$\lambda$-quiddit{\'e} sur $\mathbb{Z}[\alpha]$ avec $\alpha$ transcendant

TL;DR

This paper classifies all irreducible $ ext{lambda}$-quiddity cycles over the ring $ ext{Z}[ ext{alpha}]$ where $ ext{alpha}$ is a transcendental complex number, contributing to the understanding of Coxeter friezes.

Contribution

It provides a complete list of irreducible $ ext{lambda}$-quiddity cycles on $ ext{Z}[ ext{alpha}]$ with $ ext{alpha}$ transcendental, extending prior classifications.

Findings

All irreducible $ ext{lambda}$-quiddity cycles over $ ext{Z}[ ext{alpha}]$ are identified.

The classification applies specifically to the case where $ ext{alpha}$ is transcendental.

The results deepen the understanding of Coxeter friezes and their algebraic structures.

Abstract

As part of the study of Coxeter's friezes, M. Cuntz introduced the notion of irreducible $\lambda$-quiddity cycle. The objective of this note is to list all the irreducible $\lambda$-quiddity cycles on the ring $\mathbb{Z}[\alpha]$ with $\alpha$ a transcendent complex number. ----- Dans le cadre de l'\'etude des frises de Coxeter, M.Cuntz a introduit la notion de $\lambda$-quiddit\'e irr\'eductible. L'objectif de cette note est de lister toutes les $\lambda$-quiddit\'es irr\'eductibles sur l'anneau $\mathbb{Z}[\alpha]$ dans le cas o\`u $\alpha$ est un nombre complexe transcendant.