Large Algebraic Integers

TL;DR

This paper explores the concept of large algebraic integers and ideals, analyzing their properties, relationships with field invariants, and applications in continued fractions, along with an algorithm for testing largeness.

Contribution

It introduces the notion of largeness for algebraic integers and ideals, linking it to field invariants and providing an algorithm for testing this property.

Findings

Largeness relates to the regulator and lattice of units.

Connections established with Weil height and Bogomolov property.

Algorithm developed for testing largeness.

Abstract

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.