continuous parameter.\n\n### L2 human\n\nThe theorem is the start of Arnold's classification of critical points by modality, which continues with the unimodal and bimodal families.\nThe ADE pattern links it with finite subgroups of $SL(2, \\C)$, simple Lie algebras and the Du Val surface singularities.\nThe real forms of these germs with small codimension are the organising centres of the elementary catastrophes in [[Thom's classification|23RVD8K8RB]].\n",
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