A cluster theory approach from mutation invariants to Diophantine equations

TL;DR

This work develops a Diophantine perspective on cluster algebras by classifying sign-equivalent exchange matrices and linking rank-2 finite versus affine types to reductive versus non-reductive mutation invariants. It provides a concrete mechanism—cluster mutations—to generate all positive integer solutions for key invariants, notably the Markov and its Lampe-type variants, and explains why finite mutation-type cases yield finite solution sets while affine-type cases can yield infinite families. The study yields two classes of Diophantine equations with cluster-algebraic structures, showing that solutions can be obtained through finite mutation sequences from simple initial seeds. The results deepen the connection between cluster mutations, Diophantine equations, and the structure of finite mutation type versus affine-type cluster algebras, with explicit instances across two- and three-variable settings.

Abstract

In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive integer points of the Markov mutation invariant and its variant. As an application, several classes of Diophantine equations with cluster algebraic structures are exhibited.

Theorems & Definitions (64)

• Theorem 1.1: \ref{['main']}
• Theorem 1.2: \ref{['finite solution']}
• Theorem 1.3: Aigner, \ref{["Aigner's theorem"]}
• Theorem 1.4: \ref{['analogue']}
• Proposition 1.5: \ref{['Prop case']}
• Proposition 1.6: \ref{['Lampe case']}
• Definition 2.1: Cluster algebra
• Definition 2.2: Finite type
• Example 2.3
• Definition 2.4: Finite mutation type