Ak singularity

In mathematics, and in particular singularity theory, an A_k singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.

Let ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ be a smooth function. We denote by ${\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )}$ the infinite-dimensional space of all such functions. Let ${\displaystyle \operatorname {diff} (\mathbb {R} ^{n})}$ denote the infinite-dimensional Lie group of diffeomorphisms ${\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n},}$ and ${\displaystyle \operatorname {diff} (\mathbb {R} )}$ the infinite-dimensional Lie group of diffeomorphisms ${\displaystyle \mathbb {R} \to \mathbb {R} .}$ The product group ${\displaystyle \operatorname {diff} (\mathbb {R} ^{n})\times \operatorname {diff} (\mathbb {R} )}$ acts on ${\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )}$ in the following way: let ${\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ and ${\displaystyle \psi :\mathbb {R} \to \mathbb {R} }$ be diffeomorphisms and ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ any smooth function. We define the group action as follows:

${\displaystyle (\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}}$

The orbit of f , denoted orb(f), of this group action is given by

${\displaystyle {\mbox{orb}}(f)=\{\psi \circ f\circ \varphi ^{-1}:\varphi \in {\mbox{diff}}(\mathbb {R} ^{n}),\psi \in {\mbox{diff}}(\mathbb {R} )\}\ .}$

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in ⁠${\displaystyle \mathbb {R} ^{n}}$⁠ and a diffeomorphic change of coordinate in ⁠${\displaystyle \mathbb {R} }$⁠ such that one member of the orbit is carried to any other. A function f is said to have a type A_k-singularity if it lies in the orbit of

${\displaystyle f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}}$

where ${\displaystyle \varepsilon _{i}=\pm 1}$ and k ≥ 0 is an integer.

By a normal form we mean a particularly simple representative of any given orbit. The above expressions for f give normal forms for the type A_k-singularities. The type A_k-singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of f.

This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish ε_i = +1 from ε_i = −1.