The Logarithmic Asymptotic Phenomenon for Generalized Markov-Hurwitz Equations

TL;DR

This paper introduces a generalized family of Markov-Hurwitz equations and proves a logarithmic asymptotic behavior for their positive integer solutions, extending classical results to higher dimensions.

Contribution

It extends classical Markov-Hurwitz equations to n variables and establishes a new logarithmic asymptotic phenomenon for their solutions.

Findings

Introduction of a generalized family of Markov-Hurwitz equations

Proof of a logarithmic asymptotic behavior for solutions

Extension of classical results to higher dimensions

Abstract

The purpose of this paper is twofold. First, we introduce a family of generalized Markov-Hurwitz equations, extending classical Markov-Hurwitz equations with additional degree n-1 interaction terms, Gyoda and Matsushita's generalized Markov equations from 3 variables to n variables. Second, we prove a logarithmic asymptotic phenomenon for the positive integer solutions of these equations.