Laurent Polynomials and Superintegrable Maps

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.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (12)

• Singularity confinement and chaos in discrete systems
• Nonequilibrium Cooper pairing in the nonadiabatic regime
• A survey of Hirota's difference equations
• Diophantine Integrability
• Integrable dynamics of coupled Fermi-Bose condensates
• Elliptic curves and continued fractions
• Singularity confinement for maps with the Laurent property
• Bilinear recurrences and addition formulae for hyperelliptic sigma functions
• Separation of variables and Bäcklund transformations for the symmetric Lagrange top
• Perfect Matchings and the Octahedron Recurrence
• Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos $k$
• Integrality and the Laurent phenomenon for Somos 4 sequences

Cited by in corpus (13)

• Discrete integrable systems and Poisson algebras from cluster maps
• Introduction to Cluster Algebras. Chapters 1-3
• On the algebraic structure of rational discrete dynamical systems
• Simple identification of fake Lax pairs
• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• On non-QRT Mappings of the Plane
• An exercise in experimental mathematics: calculation of the algebraic entropy of a map
• A study on Diophantine equations via cluster theory
• From integrable equations to Laurent recurrences
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
• A Lax pair for the complete QRT mapping