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Degree Growth of Rational Maps with Laurent Property

This paper investigates the degree growth of rational maps with the Laurent property, particularly Gale–Robinson recurrences, by developing a tropical reduction method to derive explicit growth formulas and proving a conjecture for a specific infinite family of three-term recurrences.

Original authors: Melih Emin Can

Published 2026-10-16

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CC BY 4.0

Original authors: Melih Emin Can

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how systems change over time. When a system is simple, its future is easy to predict. But when a system is complex, involving rules that feed back into themselves, its behavior can become wildly unpredictable. Mathematicians measure this complexity by watching how the "size" of the system's output grows with each step. If the size grows slowly, the system is orderly. If it explodes exponentially, the system is chaotic. A specific type of mathematical rule, known as a recurrence, generates a sequence of numbers or expressions where each new term is calculated from the ones before it. Some of these rules have a special, almost magical quality called the Laurent property. This property ensures that even though the rules involve division, the resulting terms never become messy fractions with complicated denominators; instead, they remain clean, structured expressions. This structure is a rare and valuable clue, often hinting that the system is not chaotic but is instead part of a deeper, hidden order found in nature and physics.

A researcher named Melih Emin Can has taken a close look at a family of these special rules to answer a fundamental question: exactly how fast do these systems grow? The paper focuses on a specific group of recurrences known as Gale–Robinson sequences. These are not just abstract curiosities; they appear in the study of cluster algebras and discrete integrable systems, which are mathematical frameworks used to model everything from particle physics to the geometry of space. While it was already known that these systems do not explode into chaos—meaning their growth is not exponential—no one had yet found a precise, closed formula to describe exactly how their size increases. The growth was known to be quadratic, meaning it follows a curve similar to the area of a square as its side length increases, but the exact details of that curve, including small, repeating wiggles or corrections, remained a mystery. Can's work bridges this gap by developing a direct method to calculate this growth rate straight from the rules of the recurrence itself.

The core of Can's discovery is a new way to translate the complex algebraic rules into a simpler, more rigid form. Instead of tracking the full, complicated expressions, the method strips them down to their essential skeleton, focusing only on the exponents of the variables in the denominator. This simplification turns the problem into a "max-linear" recurrence, a type of rule where the next value is determined by taking the maximum of a few previous sums. By solving this simpler version, the researcher can reconstruct the exact growth of the original, complex system. This approach allows for the derivation of a precise formula for the degree of the system's iterates. The formula reveals that the growth is indeed quadratic, but it is not a smooth curve; it is a quadratic curve decorated with a periodic correction. Think of it like a staircase that rises steadily but has a repeating pattern of steps that are slightly higher or lower than the average, creating a rhythmic fluctuation in the growth rate.

The paper provides explicit formulas for two major families of these recurrences. For the two-term family, which involves rules with two parts, the author derives a complete formula that predicts the growth for any number of steps. This formula includes a specific period for the repeating correction, a number that depends entirely on the parameters of the recurrence. For example, in the famous Somos-4 and Somos-5 sequences, which are specific instances of these rules, the correction repeats every 8 and 14 steps respectively. The author confirms that these known results fit perfectly into the new, broader formula. The work then moves to the more complex three-term family, which involves rules with three parts. Here, the author formulates a bold conjecture for the general case, predicting that the growth will always follow a quadratic pattern with a periodic correction, and providing a specific formula to calculate the length of that repeating cycle based on the three numbers that define the rule.

While the general case for three-term rules remains a conjecture, the author proves it to be true for an infinite family of these rules where the first two parameters are fixed at 1 and 2, and the third can be any number greater than or equal to 3. This proof is a significant step forward, as it establishes that the quadratic growth with periodic correction is not just a coincidence for a few special cases but a robust feature of this entire class of systems. The results confirm that the algebraic entropy—a measure of chaos—for all these systems is zero, meaning they are perfectly ordered. By providing explicit formulas for the growth rates and the periods of their corrections, the paper transforms a vague understanding of these systems into a precise, calculable reality. It offers a clear map for navigating the growth of these complex mathematical structures, showing that even in systems that look complicated, there is a hidden, rhythmic order waiting to be uncovered.

Technical Summary: Degree Growth of Rational Maps with Laurent Property

Problem Statement
The paper investigates the degree growth of iterates of rational self-maps $f: \mathbb{P}^m \to \mathbb{P}^m$ , specifically those arising from recurrences possessing the Laurent property. The algebraic entropy $E(f) = \lim_{n \to \infty} \frac{1}{n} \log d_n$ , where $d_n = \deg(f^n)$ , serves as the primary metric for this growth. While it is known that recurrences with the Laurent property (such as Somos and Gale–Robinson sequences) often exhibit zero algebraic entropy (implying sub-exponential, typically quadratic, degree growth), existing literature largely establishes this qualitative behavior without providing explicit closed formulas for the degree sequences $d_n$ . The specific challenge addressed is to derive these explicit formulas directly from the recurrence relations, particularly for the two-term and three-term Gale–Robinson families.

Methodology
The author develops a direct method to determine degree growth by analyzing the recurrence relation itself, bypassing the need for an a priori description in terms of cluster seeds. The core of the methodology involves:

Tropicalization and Valuation Analysis: Utilizing discrete valuation rings and the concept of $x_i$ -adic valuations, the paper demonstrates that the denominator exponents of the iterates (which dictate the degree) satisfy a max-linear (tropical) recurrence. This reduces the problem of computing polynomial degrees to studying sequences defined by $\max$ operations.
Direct Degree Formula (Theorem 1): For a broad class of homogeneous quadratic Laurent recurrences, the paper derives a formula expressing the degree of the $n$ -th iterate, $d_n$ , as a sum of the maximum values of auxiliary sequences $\{k^{(i)}_n\}$ over a sliding window, plus one. These auxiliary sequences satisfy a max-linear recurrence derived directly from the coefficients of the original polynomial recurrence.
Explicit Computation of Max-Linear Sequences: The paper focuses on solving specific max-linear recurrences associated with Gale–Robinson systems. By analyzing the structure of these sequences, the author identifies that they follow a quadratic growth pattern $n^2/2N$ corrected by a periodic term.

Key Contributions and Results

General Degree Formula (Theorem 1): The paper establishes a closed formula for the degree of iterates for homogeneous quadratic Laurent recurrences. It proves that $d_n$ can be computed by summing the maxima of sequences $\{k^{(i)}_n\}$ , which are defined by a max-linear recurrence:
$k^{(i)}_{n+m} = \max_{(s,t) \in C} (k^{(i)}_{n+s} + k^{(i)}_{n+t}) - k^{(i)}_n$
where $C$ depends on the non-zero coefficients of the recurrence. This formulation allows for the direct computation of degree growth without prior knowledge of the cluster algebra structure.
Two-Term Gale–Robinson Recurrences (Theorem 2): For the two-term family defined by $x_{n+m}x_n = x_{n+1}x_{n+m-1} + x_{n+2}x_{n+m-2}$ , the author derives an explicit quadratic growth formula for the associated tropical sequence $\{k_n\}$ :
$k_n = \frac{n^2 + c_n}{2N}$
where $N = (m-2)(m-1) + 2$ and $\{c_n\}$ is a periodic sequence with period $N$ . This result recovers and generalizes previous case-by-case computations for Somos-4 ( $m=4$ ) and Somos-5 ( $m=5$ ).
Three-Term Gale–Robinson Recurrences (Conjecture 1 and Theorem 3):
- Conjecture: The paper formulates a conjecture for the general three-term family ( $x_{n+m}x_n = x_{n+p}x_{n+m-p} + x_{n+q}x_{n+m-q} + x_{n+r}x_{n+m-r}$ ). It predicts that the growth rate sequence is quadratic with a periodic correction, where the period $N$ is given explicitly by $N = \gcd(p,q)r^2 + \gcd(p,r)q^2 + \gcd(q,r)p^2 + pqr$ .
- Proof: The author proves this conjecture for the infinite family where parameters are $p=1, q=2$ , and $r \ge 3$ . The proof relies on combinatorial lemmas establishing the symmetry and periodicity of the max-linear sequence.
- Corollaries: Applying these results, the paper confirms that Somos-6 and Somos-7 maps have zero algebraic entropy and provides explicit degree formulas for their iterates. Specifically, for Somos-6 (corresponding to parameters $p=1, q=2, r=3$ and $m=6$ ), the period is $N=20$ . For Somos-7 (corresponding to parameters $p=1, q=2, r=4$ and $m=7$ ), the period is $N=30$ .

Significance and Claims
The paper claims its primary contribution is the development of a direct method for determining degree growth from recurrence relations, reducing the problem to the study of associated max-linear recurrences.

Explicit Formulas: Unlike previous works that established quadratic growth qualitatively (e.g., via combinatorial formulas for octahedron or cube recurrences), this paper provides explicit closed formulas for the degree sequences, including the precise leading coefficients and the periods of the correction terms.
Refinement of Zero-Entropy Results: While confirming the known result that these systems have zero algebraic entropy, the paper refines this by characterizing the exact asymptotic behavior of the degree sequence.
Uniformity: The derived formulas offer a uniform description for infinite families of recurrences (specifically the $p=1, q=2$ family for three-term recurrences) where such uniform formulas, including the explicit period of the correction term, had not previously been established.

The work is presented as a continuation of the author's master's studies, aiming to bridge the gap between the qualitative understanding of integrability in Laurent systems and the quantitative analysis of their degree growth.

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