Positivity in coefficient-free rank two cluster algebras

TL;DR

This paper proves the positivity conjecture for coefficient-free rank two cluster algebras by showing that all variables can be expressed as subtraction-free Laurent polynomials in initial variables.

Contribution

It establishes the positivity of cluster variables in rank two cluster algebras without coefficients, confirming a key conjecture in the field.

Findings

All cluster variables are subtraction-free Laurent polynomials.

The positivity conjecture is verified for rank two coefficient-free cases.

Provides explicit formulas for expressing variables as Laurent polynomials.

Abstract

Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over $\Z$ and $x_m, m \in \mathbb Z$ be rational functions defined by $x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k \in \Z$, $x_k$ can be expressed as a substraction-free Laurent polynomial in $\Z[x_m^{\pm 1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.