Ak singularity

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

Let f : Rⁿ → R 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 Rⁿ → Rⁿ, and diff(R) the infinite-dimensional Lie group of diffeomorphisms R → R. The product group diff(Rⁿ) × diff(R) acts on Ω(Rⁿ,R) in the following way: let φ : Rⁿ → Rⁿ and ψ : R → R be diffeormorphisms and f : Rⁿ → R any smooth function. We define the group action as follows:

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

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

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

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 A_k-singularity if it lies in the orbit of

$f(x_1,\ldots,x_n) = 1 + \varepsilon_1x_1^2 + \cdots + \varepsilon_{n-1}x^2_{n-1} \pm x_n^{k+1}$

where $\varepsilon_i = \pm 1$ 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.

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