Additively indecomposable quadratic forms over totally real number fields

TL;DR

This paper establishes bounds on the determinants of additively indecomposable quadratic forms over totally real number fields and classifies certain binary forms, advancing understanding of universal quadratic forms in number theory.

Contribution

It provides new bounds for determinants of indecomposable forms and classifies binary forms over specific real quadratic fields, offering insights into minimal ranks of universal quadratic forms.

Findings

Bounds for determinants of indecomposable forms

Classification of binary forms over specific fields

Bounds for minimal ranks of n-universal quadratic forms

Abstract

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper bounds for the minimal ranks of $n$-universal quadratic forms. For $\mathbb{Q}(\sqrt{2}),~\mathbb{Q}(\sqrt{3}),~\mathbb{Q}(\sqrt{5}),~\mathbb{Q}(\sqrt{6})$, and $\mathbb{Q}(\sqrt{21})$, we classify, up to equivalence, all classical, additively indecomposable binary quadratic forms.