# Integrality and the Laurent phenomenon for Somos 4 sequences

@article{Swart2005IntegralityAT, title={Integrality and the Laurent phenomenon for Somos 4 sequences}, author={Christine Swart and Andrew Hone}, journal={arXiv: Number Theory}, year={2005} }

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…

## 4 Citations

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Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic…

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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…

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In this report I sanitise (in the sense of ‘bring some sanity to’) the arguments of earlier reports detailing the correspondence between sequences (M+hS)−−∞<h<∞ of divisors on elliptic and genus two…

## References

SHOWING 1-10 OF 46 REFERENCES

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

- Mathematics, Physics
- 2005

Abstract The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be…

Sigma function solution of the initial value problem for Somos 5 sequences

- Mathematics
- 2005

The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer…

Recurrence relations for elliptic sequences: Every somos 4 is a somos k

- Mathematics
- 2004

In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist…

Elliptic curves and related sequences

- Mathematics
- 2003

A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h 2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic…

LETTER TO THE EDITOR: Diophantine integrability

- Physics, Mathematics
- 2005

The heights of iterates of the discrete Painlevé equations over number fields appear to grow no faster than polynomials while the heights of generic solutions of non-integrable discrete equations…

The Laurent Phenomenon

- Computer Science, MathematicsAdv. Appl. Math.
- 2002

A unified treatment of the phenomenon of birational maps given by Laurent polynomials is suggested, which covers a large class of applications and settles in the affirmative a conjecture of D. Gale on integrality of generalized Somos sequences.

Perfect matchings and the octahedron recurrence

- Mathematics
- 2004

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was…

The Cube Recurrence

- Mathematics, Computer ScienceElectron. J. Comb.
- 2004

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in…

Recurrence Sequences

- Computer ScienceMathematical surveys and monographs
- 2003

This work surveys a selection of number-theoretic properties of linear recurrence sequences together with their direct generalizations and describes applications of these properties which are important for a variety of applications.

Advanced Topics in the Arithmetic of Elliptic Curves

- Mathematics
- 1994

In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational…