Universality criterion sets for quadratic forms over number fields

TL;DR

This paper introduces a universal criterion set for quadratic forms over number fields, establishing their existence, uniqueness, and explicit elements, inspired by the 290-Theorem, and extends these results to broader representation subsets.

Contribution

It provides a novel characterization of minimal criterion sets for quadratic forms over number fields, proving their existence, uniqueness, and explicit structure.

Findings

Criterion sets always exist and are unique.

Minimal criterion sets contain certain explicit elements.

Extension of uniqueness to broader representation subsets.

Abstract

In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it necessarily represents all totally positive integers, i.e., is universal. We use a novel characterization of minimal criterion sets to show that they always exist and are unique, and that they must contain certain explicit elements. We also extend the uniqueness result to the more general setting of representations of a given subset of the integers.