Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T16:06:19.066Z Has data issue: false hasContentIssue false

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Published online by Cambridge University Press:  01 July 2008

ANDREW N. W. HONE
Affiliation:
Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7NF.
CHRISTINE SWART
Affiliation:
Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch, 7700, South Africa.

Abstract

Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Braden, H. W., Enolskii, V. Z. and Hone, A. N. W.. Bilinear recurrences and addition formulae for hyperelliptic sigma functions. J. Nonlin. Math. Phys. 12, Supplement 2 (2005), 4662.CrossRefGoogle Scholar
[2]Buchholz, R. H. and Rathbun, R. L.. An infinite set of Heron triangles with two rational medians. Amer. Math. Monthly 104 (1997), 107115.CrossRefGoogle Scholar
[3]Cantor, D.. On the analogue of the division polynomials for hyperelliptic curves. J. Reine Angew. Math. 447 (1994), 91145.Google Scholar
[4]Carroll, G. and Speyer, D.. The cube recurrence. Electron J. Combin. 11 (2004), #R73.CrossRefGoogle Scholar
[5]Doliwa, A., Grinevich, P., Nieszporski, M. and Santini, P. M.. Integrable lattices and their sub-lattices: from the discrete Moutard (discrete Cauchy–Riemann) 4-point equation to the self-adjoint 5-point scheme. J. Math. Phys. 48 (2007), 013513.CrossRefGoogle Scholar
[6]Einsiedler, M., Everest, G. and Ward, T.. Primes in elliptic divisibility sequences. LMS Journal of Computation and Mathematics 4 (2001), 113.CrossRefGoogle Scholar
[7]Everest, G., Miller, V. and Stephens, N.. Primes generated by elliptic curves. Proc. Amer. Math. Soc. 132 (2003), 955963.CrossRefGoogle Scholar
[8]Everest, G., Poorten van der, A., Shparlinski, I. and Ward, T.. Recurrence Sequences, AMS Mathematical Surveys and Monographs, vol. 104 (2003).CrossRefGoogle Scholar
[9]Everest, G. and Shparlinski, I.. Prime divisors of sequences associated to elliptic curves. Glasgow Math. J. 47 (2005), 115122.CrossRefGoogle Scholar
[10]Everest, G. and King, H.. Prime powers in elliptic divisibility sequences. Math. Comp. 74 (2005), 20612071.CrossRefGoogle Scholar
[11]Everest, G., McLaren, G. and Ward, T.. Primitive divisors of elliptic divisibility sequences. J. Number Theory 118 (2006), 7189.CrossRefGoogle Scholar
[12]Everest, G., Stevens, S., Tamsett, D. and Ward, T.. Primes generated by recurrence sequences. Amer. Math. Monthly 114 (2007), 417431.CrossRefGoogle Scholar
[13]Fomin, S. and Zelevinsky, A.. The Laurent phenomenon. Adv. Appl. Math. 28 (2002), 119144.CrossRefGoogle Scholar
[14]Gale, D.. The strange and surprising saga of the Somos sequences. Mathematical Intelligencer 13 (1) (1991), 4042; Somos sequence update. Mathematical Intelligencer 13(4) (1991), 49–50; reprinted in Tracking the Automatic Ant (Springer, 1998).Google Scholar
[15]Grammaticos, B., Ramani, A. and Papageorgiou, V.. Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 (1991), 18251828.CrossRefGoogle ScholarPubMed
[16]Halburd, R. G.. Diophantine integrability. J. Phys. A: Math. Gen. 38 (2005), L263L269.CrossRefGoogle Scholar
[17]Hietarinta, J. and Viallet, C.. Singularity confinement and chaos in discrete systems. Phys. Rev. Lett. 81 (1998), 325328.CrossRefGoogle Scholar
[18]Hirota, R.. Exact solution of the Korteweg–deVries equation for multiple collision of solitons. Phys. Rev. Lett. 27 (1972), 11921194.CrossRefGoogle Scholar
[19]Hone, A. N. W.. Elliptic curves and quadratic recurrence sequences. Bull. Lond. Math. Soc. 37 (2005), 161171; Corrigendum 38 (2006), 741–742.CrossRefGoogle Scholar
[20]Hone, A. N. W.. Sigma function solution of the initial value problem for Somos 5 sequences. Trans. Amer. Math. Soc. 359 (2007), 50195034.CrossRefGoogle Scholar
[21]Hone, A. N. W.. Singularity confinement for maps with the Laurent property. Phys. Lett. A 261 (2007), 341345.CrossRefGoogle Scholar
[22]Hone, A. N. W.. Laurent polynomials and superintegrable maps. SIGMA 3 (2007) 022, 18 pages.Google Scholar
[23]Kanayama, N.. Division polynomials and multiplication formulae of Jacobian varieties of dimension 2. Math. Proc. Camb. Phil. Soc. 139 (2005), 399409.CrossRefGoogle Scholar
[24]Kanayama, N.. Pivate communication (2005).Google Scholar
[25]Lang, S.. Elliptic Curves: Diophantine Analysis (Springer-Verlag, 1978).CrossRefGoogle Scholar
[26]Matsutani, S.. Recursion relation of hyperelliptic PSI-functions of genus two. Int. Transforms Spec. Func. 14 (2003), 517527.CrossRefGoogle Scholar
[27]van der Poorten, A. J.. Elliptic curves and continued fractions. J. Integer Sequences 8 (2005), Article 05.2.5.Google Scholar
[28]van der Poorten, A. J.. Curves of genus 2, continued fractions and Somos Sequences. J. Integer Sequences 8 (2005), Article 05.3.4.Google Scholar
[29]van der Poorten, A. J.. Hyperelliptic curves, continued fractions, and Somos sequences. IMS Lecture Notes-Monograph Series. Dynamics & Stochastics 48 (2006), 212224.Google Scholar
[30]van der Poorten, A. J. and Swart, C. S.. Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos k. Bull. Lond. Math. Soc. 38 (2006), 546554.CrossRefGoogle Scholar
[31]Propp, J.. The “bilinear” forum, and the Somos Sequence Site. http://www.math.wisc.edu/~proppGoogle Scholar
[32]Quispel, G. R. W., Roberts, J. A. G. and Thompson, C. J.. Integrable mappings and soliton equations II. Physica D 34 (1989), 183192.CrossRefGoogle Scholar
[33]Robinson, R.. Periodicity of Somos sequences. Proc. Amer. Math. Soc. 116 (1992), 613619.CrossRefGoogle Scholar
[34]Scott, J. S.Grassmanians and cluster algebras. Proc. Lond. Math. Soc. 92 (2006), 345380.CrossRefGoogle Scholar
[35]Shipsey, R.. Elliptic divisibility sequences. PhD thesis, Goldsmiths College, University of London (2000).Google Scholar
[36]Silverman, J. H.. The Arithmetic of Elliptic Curves (Springer, 1986).Google Scholar
[37]Silverman, J. H.. Advanced Topics in the Arithmetic of Elliptic Curves (Springer, 1994).CrossRefGoogle Scholar
[38]Silverman, J. H.. p-adic properties of division polynomials and elliptic divisibility sequences. Math. Annal. 332 (2005), 443471; Addendum 473–474.CrossRefGoogle Scholar
[39]Sloane, N. J. A.. On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequencesGoogle Scholar
[40]Speyer, D.. Perfect Matchings and the Octahedron Recurrence. J. Alg. Comb. (to appear); math.CO/0402452.Google Scholar
[41]Swart, C. S.. Elliptic curves and related sequences. PhD thesis, Royal Holloway, University of London (2003).Google Scholar
[42]Veselov, A. P.. Integrable maps. Russian Math. Surveys 46 (1991), 151.CrossRefGoogle Scholar
[43]Ward, M.. Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948), 3174.CrossRefGoogle Scholar
[44]Ward, M.. The law of repetition of primes in an elliptic divisibility sequence. Duke Math. J. 15 (1948), 941946.CrossRefGoogle Scholar
[45]Whittaker, E. T. and Watson, G. N.. A Course of Modern Analysis (4th edition). (Cambridge University Press, 1965).Google Scholar
[46]Zabrodin, A.. A survey of Hirota's difference equations. Teor. Mat. Fiz. 113 (1997), 179230 (Russian); solv-int/9704001 (English).CrossRefGoogle Scholar
[47]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.htmlGoogle Scholar