nLab Laurent phenomenon

While the composition of birational maps given by Laurent polynomials is in general not a Laurent polynomial there is a number of systematic situations when this is true. In particular, in a cluster algebra, any cluster variable is expressed in terms of any given cluster as a Laurent polynomial with coefficients in the integral group ring $\mathbf{Z}\mathbf{P}$ where $\mathbf{P}$ is the coefficient group. An explanation in the quantum case, from a Hodge-theoretic point of view, has recently being offered by A. Efimov.

• Sergey Fomin, Andrei Zelevinsky, The Laurent phenomenon, math.CO/0104241
• Sergey Fomin, Andrei Zelevinsky, Cluster algebras I: Foundations, arxiv/0104151
• Alexander I. Efimov, Quantum cluster variables via vanishing cycles, arxiv/1112.3601