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Reduction of binary forms via the hyperbolic center of mass

Artur Elezi, Tony Shaska

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.

Reduction of binary forms via the hyperbolic center of mass

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.

Paper Structure

This paper contains 9 sections, 13 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The hyperbolic geometry of positive definite binary forms
  3. The hyperbolic plane $\mathcal{H}_2$
  4. The upper half-plane $\mathcal{H}_2$ as a parameter space for positive definite quadratics.
  5. The hyperbolic three dimensional space $\mathcal{H}_3$
  6. The upper half-space $\mathcal{H}_3$ as a parameter space for positive definite Hermitian quadratics.
  7. Reduction of binary forms via the Julia quadratic
  8. The reduction of real forms via the hyperbolic center of mass
  9. The center of mass via the hyperboloid model of the hyperbolic plane

Key Result

Proposition 1

Let $A$ be one of the ideal points of a geodesic that passes through $z=x+y{\bf i},w=u+v{\bf i}\in \mathcal{H}_2$. Then $d_H(z,w)=|d_H(A,z)-d_H(A,w)|$.

Figures (5)

  • Figure 1: Geodesics and their ideal points
  • Figure 2: The hyperbolic distance between two points $z$ and $w$ with $\Re (z) \neq \Re(w)$ and $\Re (z) = \Re(w)$
  • Figure 3: The distance between $z\in \mathcal{H}_2$ and a boundary point $A$.
  • Figure 4: The additive property of the boundary distance
  • Figure 5: Geodesics in upper half-space $H$

Theorems & Definitions (27)

  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • Lemma 1
  • proof
  • ...and 17 more