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

Basil Grammaticos, Alfred Ramani, Adrian-Stefan Carstea and Ralph Willox ()
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Basil Grammaticos: CNRS/IN2P3, IJCLab, Université Paris-Saclay and Université Paris-Cité, 91405 Orsay, France
Alfred Ramani: CNRS/IN2P3, IJCLab, Université Paris-Saclay and Université Paris-Cité, 91405 Orsay, France
Adrian-Stefan Carstea: Department of Theoretical Physics, NIPNE, 077125 Magurele, Romania
Ralph Willox: School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Mathematics, 2025, vol. 13, issue 5, 1-21

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.

Keywords: mappings; integrability; singularities; degree growth; dynamical degree (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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