Positivity in Coefficient-Free Rank Two Cluster Algebras

Let b, c be positive integers, x1, x2 be indeterminates over Z and xm,m ∈ Z be rational functions defined by xm−1xm+1 = x b m+1 if m is odd and xm−1xm+1 = x c m+1 if m is even. In this short note, we prove that for any m,k ∈ Z, xk can be expressed as a substraction-free Laurent polynomial in Z[x±1 m , x ±1 m+1]. This proves FominZelevinsky’s positivity conjecture for coefficient-free rank two cluster algebras. Introduction A combinatorial result Let b, c be positive integers and x1, x2 be indeterminates over Z. The (coefficient-free) cluster algebra A(b, c) is the subring of the field Q(x1, x2) generated by the elements xm, m ∈ Z satisfying the recurrence relations: