D.14.1.3 PH_ais

Usage:

PH_ais(I); I ideal of coordinates of the vector field.

Return:

the Poincare-Hopf index of type int.

Note:

the isolated singularity must be algebraically isolated.

Theory:

The Poincare-Hopf index of a real vector field X at the isolated singularity 0 is the degree of the map (X/|X|) : S_epsilon ---> S, where S is the unit sphere, and the spheres are oriented as (n-1)-spheres in R^n. The degree depends only on the germ, X, of X at 0. If the vector field X is real analytic, then an invariant of the germ is its local ring
Qx=R[[x1..xn]]/Ix
where R[[x1,..,xn]] is the ring of germs at 0 of real-valued analytic functions on R^n, and Ix is the ideal generated by the components of X. The isolated singularity of X is algebraically isolated if the algebra Qx is finite dimensional as real vector space, geometrically this mean that 0 is also an isolated singularity for the complexified vector field. In this case the Poincare-Hopf index is the signature of the non degenerate bilinear form <,> obtained by composition of the product in the algebra Qx with a linear functional map
<,> : (Qx)x(Qx) ---(.)--> Qx ---(L)--> R
with L(Jo)>0, where Jo is the residue class of the Jacobian determinant in Qx. Here, we use a natural linear functional defined as follows. Suppose that E={E_1,..E_r} is a basis of Qx, then Jo can be written as
Jo=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r, where a_s are constant. The linear functional L:Qx--->R is defined as L(E_{j1})=(a_1)/|a_1|=sign of a_1,
the other elements of the base are sent to 0.
Refs. -Eisenbud & Levine, An algebraic formula for the degree of a C^\infty map germ, Ann. Math., 106, (1977), 19-38.
-Khimshiashvili, On a local degree of a smooth map, trudi Tbilisi Math. Inst., (1980), 105-124.

LIB "phindex.lib";
ring r=0,(x,y,z),ds;
ideal I=x3-3xy2,-y3+3yx2,z3;
PH_ais(I);
==> 3