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Diophantine non-integrability of a third order recurrence with the Laurent property

Hone, Andrew N.W. (2006) Diophantine non-integrability of a third order recurrence with the Laurent property. Journal of Physics A: Mathematical and General, 39 (12). L171-L177. ISSN 0305-4470. (doi:10.1088/0305-4470/39/12/L01) (KAR id:41492)

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http://dx.doi.org/10.1088/0305-4470/39/12/L01

Abstract

We consider a one-parameter family of third order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd. Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a certain family of Diophantine equations, studied by Mordell, that includes Markoff's equation.

Item Type: Article
DOI/Identification number: 10.1088/0305-4470/39/12/L01
Subjects: Q Science > QA Mathematics (inc Computing science) > QA101 Arithmetic
Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA150 Algebra > QA241 Number theory
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 21 Jun 2014 01:15 UTC
Last Modified: 06 Feb 2020 04:09 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/41492 (The current URI for this page, for reference purposes)
Hone, Andrew N.W.: https://orcid.org/0000-0001-9780-7369
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