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{{DISPLAYTITLE:A{{ssub|k}} singularity}}
In '''mathematics''', and in particular '''[[singularity theory]]''' an ''A''<sub>''k''</sub>, where ''k''&nbsp;≥&nbsp;0 is an [[integer]], describes a level of degeneracy of a [[Function (mathematics)|function]]. The notation was introduced by [[V. I. Arnold]].


Let ''f'' : '''R'''<sup>''n''</sup> → '''R''' be a [[smooth function]]. We denote by Ω('''R'''<sup>''n''</sup>,'''R''') the infinite-dimensional [[function space|space]] of all such functions. Let diff('''R'''<sup>''n''</sup>) denote the infinite-dimensional [[Lie group]] of [[diffeomorphism]]s '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>, and diff('''R''') the infinite-dimensional Lie group of diffeomorphisms '''R''' → '''R'''. The [[Direct product of groups|product group]] diff('''R'''<sup>''n''</sup>) × diff('''R''') [[group action|acts]] on Ω('''R'''<sup>''n''</sup>,'''R''') in the following way: let φ : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> and ψ : '''R''' → '''R''' be diffeormorphisms and ''f'' : '''R'''<sup>''n''</sup> → '''R''' any smooth function. We define the group action as follows:
:<math> (\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^{-1}</math>
The [[group orbit|orbit]] of ''f'', denoted orb(''f''), of this group action is given by
:<math> \mbox{orb}(f) = \{ \psi \circ f \circ \varphi^{-1} : \varphi \in \mbox{diff}({\bold R}^n), \psi \in \mbox{diff}({\bold 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 '''R'''<sup>''n''</sup> 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''<sub>''k''</sub>-singularity if it lies in the orbit of
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:<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 ''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''<sub>''k''</sub>-singularities. The type ''A''<sub>''k''</sub>-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 (topology)|neighbourhood]] of the orbit of&nbsp;''f''.
 
This idea extends over the [[complex number]]s where the normal forms are much simpler; for example: there is no need to distinguish  ε<sub>''i''</sub>&nbsp;=&nbsp;+1 from ε<sub>''i''</sub>&nbsp;=&nbsp;−1.
 
== References ==
 
*{{Citation|first=V. I.|last=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=Birkhäuser|year=1985|ISBN=0-8176-3187-9}}
 
[[Category:Singularity theory]]

Latest revision as of 09:52, 2 December 2014


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