# Ak singularity

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Description of the degeneracy of a function

In mathematics, and in particular [singularity theory](/source/Singularity_theory), an **Ak singularity**, where *k* ≥ 0 is an [integer](/source/Integer), describes a level of [degeneracy](/source/Degeneracy_(mathematics)) of a [function](/source/Function_(mathematics)). The notation was introduced by [V. I. Arnold](/source/V._I._Arnold).

Let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a [smooth function](/source/Smooth_function). We denote by Ω ( R n , R ) {\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )} the infinite-dimensional [space](/source/Function_space) of all such functions. Let diff ⁡ ( R n ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})} denote the infinite-dimensional [Lie group](/source/Lie_group) of [diffeomorphisms](/source/Diffeomorphism) R n → R n , {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n},} and diff ⁡ ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} )} the infinite-dimensional Lie group of diffeomorphisms R → R . {\displaystyle \mathbb {R} \to \mathbb {R} .} The [product group](/source/Direct_product_of_groups) diff ⁡ ( R n ) × diff ⁡ ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})\times \operatorname {diff} (\mathbb {R} )} [acts](/source/Group_action_(mathematics)) on Ω ( R n , R ) {\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )} in the following way: let φ : R n → R n {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} ^{n}} and ψ : R → R {\displaystyle \psi :\mathbb {R} \to \mathbb {R} } be diffeomorphisms and f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } any smooth function. We define the group action as follows:

- ( φ , ψ ) ⋅ f := ψ ∘ f ∘ φ − 1 {\displaystyle (\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}}

The [orbit](/source/Group_orbit) of f , denoted orb(*f*), of this group action is given by

- orb ( f ) = { ψ ∘ f ∘ φ − 1 : φ ∈ diff ( R n ) , ψ ∈ diff ( R ) } . {\displaystyle {\mbox{orb}}(f)=\{\psi \circ f\circ \varphi ^{-1}:\varphi \in {\mbox{diff}}(\mathbb {R} ^{n}),\psi \in {\mbox{diff}}(\mathbb {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 ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ and a diffeomorphic change of coordinate in ⁠ R {\displaystyle \mathbb {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 , … , x n ) = 1 + ε 1 x 1 2 + ⋯ + ε n − 1 x n − 1 2 ± x n k + 1 {\displaystyle f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}}

where ε i = ± 1 {\displaystyle \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](/source/Neighbourhood_(topology)) of the orbit of f.

This idea extends over the [complex numbers](/source/Complex_number) 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](/source/ISBN_(identifier)) [0-8176-3187-9](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-3187-9)

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