Tropicalization and cluster asymptotic phenomenon of generalized Markov equations
Zhichao Chen, Zelin Jia
TL;DR
The paper develops a bridge between generalized Markov equations and Euclid-type recursion through generalized cluster algebras. It introduces a deformed Fock-Goncharov tropicalization that renders the generalized Markov tree equivalent to the classical Euclid tree, and proves an explicit asymptotic convergence between generalized Euclid trees and the Euclid tree under mutations. It defines a ratio-number sequence whose limit is tied to the triple of parameters $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ via $k_{\lambda}=3+\lambda_1+\lambda_2+\lambda_3$, and extends these ideas to the logarithmic GM-tree, proposing a rationality conjecture for the limiting scalar and a generalized Markov uniqueness conjecture with practical verification implications. The work is illustrated with Lampe’s Diophantine equation and culminates in an approximate method for detecting potential counterexamples to the generalized uniqueness conjecture, leveraging the Euclid-tree framework for computational efficiency.
Abstract
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.