Generalization of Markov Diophantine equation via generalized cluster algebra

2022/01/26 by Yasuaki Gyoda, Gyoda, Yasuaki, Kodai Matsushita +1 · 4 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Commutative Algebra and Its Applications #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2201.10919

Abstract

In this paper, we deal with two classes of Diophantine equations, x²+y²+z²+k₁yz+k₂zx+k₃xy=(3+k₁+k₂+k₃)xyz and x²+y⁴+z⁴+ky²z²+2xz²+2xy²=(7+k)xy²z², where k₁,k₂,k₃,k are nonnegative integers. The former is known as the Markov Diophantine equation if k₁=k₂=k₃=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.

Cited by

• Generalized discrete Markov spectra
• Orderings of k-Markov Numbers
• Sign-coherence and tropical sign pattern for rank 3 real cluster-cyclic exchange matrices
• Divisibility by p for Markoff-like Surfaces

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