Ak singularity

In mathematics, and in particular singularity theory, an Template:Mvar singularity, where Template:Math is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.

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

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

The orbit of Template:Mvar, denoted Template:Math, of this group action is given by

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

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in Template:Tmath and a diffeomorphic change of coordinate in Template:Tmath such that one member of the orbit is carried to any other. A function Template:Mvar is said to have a type Template:Mvar-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 Template:Math is an integer.

By a normal form we mean a particularly simple representative of any given orbit. The above expressions for Template:Mvar give normal forms for the type Template:Mvar-singularities. The type Template:Mvar-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 Template:Mvar.

This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish Template:Math from Template:Math.

References

• Template:Citation

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