Ak singularity

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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:

 (\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 Ak-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 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[edit]

  • 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