$\lambda$-quiddity and subgroups generated by an algebraic number

TL;DR

This paper explores solutions to matrix equations called λ-quiddities over cyclic subgroups of complex numbers generated by algebraic numbers, providing new insights into their structure and properties.

Contribution

It extends the study of λ-quiddities to specific cyclic subgroups generated by algebraic numbers like a+b√k, addressing a problem posed by M. Cuntz.

Findings

Characterization of λ-quiddities over subgroups generated by algebraic numbers

Identification of solution structures for specific algebraic generators

Insights into the algebraic properties influencing solutions

Abstract

During his work devoted to Coxeter's friezes, M. Cuntz initiated the study of the notion of $\lambda$-quiddity and raised the problem of the study of this over some subsets of $\mathbb C$. More specifically, $\lambda$-quiddities are the solutions to a matrix equation, related to various mathematical objects, which we seek to solve over different sets. The aim of this text is to provide some new insights into the problem raised by M. Cuntz in the case of some cyclic subgroups of ($\mathbb{C},+$) generated by an algebraic number. In particular, we will study the cases of subgroups generated by $a+b\sqrt{k}$.