Recently a large class of rational maps or recurrences has been found which has the surprising property that the iterates are Laurent polynomials in the initial data. This Laurent phenomenon occurs in number theory (elliptic divisibility and Somos sequences), combinatorics (Dodgson condensation and perfect matchings), integrable systems (discrete Hirota equation), with a lot of these examples being explained via Fomin and Zelevinsky's theory of cluster algebras. We show how rational recurrences with the Laurent property can lead to solutions of Diophantine equations, concentrating on particular examples of symplectic maps related to cubic surfaces and quartic threefolds.
Contact A.P. Fordy (email: allan@maths.leeds.ac.uk ) for further details.
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Last updated February 8th, 2007.