2024/07/11 by Yasuaki Gyoda, Gyoda, Yasuaki, Shuhei Maruyama +3 · 3 citations
Computer Science · #05C70 #11A55 #11D25 #14B05 #14M25 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2407.08203
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For k≥ 0, a k-generalized Markov number is an integer which appears in some positive integer solution to the k-generalized Markov equation x2 + y2 + z2 + k(yz + zx + xy) = (3 + 3k)xyz. In this paper, we discuss a combinatorial structure of generalized Markov numbers. To investigate this structure in detail, we use two families of matrices: the k-generalized Cohn matrices and the k-Markov-monodromy matrices, which are elements of SL(2, ℤ) whose (1,2)-entries are k-generalized Markov numbers. We show that these two families of matrices recover the tree structure of the positive integer solutions to the generalized Markov equation, and we give geometric interpretations and a combinatorial interpretation of k-generalized Markov numbers. As an application, we provide a computation algorithm of classical Markov number from a one-dimensional dynamical viewpoint. Moreover, we clarify a relation between k-generalized Markov numbers and toric surface singularities via continued fractions.