# Ak singularity

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

Let f : RnR be a smooth function. We denote by Ω(Rn,R) the infinite-dimensional space of all such functions. Let diff(Rn) denote the infinite-dimensional Lie group of diffeomorphisms RnRn, and diff(R) the infinite-dimensional Lie group of diffeomorphisms RR. The product group diff(Rn) × diff(R) acts on Ω(Rn,R) in the following way: let φ : RnRn and ψ : RR be diffeormorphisms and f : RnR 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

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

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in Rn and a diffeomorphic change of coordinate in R such that one member of the orbit is carried to any other. A function f is said to have a type Ak-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 Ak-singularities. The type Ak-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 εi = +1 from εi = −1.

## References

• Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985), The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1, Birkhäuser, ISBN 0-8176-3187-9