Ak singularity

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

Let $$f: \R^n \to \R$$ be a smooth function. We denote by $$\Omega (\R^n,\R)$$ the infinite-dimensional space of all such functions. Let $$\operatorname{diff}(\R^n)$$ denote the infinite-dimensional Lie group of diffeomorphisms $$\R^n \to \R^n,$$ and $$\operatorname{diff}(\R)$$ the infinite-dimensional Lie group of diffeomorphisms $$\R \to \R.$$ The product group $$\operatorname{diff}(\R^n) \times \operatorname{diff}(\R)$$ acts on $$\Omega (\R^n,\R)$$ in the following way: let $$\varphi : \R^n \to \R^n$$ and $$\psi : \R \to \R$$ be diffeomorphisms and $$f: \R^n \to \R$$ any smooth function. We define the group action as follows:
 * $$ (\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^{-1}$$

The orbit of $f$, denoted $orb(f)$, of this group action is given by
 * $$ \mbox{orb}(f) = \{ \psi \circ f \circ \varphi^{-1} : \varphi \in \mbox{diff}(\R^n), \psi \in \mbox{diff}(\R ) \} \ . $$

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in $\R^n$ and a diffeomorphic change of coordinate in $\R$ 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
 * $$ f(x_1,\ldots,x_n) = 1 + \varepsilon_1x_1^2 + \cdots + \varepsilon_{n-1}x^{2}_{n-1} \pm x_n^{k+1}$$

where $$\varepsilon_i = \pm 1$$ 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 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 $εi = +1$ from $εi = −1$.