A step towards the cluster positivity conjecture

Abstract

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xy{x}^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.

Comments welcome, v2:16 pages. introduction expanded. section 4 added in order to compare with the known formula when r=2. references added. thank you list added. definitions clarified. This paper is superseded by a joint paper with Ralf Schiffler (http://arxiv.org/abs/1109.5130)

References in corpus (3)

• Quiver varieties and cluster algebras
• Positivity for cluster algebras from surfaces
• Non-commutative Laurent phenomenon for two variables

Cited by in corpus (1)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems