Universal quadratic forms, small norms and traces in families of number fields

TL;DR

This paper investigates universal quadratic forms over specific families of totally real number fields, providing estimates on their ranks by characterizing indecomposables and elements of small trace, and analyzing ideal norms.

Contribution

It introduces a method to estimate ranks of universal quadratic forms by characterizing indecomposables and small trace elements in these fields.

Findings

Good estimates on ranks of universal quadratic forms.

Complete characterization of indecomposable integers.

Asymptotic behavior of principal ideals with small norm.

Abstract

We obtain good estimates on the ranks of universal quadratic forms over Shanks' family of the simplest cubic fields and several other families of totally real number fields. As the main tool we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.