Ak singularity explained

In mathematics, and in particular singularity theory, an singularity, where 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)} x \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) ⋅ f : = \psi \circ f \circ \varphi ^{-1} The orbit of, denoted, of this group action is given byorb (f) = \{ \psi \circ f \circ \varphi ^{-1} : \varphi \in diff (\R ^{n),} \psi \in 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 and a diffeomorphic change of coordinate in such that one member of the orbit is carried to any other. A function is said to have a type -singularity if it lies in the orbit of

f(x _{1,\ldots,x} _{n)} = 1 + \varepsilon _{1x} ^{2} ^{1} + … + \varepsilon _{n-1} x ^{2} _{n -1} \pm x ^{k +1} _{n} where \varepsilon _{i} = \pm 1 and is an integer.

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

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