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

SIGMA 3 (2007), 022, 18 pages      math.NT/0702280      https://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.

pdf (1343 kb)   ps (285 kb)   tex (1104 kb)

References

  1. Braden H.W., Enolskii V.Z., Hone A.N.W., Bilinear recurrences and addition formulae for hyperelliptic sigma functions, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 46-62, math.NT/0501162.
  2. Bressoud D.M., Proofs and confirmations: the story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge, 1999.
  3. Bruschi M., Ragnisco O., Santini P.M., Tu G.-Z., Integrable symplectic maps, Phys. D 49 (1991), 273-294.
  4. Buchholz R.H., Rathbun R.L., An infinite set of heron triangles with two rational medians, Amer. Math. Monthly 104 (1997), 107-115.
  5. Buchstaber V.M., Enolskii V.Z., Leykin D.V., Hyperelliptic Kleinian functions and applications, in Solitons, Geometry and Topology: On the Crossroad, Editors V.M. Buchstaber and S.P. Novikov, AMS Translations Series 2, Vol. 179, AMS, 1997, 1-34, solv-int/9603005.
  6. Cantor D., On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. 447 (1994), 91-145.
  7. Carroll G., Speyer D., The cube recurrence, Electron. J. Combin. 11 (2004), # R73, math.CO/0403417.
  8. Common A., Hone A.N.W., Musette M., A new discrete Hénon-Heiles system, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 27-40.
  9. Elaydi S., Discrete chaos, Chapman and Hall/CRC, Boca Raton, 2000.
  10. Einsiedler M., Everest G., Ward T., Primes in elliptic divisibility sequences, LMS J. Comput. Math. 4 (2001), 1-13.
  11. Everest G., Miller V., Stephens N., Primes generated by elliptic curves, Proc. Amer. Math. Soc. 132 (2003), 955-963.
  12. Everest G., van der Poorten A., Shparlinski I., Ward T., Recurrence sequences, AMS Mathematical Surveys and Monographs, Vol. 104, Amer. Math. Soc., Providence, RI, 2003.
  13. Fedorov Y., Bäcklund transformations on coadjoint orbits of the loop algebra gl(r), J. Nonlinear Math. Phys. 9 (2002), suppl. 1, 29-46.
  14. Fomin S., Zelevinsky A., The Laurent phenomenon, Adv. Appl. Math. 28 (2002), 119-144, math.CO/0104241.
  15. Fomin S., Zelevinsky A., Cluster algebras IV: coefficients, Compos. Math., to appear, math.RA/0602259.
  16. Fordy A.P., Shabat A.B., Veselov A.P., Factorization and Poisson correspondences, Theor. Math. Phys. 105 (1995), 1369-1386.
  17. Gale D., The strange and surprising saga of the Somos sequences, Math. Intelligencer 13 (1991), no. 1, 40-42.
    Gale D., Somos sequence update, Math. Intelligencer 13 (1991), no. 4, 49-50 (Reprinted in Tracking the Automatic Ant., Springer, 1998).
  18. Gekhtman M., Shapiro M., Vainshtein A., Cluster algebras and Poisson geometry, Moscow Math. J. 3 (2003), 899-934, math.QA/0208033.
  19. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 (1991), 1825-1828.
  20. Halburd R.G., Diophantine integrability, J. Phys. A: Math. Gen. 38 (2005), L263-L269, nlin.SI/0504027.
  21. Hietarinta J., Viallet C., Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1998), 325-328, solv-int/9711014.
  22. Hone A.N.W., Non-autonomous Hénon-Heiles systems, Phys. D 118 (1998), 1-16, solv-int/9703005.
  23. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the Hénon-Heiles and Garnier systems, CRM Proceedings and Lecture Notes, Vol. 25, Amer. Math. Soc., 2000, 231-235.
  24. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for many-body systems related to KdV, J. Phys. A: Math. Gen. 32 (1999), L299-L306, solv-int/9904003.
  25. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the sl(2) Gaudin magnet, J. Phys. A: Math. Gen. 34 (2001), 2477-2490, nlin.SI/0007041.
  26. Hone A.N.W., Exact discretization of the Ermakov-Pinney equation, Phys. Lett. A 263 (1999), 347-354.
  27. Hone A.N.W., Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005), 161-171, Corrigendum, Bull. Lond. Math. Soc., 38 (2006), 741-742.
  28. Hone A.N.W., Sigma function solution of the initial value problem for Somos 5 sequences, Trans. Amer. Math. Soc., to appear, math.NT/0501554.
  29. Hone A.N.W., Diophantine non-integrability of a third-order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), L171-L177, math.NT/0601324.
  30. Hone A.N.W., Singularity confinement for maps with the Laurent property, Phys. Lett. A 361 (2007), 341-345, nlin.SI/0602007.
  31. Hone A.N.W., Discrete dynamics, integrability and integer sequences, Imperial College Press, in preparation.
  32. Kuznetsov V.B., Sklyanin E.K., On Bäcklund transformations for many-body systems, J. Phys. A: Math. Gen. 31 (1998), 2241-2251, solv-int/9711010.
  33. Kuznetsov V.B., Salerno M., Sklyanin E.K., Quantum Bäcklund transformation for the integrable DST model, J. Phys. A: Math. Gen. 33 (2000), 171-189, solv-int/9908002.
  34. Kuznetsov V.B., Vanhaecke P., Bäcklund transformations for finite-dimensional integrable systems: a geometric approach, J. Geom. Phys. 44 (2002), 1-40, nlin.SI/0004003.
  35. Kuznetsov V.B., Mangazeev V.V., Sklyanin E.K., Q-operator and factorised separation chain for Jack polynomials, Indag. Math. 14 (2003), 451-482, math.CA/0306242.
  36. Mills W.H., Robbins D.P., Rumsey H., Alternating-sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
  37. Nekhoroshev N.N., On action-angle variables and their generalizations, Tr. Moscow Math. Soc. 26 (1972), 181-198 (in Russian).
  38. Pasquier V., Gaudin M., The periodic Toda chain and a matrix generalization of the Bessel function recursion relations, J. Phys. A: Math. Gen. 25 (1992), 5243-5252.
  39. Kuznetsov V.B., Petrera M., Ragnisco O., Separation of variables and Bäcklund transformations for the symmetric Lagrange top, J. Phys. A: Math. Gen. 37 (2004), 8495-8512, nlin.SI/0403028.
  40. Matsutani S., Recursion relation of hyperelliptic PSI-functions of genus two, Int. Transforms Spec. Func. 14 (2003), 517-527, math-ph/0105031.
  41. van der Poorten A.J., Elliptic curves and continued fractions, J. Integer Sequences 8 (2005), Article 05.2.5, 19 pages, math.NT/0403225.
  42. van der Poorten A.J., Swart C.S., Recurrence relations for elliptic sequences: every Somos 4 is a Somos k, Bull. Lond. Math. Soc. 38 (2006), 546-554, math.NT/0412293.
  43. Propp J., The many faces of alternating-sign matrices, Disc. Math. Theoret. Comp. Sci. Proc. AA (DM-CCG) (2001), 43-58, math.CO/0208125.
  44. Propp J., The "bilinear" forum, http://www.math.wisc.edu~propp/.
  45. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations II, Phys. D 34 (1989), 183-192.
  46. Ramani A., Grammaticos B., Satsuma J., Bilinear discrete Painlevé equations, J. Phys. A: Math. Gen. 28 (1995), 4655-4665.
  47. Robinson R., Periodicity of Somos sequences, Proc. Amer. Math. Soc. 116 (1992), 613-619.
  48. Silverman J.H., The arithmetic of elliptic curves, Springer, 1986.
  49. Silverman J.H., p-adic properties of division polynomials and elliptic divisibility sequences, Math. Annal. 332 (2005), 443-471, Addendum, 473-474, math.NT/0404412.
  50. Sloane N.J.A., On-line encyclopedia of integer sequences, http://www.research.att.com/~njas/sequences, sequence A006720.
  51. Speyer D., Perfect matchings and the octahedron recurrence, 2004, math.CO/0402452.
  52. Suris Y.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser, Basel, 2003.
  53. Swart C.S., Elliptic curves and related sequences, PhD thesis, Royal Holloway, University of London, 2003.
  54. Swart C.S., Hone A.N.W., Integrality and the Laurent phenomenon for Somos 4 sequences, math.NT/0508094.
  55. Takenawa T., A geometric approach to singularity confinement and algebraic entropy, J. Phys. A: Math. Gen. 34 (2001), L95-L102, nlin.SI/0011037.
  56. Vanhaecke P., Integrable systems in the realm of algebraic geometry, 2nd ed., Springer, 2005.
  57. Veselov A.P., Integrable maps, Russ. Math. Surveys 46 (1991), 1-51.
  58. Veselov A.P., What is an integrable mapping? in What is Integrability? Editor V.E. Zakharov, Springer-Verlag, 1991, 251-272.
  59. Veselov A.P., Growth and integrability in the dynamics of mappings, Comm. Math. Phys. 145 (1992), 181-193.
  60. Veselov A.P., Shabat A.B., A dressing chain and the spectral theory of the Schrödinger operator, Funct. Anal. Appl. 27 (1993), 1-21.
  61. Ward M., Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74.
  62. Ward M., The law of repetition of primes in an elliptic divisibility sequence, Duke Math. J. 15 (1948), 941-946.
  63. Weiss J., Periodic fixed points of Bäcklund transformations and the Korteweg-de Vries equation, J. Math. Phys. 27 (1986), 2647-2656.
  64. Wojciechowski S., Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983), 279-281.
  65. Yuzbashyan E.A., Kuznetsov V.B., Altshuler B.L., Integrable dynamics of coupled Fermi-Bose condensates, Phys. Rev. B 72 (2005), 144524, 9 pages, cond-mat/0506782.
  66. Yuzbashyan E.A., Altshuler B.L., Kuznetsov V.B., Enolskii V.Z., Nonequilibrium Cooper pairing in the nonadiabatic regime, Phys. Rev. B 72 (2005), 220503(R), 4 pages, cond-mat/0505493.
  67. Zagier D., Problems posed at the St. Andrews Colloquium, 1996, Solutions, 5th day, available at http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html.
  68. Zabrodin A., A survey of Hirota's difference equations, Teor. Mat. Fiz. 113 (1997), 179-230 (in Russian), solv-int/9704001.

Previous article   Next article   Contents of Volume 3 (2007)