Height growth of solutions and a discrete Painlevé equation

Abstract

Consider the discrete equation y n+1 + y n−1 = a n + b n y n + c n y 2where the right side is of degree two in y n and where the coefficients a n , b n and c n are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, y n ∈ Q and the height of y n dominates the height of the coefficient functions a n , b n and c n . We show that if the logarithmic height of y n grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dP II or its autonomous version or y n is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

Elementary exact calculations of degree growth and entropy for discrete equations

“…The logarithmic height can be determined by knowledge of all absolute values. In this way, lower bounds on the height growth were determined in [ 22 , 23 ]. Connections between Nevanlinna theory, Diophantine integrability and the degree growth described in this paper are studied in [ 14 ] in analogues of the singularity confinement calculations are described in each setting for the same class of equations.…”

“…To calculate the degree of y n , the only extra information required from the equation is an analysis of some other singular initial conditions, which is often trivial. This measure of complexity has also been used in [ 14 , 15 ] where lower bounds on the degrees of iterates were obtained to show that many equations had exponential growth of degrees. In this paper, we are able to calculate the degrees exactly.…”

“…In both these settings, one can obtain quite strong estimates on the degrees of various rational functions of the dependent variables in integrable equations, as shown in [18,19]. However, in order to obtain the precise forms of equations, including the dependence on the independent variable, it has been shown in the examples considered in [20][21][22][23] that singularity confinement is a necessary condition for slow growth of the relevant measure of complexity.…”

Three Approaches to Detecting Discrete Integrability

“…A similar result leading to the discrete Painlevé II equation has been derived in [3]. Heights were first used in Abarenkova, Anglès d'Auriac, Boukraa, Hassani and Maillard [1] to estimate entropy.…”

A lemma on the difference quotients

“…Halburd [15] has shown, assuming that the heights of the coefficients are small compared to the height of the solution, that the heights of iterates of the discrete equation (5.1) over number fields grow exponentially, unless deg y0 (R) = 1. Using this idea of Diophantine integrability, Al-Ghassani and Halburd obtained an extension of this result to the second order case by singling out the discrete Painlevé II equation [2].…”