Essential Minimum in Families of lines

TL;DR

This paper develops methods to bound the essential minimum of certain height functions, including the Zhang-Zagier height, and identifies intervals where their images are dense, using a refined Fekete-Szego theorem.

Contribution

It introduces a refined approach to estimate bounds and density intervals for the essential minimum of specific height functions, expanding on classical theorems.

Findings

Established upper and lower bounds for the Zhang-Zagier height.

Identified intervals where the height images are dense.

Applied a refined Fekete-Szego theorem to these problems.

Abstract

We apply general methods to generate upper and lower bounds for the essential minimum of a specific family of height functions. In particular, the results shown in this article apply to the case of the Zhang-Zagier height. Furthermore, we can find intervals, where the images of these heights are dense. Our main tool to find upper bounds and intervals of density, is a refinement of the classical Fekete-Szego theorem due to Burgos Gil, Philippon, Rivera-Letelier and Sombra.