Height growth of solutions and a discrete Painlev\'e equation

TL;DR

This paper investigates the height growth of solutions to a specific discrete equation, showing that slow growth indicates the equation is either a known discrete Painlevé equation or related to a Riccati equation, supporting the idea that slow height growth signals integrability.

Contribution

It characterizes solutions with slow height growth for a class of discrete equations, linking such growth to discrete Painlevé equations or Riccati equations, thus providing evidence for height growth as an integrability detector.

Findings

Solutions with slow height growth are either discrete Painlevé II or its autonomous version.

Such solutions are also solutions to a discrete Riccati equation.

Supports the hypothesis that slow height growth indicates integrability.

Abstract

Consider the discrete equation $$ y_{n+1}+y_{n-1}=\frac{a_n+b_ny_n+c_ny_n^2}{1-y_n^2}, $$ where 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 $n$, $y_n\in\mathbb{Q}$ 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 Painlev\'e equation ${\rm dP}_{\!\rm II}$ 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.