Three approaches to detecting discrete integrability

TL;DR

This paper explores three different methods to identify discrete integrability in equations by analyzing solution complexity through Nevanlinna theory, degree growth, and height growth, linking low complexity to discrete Painlevé equations.

Contribution

It introduces three distinct approaches to measure solution complexity and demonstrates their connection to singularity confinement and discrete Painlevé equations.

Findings

Low complexity correlates with singularity confinement.

Each approach identifies discrete Painlevé equations.

Different measures of complexity provide consistent integrability indicators.

Abstract

A class of discrete equations is considered from three perspectives corresponding to three measures of the complexity of solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a function field and the height growth of iterates over the rational numbers. In each case, low complexity implies a form of singularity confinement which results in a known discrete Painlev\'e equation.