Reduction of binary forms via the hyperbolic center of mass

TL;DR

This paper introduces a geometric reduction method for real binary forms without real roots using hyperbolic center of mass, offering advantages over existing theories in certain contexts.

Contribution

It presents a novel geometric approach based on hyperbolic geometry for reducing binary forms, emphasizing the hyperbolic center of mass concept.

Findings

Model compares favorably with existing reduction theories

Utilizes hyperbolic geometry tools in reduction processes

Provides a self-contained treatment of hyperbolic geometric concepts

Abstract

In this paper we provide an alternative reduction theory for real, binary forms with no real roots. Our approach is completely geometric, making use of the notion of hyperbolic center of mass in the upper half-plane. It appears that our model compares favorably with existing reduction theories, at least in certain aspects related to the field of definition. Various tools and features of hyperbolic geometry that are interesting in themselves, but also relevant for our and various other reduction theories papers (\cite{julia} and \cite{SC}), are also treated in detail and in a self-contained way here.