Degree Growth of Rational Maps with Laurent Property

Abstract

Algebraic entropy measures the exponential growth of the degrees of iterates of rational self-maps. It is introduced by Bellon and Viallet in \cite{BellonandViallet}. We study degree growth for rational maps arising from recurrences with the Laurent property, with particular emphasis on Gale--Robinson recurrences. We develop a direct method for determining the degree growth from the recurrence relation itself, reducing the problem to the study of associated max-linear, or tropical recurrences. We first obtain a closed formula for the degrees of iterates for a broad class of homogeneous quadratic Laurent recurrences. We then apply this result to the two-term Gale--Robinson family and derive an explicit quadratic growth formula with a periodic correction term. For three-term Gale--Robinson recurrences, we express the algebraic entropy in terms of the associated tropical growth rate sequence. Based on explicit computations, we formulate a conjecture for the general three-term family predicting quadratic growth with a periodic correction. We prove the conjecture for the infinite family with parameters p=1, q=2, and r≥3.

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BibTeX RIS

BibTeX

@misc{can2026degree,
  title = {{Degree Growth of Rational Maps with Laurent Property}},
  author = {Melih Emin Can},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21627},
  eprint = {2610.21627},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21627},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Degree Growth of Rational Maps with Laurent Property
AU  - Melih Emin Can
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21627
UR  - https://arxiv.org/abs/2610.21627
AB  - Algebraic entropy measures the exponential growth of the degrees of iterates of rational self-maps. It is introduced by Bellon and Viallet in \cite{BellonandViallet}. We study degree growth for rational maps arising from recurrences with the Laurent property, with particular emphasis on Gale--Robinson recurrences. We develop a direct method for determining the degree growth from the recurrence relation itself, reducing the problem to the study of associated max-linear, or tropical recurrences. We first obtain a closed formula for the degrees of iterates for a broad class of homogeneous quadratic Laurent recurrences. We then apply this result to the two-term Gale--Robinson family and derive an explicit quadratic growth formula with a periodic correction term. For three-term Gale--Robinson recurrences, we express the algebraic entropy in terms of the associated tropical growth rate sequence. Based on explicit computations, we formulate a conjecture for the general three-term family predicting quadratic growth with a periodic correction. We prove the conjecture for the infinite family with parameters p=1, q=2, and r≥3.
ER  -

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