Eisenbud–Levine–Khimshiashvili signature formula

In mathematics, and especially

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 x = (x₁, …, x_n).

Let X be a vector field on Rⁿ. For 1 ≤ k ≤ n there exist functions ƒ_k : Rⁿ → R such that one may express X as

X=f_{1}({\mathbf {x} })\,{\frac {\partial }{\partial x_{1}}}+\cdots +f_{n}({\mathbf {x} })\,{\frac {\partial }{\partial x_{n}}}.

To say that X is an analytic vector field means that each of the functions ƒ_k : Rⁿ → R is an

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 ƒ_k from above are function germs, i.e. ƒ_k : (Rⁿ,0) → (R,0). In turn, one may call X a vector field germ.

Construction

Let A_n,0 denote the ring of analytic function germs (Rⁿ,0) → (R,0). Assume that X is a vector field germ of the form

X=f_{1}({\mathbf {x} })\,{\frac {\partial }{\partial x_{1}}}+\cdots +f_{n}({\mathbf {x} })\,{\frac {\partial }{\partial x_{n}}}

with an algebraically isolated singularity at 0. Where, as mentioned above, each of the ƒ_k are function germs (Rⁿ,0) → (R,0). Denote by I_X the

local algebra, B_X, given by the quotient

B_{X}:=A_{n,0}/I_{X}\,.

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 B_X.^[2]^[4]^[5]

The dimension of $B_{X}$ is finite if and only if the

real algebra

.

Definition of the bilinear form

Using the analytic components of X, one defines another analytic germ F : (Rⁿ,0) → (Rⁿ,0) given by

F({\mathbf {x} }):=(f_{1}({\mathbf {x} }),\ldots ,f_{n}({\mathbf {x} })),

for all x ∈ Rⁿ. Let J_F ∈ A_n,0 denote the

Jacobian matrix of F with respect to the basis {∂/∂x₁, …, ∂/∂x_n}. Finally, let [J_F] ∈ B_X denote the equivalence class of J_F, modulo I_X. Using ∗ to denote multiplication in B_X one is able to define a non-degenerate bilinear form β as follows:^[2]^[4]

\beta :B_{X}\times B_{X}{\stackrel {*}{\longrightarrow }}B_{X}{\stackrel {\ell }{\longrightarrow }}\mathbb {R} ;\ \ \beta (g,h)=\ell (g*h),

where $\scriptstyle \ell$ is any linear function such that

\ell \left(\left[J_{F}\right]\right)>0.

As mentioned: the signature of β is exactly the index of X at 0.

Example

Consider the case n = 2 of a vector field on the plane. Consider the case where X is given by

X:=(x^{3}-3xy^{2})\,{\frac {\partial }{\partial x}}+(3x^{2}y-y^{3})\,{\frac {\partial }{\partial y}}.

Clearly X has an algebraically isolated singularity at 0 since X = 0 if and only if x = y = 0. The ideal I_X is given by (x³ − 3xy², 3x²y − y³), and

B_{X}=A_{2,0}/(x^{3}-3xy^{2},3x^{2}y-y^{3})\cong \mathbb {R} \langle 1,x,y,x^{2},xy,y^{2},xy^{2},y^{3},y^{4}\rangle .

The first step for finding the non-degenerate, bilinear form β is to calculate the multiplication table of B_X; reducing each entry modulo I_X. 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 J_F = 9(x⁴ + 2x²y² + y⁴), and so [J_F] = 24y⁴. Next one assigns values for $\scriptstyle \ell$ . One may take

\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.

This choice was made so that $\scriptstyle \ell \left(\left[J_{F}\right]\right)\,>\,0$ 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:

\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]

The

eigenvalues

of this matrix are −3, −3, −1, 1, 1, 2, 3, 3 and 4 There are 3 negative eigenvalues (#N = 3), and six positive eigenvalues (#P = 6); meaning that the signature of β is #P − #N = 6 − 3 = +3. 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.

polar coordinates