A fast algorithmic way to calculate the degree growth of birational mappings

TL;DR

This paper introduces a rapid, integer-based algorithm for calculating the degree growth of iterates in birational mappings, applicable to both integrable and non-integrable cases, aligning with existing methods.

Contribution

The authors develop a fast, integer arithmetic algorithm for degree calculation in birational mappings, improving computational efficiency over traditional approaches.

Findings

The method is extremely fast and uses only simple arithmetic operations.

It accurately computes the dynamical degree for various mappings.

Results agree with previously known methods for non-integrable cases.

Abstract

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.