2025/11/05 by Chen, Zhichao, Jia, Zelin · 1 citation
Computer Science · Mathematics · #11B39 #11D09 #13F60 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2511.03428
openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28
The generalized Markov equations are deeply connected with the generalized cluster algebras of Markov type. We construct a deformed Fock-Goncharov tropicalization for the generalized Markov equations and prove that their tropicalized tree structure is essentially the same as that of the classical Euclid tree. We then define the generalized Euclid tree and prove that it converges to the classical Euclid tree up to a scalar multiple. Moreover, by means of cluster mutations, we exhibit an asymptotic phenomenon, up to some limit q, between the logarithmic generalized Markov tree and the classical Euclid tree. A rationality conjecture of q is then put forward. We also propose a generalized Markov uniqueness conjecture for the generalized Markov equations, which illustrates an application of the asymptotic phenomenon.