Ak singularity

In mathematics, and in particular singularity theory an "A""k", 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" → R be a smooth function. We denote by Ω(R"n",R) the infinite-dimensional space of all such functions. Let diff(R"n") denote the infinite-dimensional Lie group of diffeomorphisms R"n" → R"n", and diff(R) the infinite-dimensional Lie group of diffeomorphisms R → R. The product group diff(R"n") × diff(R) acts on Ω(R"n",R) in the following way: let φ : R"n" → R"n" and ψ : R → R be diffeomorphisms and "f" : R"n" → R any smooth function. We define the group action as follows:
The orbit of "f", denoted orb("f"), of this group action is given by
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 "A""k"-singularity if it lies in the orbit of
where formula_4 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.