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 $r\geq3$.