📝 SummarXiv

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 of this framework, we solve Markov-cluster equations, describe three classes of invariant Laurent polynomial rings, resolve two questions posed by Gyoda and Matsushita, and lastly give two MATLAB programs about our main theorems.