Ak singularity

Short description: Description of the degeneracy of a function

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

$(\phi ,\psi )\cdot f:=\psi \circ f\circ {\phi }^{-1}$

The orbit of f , denoted orb(f), of this group action is given by

$\text{orb}(f)=\{\psi \circ f\circ {\phi }^{-1}:\phi \in \text{diff}({\mathbb{ℝ}}^n),\psi \in \text{diff}(\mathbb{ℝ})\}.$

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in ${\mathbb{ℝ}}^n$ and a diffeomorphic change of coordinate in $\mathbb{ℝ}$ 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

$f(x_1,\dots ,x_n)=1+{\epsilon }_1{x}_1^2+\cdots +{\epsilon }_{n-1}{x}_{n-1}^2\pm x_n^{k+1}$

where ${\epsilon }_i=\pm 1$ 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.

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 0-8176-3187-9