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{{DISPLAYTITLE:''A{{ssub|k}}'' singularity}}
{{short description|Description of the degeneracy of a function}}

In mathematics, and in particular [[singularity theory]], an '''{{mvar|A{{sub|k}}}} singularity''', where {{math|''k'' ≥ 0}} is an [[integer]], describes a level of [[Degeneracy (mathematics)|degeneracy]] of a [[Function (mathematics)|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 [[function space|space]] of all such functions. Let <math>\operatorname{diff}(\R^n)</math> denote the infinite-dimensional [[Lie group]] of [[diffeomorphism]]s <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 [[Direct product of groups|product group]] <math>\operatorname{diff}(\R^n) \times \operatorname{diff}(\R)</math> [[Group action (mathematics)|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 [[group orbit|orbit]] of {{mvar|f }}, denoted {{math|orb(''f'')}}, 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 {{tmath|\R^n}} and a diffeomorphic change of coordinate in {{tmath|\R}} such that one member of the orbit is carried to any other. A function {{mvar|f }} is said to have a type {{mvar|A{{sub|k}}}}-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 {{math|''k'' ≥ 0}} is an integer.

By a '''normal form''' we mean a particularly simple representative of any given orbit. The above expressions for {{mvar|f }} give normal forms for the type {{mvar|A{{sub|k}}}}-singularities. The type {{mvar|A{{sub|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 (topology)|neighbourhood]] of the orbit of&nbsp;{{mvar|f}}.

This idea extends over the [[complex number]]s where the normal forms are much simpler; for example: there is no need to distinguish {{math|1=ε{{sub|''i''}} = +1}} from {{math|1=ε{{sub|''i''}} = −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}}

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