Arithmetic and geometry of Markov polynomials

TL;DR

This paper explores the structure and properties of Markov polynomials, focusing on their coefficients and geometric aspects, and proposes new conjectures with proofs for special cases like Fibonacci and Pell numbers.

Contribution

It introduces new conjectures about the coefficients of Markov polynomials and proves some for special cases, advancing understanding of their arithmetic and geometric properties.

Findings

Coefficients are non-negative integers.

Conjectures proposed about coefficients and Newton polygons.

Proofs provided for Fibonacci and Pell number cases.

Abstract

Markov polynomials are the Laurent-polynomial solutions of the generalised Markov equation $$X^2 + Y^2 + Z^2 = kXYZ, \quad k=\frac{x^2 + y^2 + z^2}{x y z}$$ which are the results of cluster mutations applied to the initial triple $(x, y, z)$. They were first introduced and studied by Itsara, Musiker, Propp and Viana, who proved, in particular, that their coefficients are non-negative integers. We study the coefficients of Markov polynomials as functions on the corresponding Newton polygons, proposing several new conjectures. Some of these conjectures are proved for the special cases of Markov polynomials corresponding to Fibonacci and Pell numbers.