6 papers
An Introduction to the Lagrange and Markov Spectra through the Lens of Generalized Markov Numbers
Yasuaki Gyoda
This text is a self-contained expository survey of the Lagrange and Markov spectra, centered on a comprehensive exposition of Markov's theorem and its generalizations. Its purpose…
Words for generalized Markov numbers
Yasuaki Gyoda
We construct a word-theoretic framework for generalized Markov numbers, that is, positive integers appearing in positive integer solutions of the generalized Markov equation $x^2+y…
Generalized discrete Markov spectra
Yasuaki Gyoda
We develop a generalized Markov theory for the Markov--Lagrange and Markov spectra. The classical discrete Markov spectrum is governed by Markov numbers, the positive integers occu…
Cluster algebraic interpretation of generalized Markov numbers and their matrixizations
Esther Banaian, Yasuaki Gyoda
Markov numbers, i.e. positive integers appearing in solutions to , can be viewed as specializations of cluster variables. The second author and Matsushita g…
SL(2,Z)-matrixizations of generalized Markov numbers
Yasuaki Gyoda, Shuhei Maruyama, Yusuke Sato
For , a -generalized Markov number is an integer which appears in some positive integer solution to the -generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + x…
Uniqueness theorem of generalized Markov numbers that are prime powers
Yasuaki Gyoda, Shuhei Maruyama
In this paper, we study positive integer solutions to a generalized form of the Markov equation, given as . This equation extends t…