---
tag: M5437FP73G
kind: statement
role: theorem
name: Arnold's classification of simple critical points
msc: [58K40, 32S05, 14B05]
uses: [VZXF7RNFKS, 0YTCMFB4HC, 0NAHXR1FRH, 8CFTJP1WN8]
about: [VZXF7RNFKS, 0YTCMFB4HC]
sources:
  - url: https://en.wikipedia.org/wiki/ADE_classification
---

## Statement

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with a [[critical point|0YTCMFB4HC]] at $0$.
Call $f$ *simple* if, in a jet space of high enough order, some neighbourhood of $f$ meets only finitely many right equivalence classes.
Equivalently, $f$ has [[modality|8CFTJP1WN8]] $0$.
Then $f$ 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 $0$.
It is one of the following germs plus $z_1^2 + \cdots + z_{n-2}^2$, with the obvious changes when $n \le 2$:

1. $A_k$, $k \ge 1$: $x^{k+1} + y^2$;
2. $D_k$, $k \ge 4$: $x^2 y + y^{k-1}$;
3. $E_6$: $x^3 + y^4$;
4. $E_7$: $x^3 + x y^3$;
5. $E_8$: $x^3 + y^5$.

## Remarks

For $n = 1$ only the germs $A_k$, written $x^{k+1}$, occur.
In 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.
The proof proceeds by the corank and the lowest-order terms of $f$, using [[finite determinacy|EGPP69XA4F]] to reduce to polynomials.
It shows that every germ outside the list deforms, by arbitrarily small perturbations, into members of a unimodal family.
These 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 $a$.
Example: $x^2 y + y^3$ is a $D_4$ singularity; its cubic part factors into three distinct lines through $0$.
Over $\R$ the same list holds up to signs of the terms.
For $n = 3$ the zero sets of these germs are the [[ADE singularities|HQNXQBZ6C7]] of surfaces.

## Explanations

### L1 human

Critical points of functions of several variables can be very complicated.
Arnold asked which of them are not part of a continuous family of different shapes, and found a short list.
The list is labelled by the same letters $A$, $D$, $E$ as the simply laced Dynkin diagrams of Lie theory.
Everything else comes in families that depend on at least one continuous parameter.

### L2 human

The theorem is the start of Arnold's classification of critical points by modality, which continues with the unimodal and bimodal families.
The ADE pattern links it with finite subgroups of $SL(2, \C)$, simple Lie algebras and the Du Val surface singularities.
The real forms of these germs with small codimension are the organising centres of the elementary catastrophes in [[Thom's classification|23RVD8K8RB]].