Calculating algebraic entropies: an express method

TL;DR

This paper introduces an 'express' method for quickly assessing the integrability of three-point mappings with confined singularities, offering a practical alternative to more complex degree growth calculations.

Contribution

The paper presents a novel, simplified approach inspired by Halburd's method for determining integrability without full degree growth analysis.

Findings

The 'express' method effectively identifies integrable systems.

It offers a faster alternative to traditional degree growth computations.

The method has limitations demonstrated through various examples.

Abstract

We describe a method for investigating the integrable character of a given three-point mapping, provided that the mapping has confined singularities. Our method, dubbed "express", is inspired by a novel approach recently proposed by R.G. Halburd. While the latter aims at computing the exact degree growth of a given mapping based on the structure of its singularities, we content ourselves with obtaining an answer as to whether a given system is integrable or not. We present several examples illustrating our method as well as its limitations. We also compare the present method to the full-deautonomisation approach we recently introduced.