Computing the Canonical Height of a Point in Projective Space

TL;DR

This paper presents a new algorithm for computing the canonical height of points in projective space that avoids integer factorization, applicable to morphisms of degree at least 2.

Contribution

It introduces a factorization-free algorithm for calculating canonical heights in projective space, improving computational efficiency.

Findings

Algorithm requires no integer factorization

Applicable to morphisms of degree ≥ 2

Efficient computation of canonical heights

Abstract

We give an algorithm which requires no integer factorization for computing the canonical height of a point in $\mathbb{P}^1(\mathbb{Q})$ relative to a morphism $\phi: \mathbb{P}_{\mathbb{Q}}^1 \rightarrow \mathbb{P}_{\mathbb{Q}}^1$ of degree $d \geq 2$.