Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables

TL;DR

The paper establishes a concrete link between the Jacobi--Perron algorithm and additively indecomposable integers in totally real cubic fields, focusing on Ennola's and Shanks' simplest cubic families. It provides explicit indecomposable descriptions in Ennola's cubic fields, proves a sharp Pythagoras number of 6 for the relevant order, and demonstrates periodic JPA expansions for carefully chosen vectors in both field families. The work combines deep number-theoretic constructs (codifferents, unit cones, and signatures) with multidimensional continued fractions, and it reports extensive computational experiments on indecomposability of convergents and semiconvergents to illuminate when these expansions reflect underlying arithmetic. Overall, the results advance our understanding of how multidimensional CFs encode arithmetic of totally real cubic fields and have implications for universal quadratic forms and related diophantine questions in number fields.

Abstract

We study a new connection between multidimensional continued fractions, such as Jacobi--Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions.

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: KT
• Proposition 2.2: KT
• Lemma 3.1
• proof
• Lemma 3.2
• proof