论文标题
通过广义群集代数对马尔可夫二聚体方程的概括
Generalization of Markov Diophantine equation via generalized cluster algebra
论文作者
论文摘要
在本文中，我们处理两类的双苯胺方程，$ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy =（3+k_1+k_1+k_2+k_3）xyz $和$ x^2+y^4+z^4+z^4+ky^2z^2z^2+2x+2xz^2+2+2+k） $ k_1，k_2，k_3，k $是非负整数。如果$ k_1 = k_2 = k_3 = 0 $，那么前者被称为马尔可夫二芬太汀方程，而后者是兰特（Lampe）最近研究的Diophantine方程，如果$ k = 0 $。我们提供算法来列举这些方程式的所有积极整数解，并讨论其背后的广义群集代数的结构。
In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+ky^2z^2+2xz^2+2xy^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.
