
D.14.3.3 PH_ais
Procedure from library phindex.lib (see phindex_lib).
 Usage:
 PH_ais(I); I ideal of coordinates of the vector field.
 Return:
 the PoincareHopf index of type int.
 Note:
 the isolated singularity must be algebraically isolated.
 Theory:
 The PoincareHopf 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
(n1)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 realvalued 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 PoincareHopf 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), 1938.
Khimshiashvili, On a local degree of a smooth map, trudi
Tbilisi Math. Inst., (1980), 105124.
Example:
 LIB "phindex.lib";
ring r=0,(x,y,z),ds;
ideal I=x33xy2,y3+3yx2,z3;
PH_ais(I);
==> 3

