Laurent Phenomenon

The Laurent phenomenon is the property that elements produced by certain rational recurrences are nevertheless Laurent polynomials in their initial variables, even though the recurrence formulas involve division.

For a coefficient-free cluster algebra with initial cluster , where a cluster is the -tuple of cluster variables in a seed, every cluster variable satisfies

The same assertion holds with any cluster chosen as the initial cluster. More generally, Fomin and Zelevinsky (2002) gave conditions under which subtraction-free rational recurrences have this property. Their framework proves the Laurent property for generalized Somos sequences and explains why many such recurrences produce integers after all initial values are set to 1.