Ak singularity

Article summary

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) \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 , denoted , 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 and a diffeomorphic change of coordinate in such that one member of the orbit is carried to any other.…

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