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A Fast Algorithmic Way to Calculate the Degree Growth of Birational Mappings

Author

Listed:
  • Basil Grammaticos

    (CNRS/IN2P3, IJCLab, Université Paris-Saclay and Université Paris-Cité, 91405 Orsay, France
    These authors contributed equally to this work.)

  • Alfred Ramani

    (CNRS/IN2P3, IJCLab, Université Paris-Saclay and Université Paris-Cité, 91405 Orsay, France
    These authors contributed equally to this work.)

  • Adrian-Stefan Carstea

    (Department of Theoretical Physics, NIPNE, 077125 Magurele, Romania
    These authors contributed equally to this work.)

  • Ralph Willox

    (School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
    These authors contributed equally to this work.)

Abstract

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings based on Halburd’s method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case, the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.

Suggested Citation

  • Basil Grammaticos & Alfred Ramani & Adrian-Stefan Carstea & Ralph Willox, 2025. "A Fast Algorithmic Way to Calculate the Degree Growth of Birational Mappings," Mathematics, MDPI, vol. 13(5), pages 1-21, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:737-:d:1598738
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    References listed on IDEAS

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    1. Papageorgiou, Vassilios G. & Nijhoff, Frank W., 1996. "On some integrable discrete-time systems associated with the Bogoyavlensky lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 228(1), pages 172-188.
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