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.
Categories: math.RA math.RT