Periodicity of Multidimensional Continued Fractions

TL;DR

This paper extends the understanding of periodicity in multidimensional continued fractions, demonstrating new classes of algebraic number fields and roots of specific cubic equations exhibit periodic continued fraction expansions.

Contribution

It generalizes previous results by showing periodicity for roots of certain polynomial families in various algebraic number fields using the Algebraic Jacobi-Perron algorithm.

Findings

Periodic continued fractions for roots of $ ext{Q}( oot{l}{m^l+1})$.

Roots of specific cubic equations have periodic multidimensional continued fractions.

Abstract

It is known that the continued fraction expansion of a real number is periodic if and only if the number is a quadratic irrational. In an attempt to generalize this phenomenon to other settings, Jun-Ichi Tamura and Shin-Ichi Yasutomi have developed a new algorithm for multidimensional continued fractions (Algebraic Jacobi-Perron algorithm) that involves cubic irrationals, and proved periodicity in some cubic number fields, such as $\mathbb{Q}(\sqrt[3]{m^3+1})$ where $m\in\mathbb{Z}$, and $\mathbb{Q}(\delta_m)$ where $\delta_m$ is a root of $x^3-mx+1=0,\,\,m\in\mathbb{Z},\,\, m\geq3$ with the algorithm. In this paper, we study some other types of number fields that give rise to periodic continued fractions using the Algebraic Jacobi-Perron algorithm obtaining results for $\mathbb{Q}(\sqrt[l]{m^l+1})$ for any positive integer $l$. Furthermore, we find that some families of cubic equations, such as $x^3+3ax^2+bx+ab-2a^3+1=0,\,b\leq3a^2-3,\,a,b\in\mathbb{Z}$, have roots that have periodic multidimensional continued fractions.