Английская Википедия:Ak singularity

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In mathematics, and in particular singularity theory, an Шаблон:Mvar singularity, where Шаблон:Math is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.

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

<math> (\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^{-1}</math>

The orbit of Шаблон:Mvar, denoted Шаблон:Math, of this group action is given by

<math> \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 Шаблон:Tmath and a diffeomorphic change of coordinate in Шаблон:Tmath such that one member of the orbit is carried to any other. A function Шаблон:Mvar is said to have a type Шаблон:Mvar-singularity if it lies in the orbit of

<math> 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>\varepsilon_i = \pm 1</math> and Шаблон:Math is an integer.

By a normal form we mean a particularly simple representative of any given orbit. The above expressions for Шаблон:Mvar give normal forms for the type Шаблон:Mvar-singularities. The type Шаблон:Mvar-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 Шаблон:Mvar.

This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish Шаблон:Math from Шаблон:Math.

References

• Шаблон:Citation

Шаблон:Mathanalysis-stub