Binary forms with covariant points close to the real axis

TL;DR

This paper investigates the relationship between covariant points of real binary forms and their roots' locations, providing conditions for covariant points to be near the real axis and enhancing reduction algorithms.

Contribution

It introduces new conditions linking covariant points close to the real axis with root distributions, improving binary form reduction methods.

Findings

Covariant points near the real axis imply roots on small circles.

Conditions on radius r determine proximity of covariant points to the real axis.

Results enhance existing reduction algorithms for binary forms.

Abstract

For a real binary form $F(X, Z)$, Stoll and Cremona have defined a reduction theory using the action of the modular group $SL_2(\mathbb{Z})$, and associated to each binary form a covariant point $z(F)$ located in the upper half plane. When the point $z(F)$ is close to the real axis, then at least half of the roots will be on a circle of small radius $r$. Conversely, we find conditions depending on the radius $r$ such that the covariant point $z(F)$ to be close to the real axis. The results have further applications to improving the reduction algorithm for binary forms of Stoll and Cremona.