
D.14.3.4 PH_nais
Procedure from library phindex.lib (see phindex_lib).
 Usage:
 PH_nais(I); I ideal of coordinates of the vector field.
 Return:
 the PoincareHopf index of type int.
 Note:
 the vector field must be a non algebraically isolated singularity
at 0, with reduced complex zeros of codimension 1.
 Theory:
 Suppose that 0 is an algebraically isolated singularity of the real
analytic vector field X, geometrically this corresponds to the fact that the
complexified vector field has positive dimension singular locus,
algebraically this mean that the 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 is infinite dimensional as real vector space. In the case that
X has a reduced hypersurface as complex zeros we have the next.
There exist a real analytic function f:R^n>R, and a real analytic
vector field Y s. t. X=fY. The function f does not change of sign
out of 0 and
Mx=R[[x1..xn]]/(Ix : radical(Ix))
is a finite dimensional subalgebra of Qx. The PoincareHopf index
of X at 0 is the sign of f times the signature of the non degenerate
bilinear form <,> obtained by composition of the product in the
algebra Mx with a linear functional map
<,> : (Mx)x(Mx) (.)> Mx (L)> R
with L(Jp)>0, where Jp is the residue class of the Jacobian
determinant of X, JX, over f^n, JX/(f^n) in Mx. Here, we use a
natural linear functional defined as follows. Suppose that
E={E_1,..E_r} is a basis of Mx, then Jp is writing as
Jp=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r,
where a_s are constant. The linear functional L:M>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. CastellanosVargas, V., Una formula algebraica del indice de
PoincareHopf para campos vectoriales reales con una variedad
de ceros complejos, Ph. D. thesis CIMAT (2000), chapther 1,
Guanajuato Mexico.
Castellanos Vargas, V. The index of non algebraically
isolated singularity, Bol. Soc. Mat. Mexicana, (3)
Vol. 8, 2002, 141147.
Example:
 LIB "phindex.lib";
ring r=0,(x,y,z),ds;
ideal I=x52x3y23xy4+x3z23xy2z2,3x4y2x2y3+y53x2yz2+y3z2,x2z3+y2z3+z5;
PH_nais(I);
==> 3

