Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 022, 18 pages      math.NT/0702280      http://dx.doi.org/10.3842/SIGMA.2007.022
Contribution to the Vadim Kuznetsov Memorial Issue

Laurent Polynomials and Superintegrable Maps

Andrew N.W. Hone
Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7NF, UK

Received October 26, 2006; Published online February 07, 2007

Abstract
This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations.

Key words: Laurent property; integrable maps; Somos sequences.

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