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

SIGMA 14 (2018), 065, 17 pages      arXiv:1804.02804

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

Masataka Kanki a and Takafumi Mase b and Tetsuji Tokihiro b
a) Department of Mathematics, Kansai University, Japan
b) Graduate School of Mathematical Sciences, University of Tokyo, Japan

Received April 10, 2018, in final form June 21, 2018; Published online June 27, 2018

In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.

Key words: integrability detector; coprimeness; singularity confinement; discrete Toda equation.

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  1. Bellon M.P., Viallet C.-M., Algebraic entropy, Comm. Math. Phys. 204 (1999), 425-437, chao-dyn/9805006.
  2. Beukers F., On a sequence of polynomials, J. Pure Appl. Algebra 117/118 (1997), 97-103.
  3. Bruschi M., Ragnisco O., Santini P.M., Tu G.Z., Integrable symplectic maps, Phys. D 49 (1991), 273-294.
  4. Demskoi D.K., Viallet C.-M., Algebraic entropy for semi-discrete equations, J. Phys. A: Math. Theor. 45 (2012), 352001, 10 pages, arXiv:1206.1214.
  5. Fomin S., Zelevinsky A., The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119-144, math.CO/0104241.
  6. Galashin P., Pylyavskyy P., $R$-systems, arXiv:1709.00578.
  7. Gale D., The strange and surprising saga of the Somos sequences, Math. Intelligencer 13 (1991), 40-42.
  8. Glick M., The Devron property, J. Geom. Phys. 87 (2015), 161-189, arXiv:1312.6881.
  9. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  10. Hamad K., Hone A.N.W., van der Kamp P.H., Quispel G.R.W., QRT maps and related Laurent systems, Adv. in Appl. Math. 96 (2018), 216-248, arXiv:1702.07047.
  11. Hamad K., van der Kamp P.H., From discrete integrable equations to Laurent recurrences, J. Difference Equ. Appl. 22 (2016), 789-816.
  12. Hietarinta J., Viallet C.-M., Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1998), 325-328, solv-int/9711014.
  13. Hietarinta J., Viallet C.-M., Searching for integrable lattice maps using factorization, J. Phys. A: Math. Theor. 40 (2007), 12629-12643, arXiv:0705.1903.
  14. Kamiya R., Kanki M., Mase T., Tokihiro T., Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation, J. Math. Phys. 58 (2017), 012702, 12 pages, arXiv:1610.02646.
  15. Kanki M., Mada J., Mase T., Tokihiro T., Irreducibility and co-primeness as an integrability criterion for discrete equations, J. Phys. A: Math. Theor. 47 (2014), 465204, 15 pages, arXiv:1405.2229.
  16. Kanki M., Mada J., Tokihiro T., Singularities of the discrete KdV equation and the Laurent property, J. Phys. A: Math. Theor. 47 (2014), 065201, 12 pages, arXiv:1311.0060.
  17. Kanki M., Mase T., Tokihiro T., Algebraic entropy of an extended Hietarinta-Viallet equation, J. Phys. A: Math. Theor. 48 (2015), 355202, 19 pages, arXiv:1502.02415.
  18. Kanki M., Mase T., Tokihiro T., Singularity confinement and chaos in two-dimensional discrete systems, J. Phys. A: Math. Theor. 49 (2016), 23LT01, 9 pages, arXiv:1512.09168.
  19. Mase T., The Laurent phenomenon and discrete integrable systems, in The Breadth and Depth of Nonlinear Discrete Integrable Systems, RIMS Kokyuroku Bessatsu, Vol. B41, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013, 43-64.
  20. Mase T., Investigation into the role of the Laurent property in integrability, J. Math. Phys. 57 (2016), 022703, 21 pages, arXiv:1505.01722.
  21. Mase T., Willox R., Ramani A., Grammaticos B., Integrable mappings and the notion of anticonfinement, J. Phys. A: Math. Theor. 52 (2018), 265201, 11 pages, arXiv:1511.02000.
  22. Mirimanoff D., Sur l'équation $(x+1)^l-x^l-1=0$, Nouv. Ann. Math 3 (1903), 385-397.
  23. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  24. Viallet C.-M., On the algebraic structure of rational discrete dynamical systems, J. Phys. A: Math. Theor. 48 (2015), 16FT01, 21 pages, arXiv:1501.06384.

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