arXiv subjects

Leizhen Bao

Publications and source records attributed to Leizhen Bao.

2 recordsLinked to original sources

arXiv2026

The approach of cluster symmetry to Diophantine equations

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.

math.NT↗

arXiv2023

A study on Diophantine equations via cluster theory

In this paper, we mainly answer a Lampe's question\cite{lampe} about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution $(ε,ε,ε, ε,ε)$ under the actions of the group $G$ which is defined by mutations of a cluster algebra. In order to do this, using a rational map $φ$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $φ(ε,ε,ε, ε,ε) = (3,4,4)$ under the actions of the group $\widetilde{G}$, and the set $S(3,4,4)$ is shown to be the orbit of $(ε,ε,ε, ε,ε)$ under the actions of a subgroup of $G$. Then the criterion is proved as the main conclusion.

math.NT↗