In mathematics, and particularly
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
, the Milnor number, named after
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Un ...
, 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, or is
infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group)
Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...
. It can be considered both a
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
invariant and an
algebraic invariant. This is why it plays an important role in
algebraic geometry and
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
.
Algebraic definition
Consider a holomorphic
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
function germ
:
and denote by
the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of all function germs
.
Every level of a function is a complex hypersurface in
, therefore we will call
a hypersurface
singularity.
Assume it is an
isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity
is singular at
if its
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is zero at
, a singular point is isolated if it is the only singular point in a sufficiently small
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
. 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
Isolation is the near or complete lack of social contact by an individual.
Isolation or isolated may also refer to:
Sociology and psychology
*Isolation (health care), various measures taken to prevent contagious diseases from being spread
**Is ...
critical point of ''f''.
Geometric interpretation
Milnor originally 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
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
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 we have equalities
:Milnor number = number of spheres in the
wedge
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
= middle
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
of
=
degree of the map on
= multiplicity of the gradient
Another way of looking at Milnor number is by
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbatio ...
. We say 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
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of all second order partial derivatives has zero
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
at
:
:
We assume that ''f'' has a degenerate singularity at 0. We can speak about the multiplicity of this degenerate singularity by thinking about how many points are
infinitesimally glued. If we now
perturb the image of ''f'' 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, we take another function germ ''g'' which is non-singular at the origin and consider 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
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
Here we give some worked examples in two variables. Working with only one is too simple and does not give a feel for the techniques, whereas working with three variables can be quite tricky. Two is a nice number. Also we stick to polynomials. If ''f'' is only
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
and not a polynomial, then we could have worked with the
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
expansion of ''f''.
1
Consider a function germ with a non-degenerate singularity at 0, say
. The Jacobian ideal is just
. We next compute the local algebra:
:
To see why this is true we can use
Hadamard's lemma which says that we can write any function
as
:
for some constant ''k'' and functions
and
in
(where either
or
or both may be exactly zero). So, modulo functional multiples of ''x'' and ''y'', we can write ''h'' 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 we get ''μ''(''g'') = 1.
Note that applying this method to a non-singular function germ ''g'' we get ''μ''(''g'') = 0.
2
Let
, then
:
So in this case
.
3
One can show that if
then
This can be ''explained'' by the fact that ''f'' is singular at every point of the ''x''-axis.
Versal Deformations
Let ''f'' have finite Milnor number ''μ'', and let
be a
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting o ...
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.
Invariance
We can collect function germs together to construct
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es. One standard equivalence is
''A''-equivalence. We say that two function germs
are ''A''-equivalent if there exist
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
germs
and
such that
: there exists a diffeomorphic change of variable in both
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
and
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
which takes ''f'' to ''g''.
If ''f'' and ''g'' are ''A''-equivalent then ''μ''(''f'') = ''μ''(''g'').
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
. We have
but ''f'' and ''g'' are clearly not ''A''-equivalent since the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
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
*
*
*
*
{{DEFAULTSORT:Milnor Number
Singularity theory
Algebraic geometry