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Height growth of solutions and a discrete Painlevé equation

Al-Ghassani, A; Halburd, RG; (2015) Height growth of solutions and a discrete Painlevé equation. Nonlinearity , 28 (7) pp. 2379-2396. 10.1088/0951-7715/28/7/2379. Green open access

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Abstract

Consider the discrete equation where the right side is of degree two in yn and where the coefficients an, bn and cn are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, yn ∈ ℚ and the height of yn dominates the height of the coefficient functions an, bn and cn. We show that if the logarithmic height of yn grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dPII or its autonomous version or yn is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

Type: Article
Title: Height growth of solutions and a discrete Painlevé equation
Open access status: An open access version is available from UCL Discovery
DOI: 10.1088/0951-7715/28/7/2379
Publisher version: http://dx.doi.org/10.1088/0951-7715/28/7/2379
Language: English
Additional information: Copyright © 2015 IOP Publishing Ltd & London Mathematical Society.
Keywords: discrete Painleve equations, discrete integrable systems, Diophantine integrability
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1412350
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