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SIGMA, 2007, Volume 3, 022, 18 pages (Mi sigma148)  

This article is cited in 17 scientific papers (total in 17 papers)

Laurent Polynomials and Superintegrable Maps

Andrew N. W. Hone

Institute of Mathematics, Statistics \& Actuarial Science, University of Kent, Canterbury CT2 7NF, UK

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.

Keywords: Laurent property; integrable maps; Somos sequences


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ArXiv: math.NT/0702280
MSC: 11B37; 33E05; 37J35
Received: October 26, 2006; Published online February 7, 2007

Citation: Andrew N. W. Hone, “Laurent Polynomials and Superintegrable Maps”, SIGMA, 3 (2007), 022, 18 pp.

Citation in format AMSBIB
\by Andrew N.~W.~Hone
\paper Laurent Polynomials and Superintegrable Maps
\jour SIGMA
\yr 2007
\vol 3
\papernumber 022
\totalpages 18

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    This publication is cited in the following articles:
    1. Hone A.N.W., Swart Ch., “Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences”, Math. Proc. Cambridge Philos. Soc., 145 (2008), 65–85  crossref  mathscinet  zmath  isi  elib  scopus
    2. Grammaticos B., Halburd R.G., Ramani A., Viallet C.-M., “How to detect the integrability of discrete systems”, J. Phys. A, 42:45 (2009), 454002, 30 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Kassotakis P., Joshi N., “Integrable Non-QRT Mappings of the Plane”, Lett. Math. Phys., 91:1 (2010), 71–81  crossref  mathscinet  zmath  isi  scopus
    4. Fordy A.P., Marsh R.J., “Cluster mutation-periodic quivers and associated Laurent sequences”, J Algebraic Combin, 34:1 (2011), 19–66  crossref  mathscinet  zmath  isi  scopus
    5. Fordy A.P., “Mutation-periodic quivers, integrable maps and associated Poisson algebras”, Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 369:1939 (2011), 1264–1279  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Rei Inoue, Tomoki Nakanishi, “Difference equations and cluster algebras I: Poisson bracket for integrable difference equations”, RIMS Kokyuroku Bessatsu B, 28 (2011), 63–88, arXiv: 1012.5574  mathscinet
    7. Fordy A.P. Hone A., “Discrete Integrable Systems and Poisson Algebras From Cluster Maps”, Commun. Math. Phys., 325:2 (2014), 527–584  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Fordy A.P., “Periodic Cluster Mutations and Related Integrable Maps”, J. Phys. A-Math. Theor., 47:47 (2014), 474003  crossref  mathscinet  zmath  isi  elib  scopus
    9. Hone A.N.W. Inoue R., “Discrete Painlevé Equations From Y-Systems”, J. Phys. A-Math. Theor., 47:47 (2014), 474007  crossref  mathscinet  zmath  isi  elib  scopus
    10. Cruz I. Esmeralda Sousa-Dias M., “Reduction of Cluster Iteration Maps”, J. Geom. Mech., 6:3 (2014), 297–318  crossref  mathscinet  zmath  isi  scopus
    11. Viallet C.-M., “on the Algebraic Structure of Rational Discrete Dynamical Systems”, J. Phys. A-Math. Theor., 48:16 (2015), 16FT01  crossref  mathscinet  zmath  isi  elib  scopus
    12. Chang X.-K. Hu X.-B. Xin G., “Hankel Determinant Solutions To Several Discrete Integrable Systems and the Laurent Property”, SIAM Discret. Math., 29:1 (2015), 667–682  crossref  mathscinet  zmath  isi  scopus
    13. Grammaticos B., Ramani A., Willox R., Mase T., Satsuma J., “Singularity Confinement and Full-Deautonomisation: a Discrete Integrability Criterion”, Physica D, 313 (2015), 11–25  crossref  mathscinet  zmath  adsnasa  isi  scopus
    14. Alman J., Cuenca C., Huang J., “Laurent phenomenon sequences”, J. Algebr. Comb., 43:3 (2016), 589–633  crossref  mathscinet  zmath  isi  scopus
    15. Hamad Kh. van der Kamp P.H., “From discrete integrable equations to Laurent recurrences”, J. Differ. Equ. Appl., 22:6 (2016), 789–816  crossref  mathscinet  zmath  isi  elib  scopus
    16. van der Kamp P.H., “Somos-4 and Somos-5 are arithmetic divisibility sequences”, J. Differ. Equ. Appl., 22:4 (2016), 571–581  crossref  mathscinet  isi  scopus
    17. Galashin P., Pylyavskyy P., “R-Systems”, Sel. Math.-New Ser., 25:2 (2019), UNSP 22  crossref  mathscinet  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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