In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ.

If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory.

Algebraic definition

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Consider a holomorphic complex function germ

 

and denote by   the ring of all function germs  . Every level of a function is a complex hypersurface in  , therefore   is dubbed a hypersurface singularity.

Assume it is an isolated singularity: in the case of holomorphic mappings it is said that a hypersurface singularity   is singular at   if its gradient   is zero at  , and it is said that   is an isolated singular point if it is the only singular point in a sufficiently small neighbourhood of  . In particular, the multiplicity of the gradient

 

is finite by an application of Rückert's Nullstellensatz. This number   is the Milnor number of singularity   at  .

Note that the multiplicity of the gradient is finite if and only if the origin is an isolated critical point of f.

Geometric interpretation

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Milnor originally[1] introduced   in geometric terms in the following way. All fibers   for values   close to   are nonsingular manifolds of real dimension  . Their intersection with a small open disc   centered at   is a smooth manifold   called the Milnor fiber. Up to diffeomorphism   does not depend on   or   if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map.

The Milnor fiber   is a smooth manifold of dimension   and has the same homotopy type as a bouquet of   spheres  . This is to say that its middle Betti number   is equal to the Milnor number and it has homology of a point in dimension less than  . For example, a complex plane curve near every singular point   has its Milnor fiber homotopic to a wedge of   circles (Milnor number is a local property, so it can have different values at different singular points).

Thus the following equalities hold:

Milnor number = number of spheres in the wedge = middle Betti number of   = degree of the map   on   = multiplicity of the gradient  

Another way of looking at Milnor number is by perturbation. It is said that a point is a degenerate singular point, or that f has a degenerate singularity, at   if   is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at  :

 

It is assumed that f has a degenerate singularity at 0. The multiplicity of this degenerate singularity may be considered by thinking about how many points are infinitesimally glued. If the image of f is now perturbed in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate. The number of such isolated non-degenerate singularities will be the number of points that have been infinitesimally glued.

Precisely, another function germ g which is non-singular at the origin is taken and considered the new function germ h := f + εg where ε is very small. When ε = 0 then h = f. The function h is called the morsification of f. It is very difficult to compute the singularities of h, and indeed it may be computationally impossible. This number of points that have been infinitesimally glued, this local multiplicity of f, is exactly the Milnor number of f.

Further contributions[2] give meaning to Milnor number in terms of dimension of the space of versal deformations, i.e. the Milnor number is the minimal dimension of parameter space of deformations that carry all information about initial singularity.

Examples

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Given below are some worked examples of polynomials in two variables. Working with only a single variable is too simple and does not give an appropriate illustration of the techniques, whereas working with three variables can be cumbersome. Note that if f is only holomorphic and not a polynomial, then the power series expansion of f can be used.

Consider a function germ with a non-degenerate singularity at 0, say  . The Jacobian ideal is just  . Computing the local algebra:

 

Hadamard's lemma, which says that any function   may be written as

 

for some constant k and functions   and   in   (where either   or   or both may be exactly zero), justifies this. So, modulo functional multiples of x and y, the function h may be written as a constant. The space of constant functions is spanned by 1, hence  

It follows that μ(f) = 1. It is easy to check that for any function germ g with a non-degenerate singularity at 0, μ(g) = 1.

Note that applying this method to a non-singular function germ g yields μ(g) = 0.

Let  , then

 

So in this case  .

It may be shown that if   then  

This can be explained by the fact that f is singular at every point of the x-axis.

Versal deformations

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Let f have finite Milnor number μ, and let   be a basis for the local algebra, considered as a vector space. Then a miniversal deformation of f is given by

 
 

where  . These deformations (or unfoldings) are of great interest in much of science. [citation needed]

Invariance

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Function germs can be collected together to construct equivalence classes. One standard equivalence is A-equivalence. It is said that two function germs   are A-equivalent if there exist diffeomorphism germs   and   such that  : there exists a diffeomorphic change of variable in both domain and range which takes f to g. If f and g are A-equivalent then μ(f) = μ(g).[citation needed]

Nevertheless, the Milnor number does not offer a complete invariant for function germs, i.e. the converse is false: there exist function germs f and g with μ(f) = μ(g) which are not A-equivalent. To see this consider   and  . This yields   but f and g are clearly not A-equivalent since the Hessian matrix of f is equal to zero while that of g is not (and the rank of the Hessian is an A-invariant, as is easy to see).

References

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  1. ^ Milnor, John (1969). Singular points of Complex Hypersurfaces. Annals of Mathematics Studies. Princeton University Press.
  2. ^ Arnold, V.I.; Gusein-Zade, S.M.; Varchenko, A.N. (1988). Singularities of differentiable maps. Vol. 2. Birkhäuser.