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The approach of cluster symmetry to Diophantine equations

Leizhen Bao, Fang Li

TL;DR

The paper develops a systematic framework that ties cluster theory to Diophantine equations by introducing cluster-symmetric maps and invariant Laurent polynomials, enabling solution generation via mutations.A central technical contribution is a necessary-and-sufficient criterion (via homogeneous linear equations) for a Laurent polynomial to be invariant under a cluster-symmetric map, together with computational tools to construct such invariants.The authors apply the framework to Markov-type equations, proving that several Markov-cluster polynomials share the same orbit structure as the classical Markov equation and providing explicit seeds and mutations for rank-3 cases like $F_{3,10}$ and $F_{3,7}$, thereby solving corresponding Diophantine problems.Additionally, the work clarifies when a Laurent polynomial can be realized within a generalized cluster algebra, develops algorithms to identify cluster-symmetric pairs, and outlines a practical workflow for constructing invariant rings and associated Diophantine equations, highlighting applications and connections across cluster theory, invariant theory, and number theory.

Abstract

This paper aims to employ a cluster-theoretic approach to provide a class of Diophantine equations whose solutions can be obtained by starting from initial solutions through mutations. We establish a novel framework bridging cluster theory and Diophantine equations through the lens of cluster symmetry. On the one hand, we give the necessary and sufficient condition for Laurent polynomials to remain invariant under a given cluster symmetric map. On the other hand, we construct a discriminant algorithm to determine whether a given Laurent polynomial has cluster symmetry and whether it can be realized in a generalized cluster algebra. As applications, we solve Markov-cluster equations, describe some invariant Laurent polynomial rings, and resolve the questions posed by Gyoda and Matsushita.

The approach of cluster symmetry to Diophantine equations

TL;DR

The paper develops a systematic framework that ties cluster theory to Diophantine equations by introducing cluster-symmetric maps and invariant Laurent polynomials, enabling solution generation via mutations.A central technical contribution is a necessary-and-sufficient criterion (via homogeneous linear equations) for a Laurent polynomial to be invariant under a cluster-symmetric map, together with computational tools to construct such invariants.The authors apply the framework to Markov-type equations, proving that several Markov-cluster polynomials share the same orbit structure as the classical Markov equation and providing explicit seeds and mutations for rank-3 cases like $F_{3,10}$ and $F_{3,7}$, thereby solving corresponding Diophantine problems.Additionally, the work clarifies when a Laurent polynomial can be realized within a generalized cluster algebra, develops algorithms to identify cluster-symmetric pairs, and outlines a practical workflow for constructing invariant rings and associated Diophantine equations, highlighting applications and connections across cluster theory, invariant theory, and number theory.

Abstract

This paper aims to employ a cluster-theoretic approach to provide a class of Diophantine equations whose solutions can be obtained by starting from initial solutions through mutations. We establish a novel framework bridging cluster theory and Diophantine equations through the lens of cluster symmetry. On the one hand, we give the necessary and sufficient condition for Laurent polynomials to remain invariant under a given cluster symmetric map. On the other hand, we construct a discriminant algorithm to determine whether a given Laurent polynomial has cluster symmetry and whether it can be realized in a generalized cluster algebra. As applications, we solve Markov-cluster equations, describe some invariant Laurent polynomial rings, and resolve the questions posed by Gyoda and Matsushita.

Paper Structure

This paper contains 21 sections, 36 theorems, 266 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Invariant Laurent polynomials of cluster symmetric maps
  3. Cluster symmetric maps of datum
  4. Construction of a cluster symmetric polynomial
  5. Examples and practice-level discussion
  6. Cluster symmetric maps and generalized cluster algebras
  7. Generalized cluster algebra
  8. Cluster symmetric maps of a seed
  9. Existence of some cluster symmetric polynomials as examples
  10. Orbits of solution sets under action of cluster symmetric groups
  11. Orbits of solution sets of general cluster symmetric equations
  12. Uniqueness of orbits of solution sets of some special Diophantine equations
  13. Proof of Theorem \ref{['thm markov-cluster sol']}
  14. Solutions of the equation $F_{3,10}(\mathbf{x}) = F_{3,10}(\mathbf{1})$
  15. Solutions of the equation $F_{3,7}(\mathbf{x}) = F_{3,7}(\mathbf{1})$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Given a cluster symmetric map $\psi_{\sigma, s, \omega_s}$. Let $F(\mathbf{x})$ be a Laurent polynomial of type $\frac{\bm{\eta}}{\mathbf{d}}$ in $\mathbb{Q}[\mathbf{x}^{\pm}]$ and its expansion is where $\bm{\eta} \in \mathbb{Z}_{\geq 0}^n, \mathbf{d} \in \mathbb{Z}^n$ with $\mathbf{d} = \sigma{(\mathbf{d})}$ and $\eta_s = \eta_{\sigma^{-1}(s)} = 2d_s = 2d_{\sigma^{-1}(s)}$. holds, if and only

Figures (5)

  • Figure 1: Relationship between cluster theory and Diophantine equations
  • Figure 2: Result of Code \ref{['code: somos5']}
  • Figure 3: Result of Code \ref{['code: gyoda']}
  • Figure 4: Result of Code \ref{['code: pair1']}.
  • Figure 5: Result of Code \ref{['code: pair2']}.

Theorems & Definitions (111)

  • Theorem 1.1: (Theorem \ref{['thm len1_T2f']} and \ref{['thm len1']}, Remark \ref{['rmk HLE']})
  • Remark 1.2
  • Theorem 1.3: (Algorithm \ref{['algo diop2cs']}, Proposition \ref{['prop S(F)']})
  • Proposition 1.4: (Definition \ref{['def seed and 1-csm']}, \ref{['def S1 of F']}, \ref{['def 1-cs seed']}, Proposition \ref{['prop find 1-csSeed']})
  • Theorem 1.5: (Theorem \ref{['thm markov-cluster sol']}, Definition \ref{['def mc eqn']})
  • Theorem 1.6: (Corollary \ref{['cor rank3 nonconstant']}, Remark \ref{['rmk irr B']})
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • ...and 101 more