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

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Abstract

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.

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Authors and Affiliations

  1. División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Km 1. Carretera Cunaducan-Jalpa de Méndez, C. P. 86690, Cunduacan, Tabasco, Mexico

    Víctor Castellanos

  2. Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México, Apdo Postal 61-3 (Xangari), C. P. 58089, Morelia, Michoacan, Mexico

    Abel Castorena

  3. Departamento de Matemáticas, Universidad de Guanajuato, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Guanajuato, Mexico

    Manuel Cruz-López

Authors
  1. Víctor Castellanos
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  2. Abel Castorena
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  3. Manuel Cruz-López
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Correspondence to Víctor Castellanos.

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Castellanos, V., Castorena, A. & Cruz-López, M. Algebraic and Computational Formulas for the Index of Real Analytic Vector Fields. Results. Math. 59, 125–139 (2011). https://doi.org/10.1007/s00025-010-0066-9

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  • DOI: https://doi.org/10.1007/s00025-010-0066-9

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