Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universality criterion sets for quadratic forms over number fields

Vitezslav Kala, Jakub Krásenský, Giuliano Romeo

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.

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 of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of , 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.

Paper Structure

This paper contains 15 sections, 19 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. New notation
  4. Uniqueness
  5. Indecomposable elements, rational integers
  6. Variations of the criterion set definition
  7. Open problems
  8. Growth of criterion sets
  9. Explicit criterion sets
  10. Provable criterion sets
  11. Very small criterion sets
  12. Relation between different criterion sets
  13. Lifting of criterion sets
  14. Critical elements
  15. Maximal criterion sets

Key Result

Theorem 1.1

For every totally real number field $K$, there exists a unique criterion set $\mathcal{C}_K$ which is minimal with respect to inclusion. This set $\mathcal{C}_K$ is finite and consists precisely of the critical elements (see Definition de:critical).

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • ...and 30 more