标题： 聚类对称性在丢番图方程中的应用 显示英文标题

标题： The approach of cluster symmetry to Diophantine equations

摘要： 本文旨在使用一种簇理论方法，给出一种可以通过从初始解通过变异得到解的丢番图方程。我们通过簇对称性的视角，建立了一个连接簇理论和丢番图方程的新框架。一方面，我们给出了在给定的簇对称映射下，Laurent多项式保持不变的充要条件。另一方面，我们构建了一个判别算法，以确定相应的Laurent多项式是否具有簇对称对，并且是否可以在广义簇代数中实现。作为该框架的应用，我们解决了两个丢番图方程，描述了三类不变的Laurent多项式环，解决了Gyoda和Matsushita在\cite{Gyoda}中提出的两个问题，并最后分别给出了两个Matlab程序，用于构造不变的Laurent多项式和寻找簇对称对。 显示英文摘要

摘要： This paper aims to use a cluster-theoretic approach to give a kind of Diophantine equation 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 the corresponding Laurent polynomial admits a cluster symmetric pair and whether it can be realized in a generalized cluster algebra. As applications of this framework, we solve two Diophantine equations, describe three classes of invariant Laurent polynomial rings, resolve two questions posed by Gyoda and Matsushita in \cite{Gyoda}, and lastly give two Matlab programs, respectively, for constructing invariant Laurent polynomials and finding cluster symmetric pairs.