Английская Википедия:Eisenbud–Levine–Khimshiashvili signature formula
Шаблон:Short description In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré–Hopf index of a real, analytic vector field at an algebraically isolated singularity.[1][2] It is named after David Eisenbud, Harold I. Levine, and George Khimshiashvili. Intuitively, the index of a vector field near a zero is the number of times the vector field wraps around the sphere. Because analytic vector fields have a rich algebraic structure, the techniques of commutative algebra can be brought to bear to compute their index. The signature formula expresses the index of an analytic vector field in terms of the signature of a certain quadratic form.
Nomenclature
Consider the n-dimensional space Rn. Assume that Rn has some fixed coordinate system, and write x for a point in Rn, where Шаблон:Nowrap
Let X be a vector field on Rn. For Шаблон:Nowrap there exist functions Шаблон:Nowrap such that one may express X as
- <math> X = f_1({\mathbf x})\,\frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} . </math>
To say that X is an analytic vector field means that each of the functions Шаблон:Nowrap is an analytic function. One says that X is singular at a point p in Rn (or that p is a singular point of X) if Шаблон:Nowrap, i.e. X vanishes at p. In terms of the functions Шаблон:Nowrap it means that Шаблон:Nowrap for all Шаблон:Nowrap. A singular point p of X is called isolated (or that p is an isolated singularity of X) if Шаблон:Nowrap and there exists an open neighbourhood Шаблон:Nowrap, containing p, such that Шаблон:Nowrap for all q in U, different from p. An isolated singularity of X is called algebraically isolated if, when considered over the complex domain, it remains isolated.[3][4]
Since the Poincaré–Hopf index at a point is a purely local invariant (cf. Poincaré–Hopf theorem), one may restrict one's study to that of germs. Assume that each of the ƒk from above are function germs, i.e. Шаблон:Nowrap In turn, one may call X a vector field germ.
Construction
Let An,0 denote the ring of analytic function germs Шаблон:Nowrap. Assume that X is a vector field germ of the form
- <math> X = f_1({\mathbf x})\,\frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} </math>
with an algebraically isolated singularity at 0. Where, as mentioned above, each of the ƒk are function germs Шаблон:Nowrap. Denote by IX the ideal generated by the ƒk, i.e. Шаблон:Nowrap Then one considers the local algebra, BX, given by the quotient
- <math> B_X := A_{n,0} / I_X \, . </math>
The Eisenbud–Levine–Khimshiashvili signature formula states that the index of the vector field X at 0 is given by the signature of a certain non-degenerate bilinear form (to be defined below) on the local algebra BX.[2][4][5]
The dimension of <math>B_X</math> is finite if and only if the complexification of X has an isolated singularity at 0 in Cn; i.e. X has an algebraically isolated singularity at 0 in Rn.[2] In this case, BX will be a finite-dimensional, real algebra.
Definition of the bilinear form
Using the analytic components of X, one defines another analytic germ Шаблон:Nowrap given by
- <math> F({\mathbf x}) := (f_1({\mathbf x}), \ldots, f_n({\mathbf x})) , </math>
for all Шаблон:Nowrap. Let Шаблон:Nowrap denote the determinant of the Jacobian matrix of F with respect to the basis Шаблон:Nowrap Finally, let Шаблон:Nowrap denote the equivalence class of JF, modulo IX. Using ∗ to denote multiplication in BX one is able to define a non-degenerate bilinear form β as follows:[2][4]
- <math> \beta : B_X \times B_X \stackrel{*}{\longrightarrow} B_X \stackrel{\ell}{\longrightarrow} \R; \ \ \beta(g,h) = \ell(g*h) , </math>
where <math>\scriptstyle \ell</math> is any linear function such that
- <math> \ell \left( \left[ J_F \right] \right) > 0 . </math>
As mentioned: the signature of β is exactly the index of X at 0.
Example
Consider the case Шаблон:Nowrap of a vector field on the plane. Consider the case where X is given by
- <math>X := (x^3 - 3xy^2) \, \frac{\partial}{\partial x} + (3x^2y - y^3) \, \frac{\partial}{\partial y} . </math>
Clearly X has an algebraically isolated singularity at 0 since Шаблон:Nowrap if and only if Шаблон:Nowrap The ideal IX is given by Шаблон:Nowrap and
- <math> B_X = A_{2,0} / (x^3 - 3xy^2, 3x^2y - y^3) \cong \R\langle 1, x, y, x^2, xy, y^2, xy^2, y^3, y^4 \rangle . </math>
The first step for finding the non-degenerate, bilinear form β is to calculate the multiplication table of BX; reducing each entry modulo IX. Whence
| ∗ | 1 | x | y | x2 | xy | y2 | xy2 | y3 | y4 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | x | y | x2 | xy | y2 | xy2 | y3 | y4 |
| x | x | x2 | xy | 3xy3 | y3/3 | xy2 | y4/3 | 0 | 0 |
| y | y | xy | y2 | y3/3 | xy2 | y3 | 0 | y4 | 0 |
| x2 | x2 | 3xy2 | y3/3 | y4 | 0 | y4/3 | 0 | 0 | 0 |
| xy | xy | y3/3 | xy2 | 0 | y4/3 | 0 | 0 | 0 | 0 |
| y2 | y2 | xy2 | y3 | y4/3 | 0 | y4 | 0 | 0 | 0 |
| xy2 | xy2 | y4/3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| y3 | y3 | 0 | y4 | 0 | 0 | 0 | 0 | 0 | 0 |
| y4 | y4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Direct calculation shows that Шаблон:Nowrap, and so Шаблон:Nowrap Next one assigns values for <math>\scriptstyle \ell</math>. One may take
- <math> \ell(1) = \ell(x) = \ell(y) = \ell(x^2) = \ell(xy) = \ell(y^2) = \ell(xy^2) = \ell(y^3) = 0, \ \text{and} \ \ell(y^4) = 3 .</math>
This choice was made so that <math>\scriptstyle \ell\left( \left[ J_F \right] \right) \, > \, 0</math> as was required by the hypothesis, and to make the calculations involve integers, as opposed to fractions. Applying this to the multiplication table gives the matrix representation of the bilinear form β with respect to the given basis: <math display="block"> \left[ \begin{array}{ccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right] </math> The eigenvalues of this matrix are Шаблон:Nowrap There are 3 negative eigenvalues (Шаблон:Nowrap), and six positive eigenvalues (Шаблон:Nowrap); meaning that the signature of β is Шаблон:Nowrap. It follows that X has Poincaré–Hopf index +3 at the origin.
Topological verification
With this particular choice of X it is possible to verify the Poincaré–Hopf index is +3 by a direct application of the definition of Poincaré–Hopf index.[6] This is very rarely the case, and was the reason for the choice of example. If one takes polar coordinates on the plane, i.e. Шаблон:Nowrap and Шаблон:Nowrap then Шаблон:Nowrap and Шаблон:Nowrap Restrict X to a circle, centre 0, radius Шаблон:Nowrap, denoted by C0,ε; and consider the map Шаблон:Nowrap given by
- <math> G\colon X \longmapsto \frac{X}{||X||} .</math>
The Poincaré–Hopf index of X is, by definition, the topological degree of the map G.[6] Restricting X to the circle C0,ε, for arbitrarily small ε, gives
- <math> G(\theta) = (\cos(3\theta),\sin(3\theta)) , \, </math>
meaning that as θ makes one rotation about the circle C0,ε in an anti-clockwise direction; the image G(θ) makes three complete, anti-clockwise rotations about the unit circle C0,1. Meaning that the topological degree of G is +3 and that the Poincaré–Hopf index of X at 0 is +3.[6]
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