Ak singularity

This article only references primary sources. Please improve this article by adding secondary or tertiary sources. Unsourced material may be challenged and removed. Find sources: "Ak singularity" – news · newspapers · books · scholar · JSTOR (July 2025) (Learn how and when to remove this message)

In mathematics, and in particular singularity theory, an A_k singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.

Let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a smooth function. We denote by Ω ( R n , R ) {\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )} the infinite-dimensional space of all such functions. Let diff ⁡ ( R n ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})} denote the infinite-dimensional Lie group of diffeomorphisms 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 diff ⁡ ( R n ) × diff ⁡ ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})\times \operatorname {diff} (\mathbb {R} )} acts 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 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 A_k-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 A_k-singularities. The type A_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 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 ε_i = +1 from ε_i = −1.

• 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

This mathematical analysis–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e