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Sails for universal quadratic forms

Vítězslav Kala, Siu Hang Man

TL;DR

The paper forges a link between sails from the geometry of generalized continued fractions and the arithmetic of totally real fields, focusing on universal quadratic forms and indecomposable integers. By analyzing sails and unit-signature structure, it proves that in real biquadratic fields with unit-signature rank at least 3, both the minimal classical universal-lattice rank and the indecomposable count grow as powers of the discriminant, and it constructs a family with only logarithmic growth. It also develops practical tools to compute unit-signature ranks, applies these ideas to Kitaoka's conjecture, and presents a biquadratic-family with explicit indecomposables, illustrating how sails refine our understanding of universal forms beyond quadratic case. The results highlight a geometric pathway to control universal representations in higher-degree totally real fields and suggest avenues for future exploration of sails in more signatures and broader field families.

Abstract

We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, universal quadratic forms and additively indecomposable integers. Our main application is to biquadratic fields, for which we show that if their signature rank is at least 3, then ranks of universal forms and numbers of indecomposables grow as a power of the discriminant. We also construct a family in which these numbers grow only logarithmically.

Sails for universal quadratic forms

TL;DR

The paper forges a link between sails from the geometry of generalized continued fractions and the arithmetic of totally real fields, focusing on universal quadratic forms and indecomposable integers. By analyzing sails and unit-signature structure, it proves that in real biquadratic fields with unit-signature rank at least 3, both the minimal classical universal-lattice rank and the indecomposable count grow as powers of the discriminant, and it constructs a family with only logarithmic growth. It also develops practical tools to compute unit-signature ranks, applies these ideas to Kitaoka's conjecture, and presents a biquadratic-family with explicit indecomposables, illustrating how sails refine our understanding of universal forms beyond quadratic case. The results highlight a geometric pathway to control universal representations in higher-degree totally real fields and suggest avenues for future exploration of sails in more signatures and broader field families.

Abstract

We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, universal quadratic forms and additively indecomposable integers. Our main application is to biquadratic fields, for which we show that if their signature rank is at least 3, then ranks of universal forms and numbers of indecomposables grow as a power of the discriminant. We also construct a family in which these numbers grow only logarithmically.
Paper Structure (10 sections, 26 theorems, 89 equations)

This paper contains 10 sections, 26 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Sails for totally real number fields
  3. Unit signature ranks and indecomposable elements
  4. Units in biquadratic fields
  5. Continued fraction
  6. Universal lattices and indecomposable elements
  7. Proof of Theorem 1.1
  8. Finding unit signature ranks for biquadratic fields
  9. Applications towards Kitaoka conjecture
  10. A family of fields with few indecomposables

Key Result

Theorem 1.1

Let $K$ be a totally real biquadratic field with $\mathop{\mathrm{sgnrk}}\nolimits(K) \ge 3$. Then we have $R_{\mathop{\mathrm{cls}}\nolimits}(K), \iota(K) \gg \Delta_K^{1/12}$ and $R(K) \gg \Delta_K^{1/42}$ for some explicitly computable constants, where $\Delta_K$ denotes the discriminant of $K$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Example 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem $\ref{['thm:si']}$
  • ...and 33 more