On multiplicatively independent bases in cyclotomic number fields

TL;DR

This paper investigates the multiplicative independence of certain algebraic integer bases in cyclotomic number fields, providing results for cases where the difference between parameters is less than one million.

Contribution

It establishes conditions under which two such bases are multiplicatively independent, extending understanding of their algebraic properties.

Findings

Proves multiplicative independence for cases with |m-n|<10^6

Shows independence when |m|,|n|>1

Extends previous results on algebraic integer bases

Abstract

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent. We show the multiplicative independence in case that $0<|m-n|<10^6$ and $|m|,|n|> 1$.