Algebraic and Computational Formulas for the Index of Real Analytic Vector Fields

摘要

The signature formula of Eisenbud-Levine and Khimshiashvili for computing the Poincaré-Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non-algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also analyse some instances in the codimension 2 case and describe a computer implementation that permits the calculation of the index in both the algebraically and non-algebraically isolated cases.