### D.14.1 phindex_lib

Library :
  phindex.lib
Purpose:
  Procedures to compute the index of real analytic vector fields
Author:
  Victor Castellanos

Note:
  To compute the Poincare-Hopf index of a real analytic vector field with an algebraically isolated singularity at 0 (w. an a. i. s), we use the algebraic formula for the degree of the real analytic map germ found by Eisenbud-Levine in 1997. This result was also proved by Khimshiashvili. If the isolated singularity is non algebraically isolated and the vector field has similar reduced complex zeroes of codimension 1, we use a formula as the Eisenbud-Levine found by Victor Castellanos, in both cases is necessary to use a local order (ds,...). To compute the signature of a quadratic form (or symmetric matrix) we use the method of Lagrange.

Procedures:

> D.14.1.1 signatureL |  | signature of symmetric matrix M, method of Lagrange.
> D.14.1.2 signatureLqf |  | signature of quadratic form h, method of Lagrange.
> D.14.1.3 PH_ais |  | P-H index of real analytic vector field I w. an a. i. s.
> D.14.1.4 PH_nais |  | P-H index of real analytic vector field I w. a non a. i. s

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