The approach of cluster symmetry to Diophantine equations

TL;DR

This paper introduces a novel cluster-theoretic framework to solve certain Diophantine equations by leveraging cluster symmetry, providing new methods to identify invariant Laurent polynomials and solve related equations.

Contribution

It establishes a new connection between cluster theory and Diophantine equations, including criteria for Laurent polynomial invariance and algorithms for detecting cluster symmetry.

Findings

Solved Markov-cluster equations using the new framework

Described invariant Laurent polynomial rings under cluster symmetry

Resolved questions posed by Gyoda and Matsushita

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.