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Representing rational integers by generalized quadratic forms over quadratic fields

Ondřej Chwiedziuk, Matěj Doležálek, Emma Pěchoučková, Zdeněk Pezlar, Om Prakash, Giuliano Romeo, Anna Růžičková, Mikuláš Zindulka

Abstract

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.

Representing rational integers by generalized quadratic forms over quadratic fields

Abstract

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.
Paper Structure (12 sections, 43 theorems, 173 equations, 4 tables)

This paper contains 12 sections, 43 theorems, 173 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized polynomials with rational values
  4. Generalized quadratic forms
  5. Binary and ternary generalized quadratic forms
  6. Indefinite generalized quadratic forms
  7. Examples of universal binary generalized forms
  8. Preliminaries
  9. Universal quadratic forms associated to binary generalized forms for D=2
  10. Universal quadratic forms associated to binary generalized forms for D=3
  11. Universal quadratic forms associated to classical binary generalized forms for D=5
  12. Universal quadratic forms associated to binary generalized forms for D=6,7,10

Key Result

Theorem 1.1

A classical quadratic form $Q$ over $\mathbb{Z}$ is universal if and only if it represents the integers

Theorems & Definitions (81)

  • Theorem 1.1: 15-Theorem, Conway-Schneeberger
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 71 more