Smallest representatives of $\operatorname{SL}(2,\mathbb Z)$-orbits of binary forms and endomorphisms of ${\mathbb P}^1$

TL;DR

This paper presents an algorithm to find minimal representatives of binary forms and endomorphisms of the projective line under certain group actions, aiding computational tasks in algebraic geometry.

Contribution

It introduces a method to compute smallest orbit representatives of binary forms and extends this to endomorphisms of ${ m P}^1$, improving computational efficiency.

Findings

Algorithm effectively finds minimal orbit representatives.

Extension to endomorphisms of ${ m P}^1$ broadens applicability.

Facilitates computations involving minimal models of endomorphisms.

Abstract

We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\operatorname{PGL}(2,\mathbb Q)$ conjugacy class of an endomorphism of the projective line. Having a small model of such an endomorphism is useful for various computations.