Hone, Andrew N.W., Swart, Christine (2008) Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. Mathematical Proceedings of the Cambridge Philosophical Society, 145 (Part I). pp. 6585. ISSN 03050041. (doi:10.1017/s030500410800114x) (KAR id:15690)
PDF (Integrality and the Laurent Phenomenon)
Language: English 

Download (331kB)
Preview

Preview 
This file may not be suitable for users of assistive technology.
Request an accessible format


Official URL http://dx.doi.org/10.1017/s030500410800114x 
Abstract
Somos 4 sequences are a family of sequences defined by a fourthorder 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. 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 byproduct, nonperiodic 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.
Item Type:  Article 

DOI/Identification number:  10.1017/s030500410800114x 
Uncontrolled keywords:  Number Theory (math.NT); Combinatorics (math.CO 
Subjects:  Q Science > QA Mathematics (inc Computing science) 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Andrew Hone 
Date Deposited:  20 Apr 2009 14:42 UTC 
Last Modified:  16 Nov 2021 09:53 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/15690 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
 Link to SensusAccess
 Export to:
 RefWorks
 EPrints3 XML
 BibTeX
 CSV
 Depositors only (login required):