Ak singularity

Short description: Description of the degeneracy of a function

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

[math]\displaystyle{ (\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^{-1} }[/math]

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

[math]\displaystyle{ \mbox{orb}(f) = \{ \psi \circ f \circ \varphi^{-1} : \varphi \in \mbox{diff}(\R^n), \psi \in \mbox{diff}(\R ) \} \ . }[/math]

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in [math]\displaystyle{ \R^n }[/math] and a diffeomorphic change of coordinate in [math]\displaystyle{ \R }[/math] 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

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

where [math]\displaystyle{ \varepsilon_i = \pm 1 }[/math] 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