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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 : f : (\mathbb^n,0) \to (\mathbb,0) \ and denote by \mathcal_n 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 (\mathbb^n,0) \to (\mathbb,0). Every level of a function is a complex hypersurface in \mathbb^n, therefore we will call f a hypersurface singularity. Assume it is an isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity f is singular at 0 \in \mathbb^n 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 ...
\nabla f is zero at 0 , 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 : \mu(f) = \dim_ \mathcal_n/\nabla f is finite by an application of Rückert's Nullstellensatz. This number \mu(f) is the Milnor number of singularity f at 0. 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 \mu(f) in geometric terms in the following way. All fibers f^(c) for values c close to 0 are nonsingular manifolds of real dimension 2(n-1). Their intersection with a small open disc D_ centered at 0 is a smooth manifold F called the Milnor fiber. Up to diffeomorphism F does not depend on c or \epsilon if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map. The Milnor fiber F is a smooth manifold of dimension 2(n-1) and has the same homotopy type as a bouquet of \mu(f) spheres S^. 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 ...
b_(F) is equal to the Milnor number and it has homology of a point in dimension less than n-1. For example, a complex plane curve near every singular point z_0 has its Milnor fiber homotopic to a wedge of \mu_(f) 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 F = degree of the map z\to \frac on S_\epsilon = multiplicity of the gradient \nabla f 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 z_0 \in \mathbb^n if z_0 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 z_0: : \det\left( \frac \right)_^ =0. 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 f(x,y) = x^2 + y^2. The Jacobian ideal is just \langle 2x, 2y \rangle = \langle x, y \rangle . We next compute the local algebra: : \mathcal_f = \mathcal / \langle x, y \rangle = \langle 1 \rangle . To see why this is true we can use Hadamard's lemma which says that we can write any function h\in\mathcal as : h(x,y) = k + xh_1(x,y) + yh_2(x,y) for some constant ''k'' and functions h_1 and h_2 in \mathcal (where either h_1 or h_2 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 \mathcal_f = \langle 1 \rangle 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 f(x,y) = x^3 + xy^2, then : \mathcal_f = \mathcal / \langle 3x^2 + y^2, xy \rangle = \langle 1, x, y, x^2 \rangle . So in this case \mu(f) = 4.


3

One can show that if f(x,y) = x^2y^2 + y^3 then \mu(f) = \infty. 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 g_1,\ldots, g_ 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 : F : (\mathbb^n \times \mathbb^,0) \to (\mathbb,0) , : F(z,a) := f(z) + a_1g_1(z) + \cdots + a_g_(z) , where (a_1,\dots,a_)\in \mathbb^. 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 f,g : (\mathbb^n,0) \to (\mathbb,0) 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 \phi : (\mathbb^n,0) \to (\mathbb^n,0) and \psi : (\mathbb,0) \to (\mathbb,0) such that f \circ \phi = \psi \circ g: 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 f(x,y) = x^3+y^3 and g(x,y) = x^2+y^5. We have \mu(f) = \mu(g) = 4 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