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Additive structure of non-monogenic simplest cubic fields

Daniel Gil-Muñoz, Magdaléna Tinková

TL;DR

The authors classify all indecomposable totally positive integers in non-monogenic simplest cubic fields $K$ with $[\mathcal{O}_K:\mathbb{Z}[\rho]]=3$ by embedding $K$ into Minkowski space and enumerating lattice points in two key parallelepipeds generated by a proper pair of totally positive units. They obtain a complete list (up to unit multiples) with explicit parametrizations and prove their indecomposability via codifferential methods, culminating in a total count $\frac{a^2+3a}{18}+2a+2$ of indecomposables. Leveraging this structure, they derive sharp bounds on norms of indecomposables, determine the smallest and largest possible norms, and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$ for this family. The indecomposable data further yields quantitative bounds on the minimal number of variables required for universal quadratic forms over $\mathcal{O}_K$, providing concrete asymptotics in terms of the parameter $a$. Overall, the work connects the geometry of lattice points in parallelepipeds to additive and quadratic-form properties in a significant family of real cubic fields.

Abstract

We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[ρ]]$ of a root $ρ$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$.

Additive structure of non-monogenic simplest cubic fields

TL;DR

The authors classify all indecomposable totally positive integers in non-monogenic simplest cubic fields with by embedding into Minkowski space and enumerating lattice points in two key parallelepipeds generated by a proper pair of totally positive units. They obtain a complete list (up to unit multiples) with explicit parametrizations and prove their indecomposability via codifferential methods, culminating in a total count of indecomposables. Leveraging this structure, they derive sharp bounds on norms of indecomposables, determine the smallest and largest possible norms, and prove that the Pythagoras number of is for this family. The indecomposable data further yields quantitative bounds on the minimal number of variables required for universal quadratic forms over , providing concrete asymptotics in terms of the parameter . Overall, the work connects the geometry of lattice points in parallelepipeds to additive and quadratic-form properties in a significant family of real cubic fields.

Abstract

We consider Shanks' simplest cubic fields for which the index of a root of the defining parametric polynomial is . For them, we study the additive indecomposables of and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of . Mainly, we obtain good bounds on the ranks of universal quadratic forms over and prove that the Pythagoras number of is .
Paper Structure (21 sections, 29 theorems, 174 equations, 3 figures, 3 tables)

This paper contains 21 sections, 29 theorems, 174 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplest cubic fields and their monogenity
  4. Non-monogenic simplest cubic fields with index $3$
  5. The parallelepipeds of lattice points
  6. Lattice points in the first parallelepiped
  7. Lattice points in the second parallelepiped
  8. Indecomposables when $[{\mathcal{O}}_K:{\mathbb Z}[\rho]]=3$
  9. The case $s=0$
  10. The case $s=1$
  11. The case $s=2$
  12. Representing the lattice points
  13. Proof of the main theorem
  14. Proving the indecomposability
  15. Transformations of lattice points
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field with $[\mathcal{O}_K:\mathbb{Z}[\rho]]=3$. Up to multiplication by totally positive units, the indecomposable integers in $K$ are 1, The points from (ii) to (iv) have minimal trace $2$, while the remaining ones have minimal trace $1$. The total number of indecomposables in $K$ (up to multiplication by totally positive units) is $\frac{a^2+3a}{18}

Figures (3)

  • Figure 1: Lattice points with $s=0$
  • Figure 2: Lattice points with $s=1$
  • Figure 3: Lattice points with $s=2$

Theorems & Definitions (57)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 3.1
  • ...and 47 more