Universality criterion sets for quadratic forms over number fields
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.
Fields of Study & Subjects
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Coding theory and cryptography
Publication details
- Journal / source
- Advances in Mathematics
- Publisher
- Elsevier BV
- Volume
- 500
- Pages
- 111080–111080
- Publication date
- 3 June 2026
- Language
- EN