Additively indecomposable quadratic forms over totally real number fields

TL;DR

This work develops determinant- and indecomposability-based tools for quadratic forms over totally real fields, proving a determinant bound: if $N_{K/ Q}(\det(Q)) \ge \gamma_{K,n}^n C^n$, then $Q$ must decompose additively as $Q=H+\alpha L^2$; it then translates these bounds into both upper and lower limits for the ranks of $n$-universal forms via indecomposable integers and indecomposable $n$-ary forms. The framework introduces the $g$-invariant and sets of indecomposable objects, yielding explicit rank bounds and constructive decompositions, with sharper results in real quadratic fields. The paper also proves existence of additively indecomposable binary forms in every real quadratic field, derives density-type lower bounds for the number of such forms, and presents comprehensive classifications for several concrete fields ($D=2,3,5,6,21$), supported by an implemented algorithm that enumerates indecomposables and tests decomposability. Overall, these results sharpen our understanding of universality in quadratic forms over number fields and provide practical tools and data for bounding minimal ranks and constructing universal 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.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Theorem 3.1
• Theorem 3.2: =Theorem \ref{['thm:main']}