Maximal size of irreducible $\lambda$-quiddities over polynomial and formal power series rings

TL;DR

This paper investigates the maximal size of irreducible $ ext{lambda}$-quiddities over polynomial and formal power series rings, providing bounds and classifications for these solutions in various ring contexts.

Contribution

It offers a comprehensive analysis of the boundedness and structure of irreducible $ ext{lambda}$-quiddities over polynomial and formal power series rings, extending previous understanding.

Findings

Determined bounds for the size of irreducible $ ext{lambda}$-quiddities over polynomial rings.

Classified irreducible solutions over rings of formal power series.

Extended results to infinite rings and various algebraic structures.

Abstract

The study of the combinatorics of the modular group and of Coxeter's friezes naturally leads to the investigation of a matrix equation, sometimes referred to as the Conway-Coxeter equation. The solutions of size $n$ of this equation, called $\lambda$-quiddities, are $n$-tuples of elements of a given ring $B$. A detailled understanding of these objects relies on the notion of irreducible solutions, from which all $\lambda$-quiddities can be reconstructed. One of the central questions that naturally arises in this context is whether the irreducible $\lambda$-quiddities over $B$ have bounded size, and, if so, how to determine such a bound. In this paper, we aim to list results that address this question in the case of polynomial rings $A[X]$ and $\mathbb{K}[X]$, where $A$ is a finite commutative unitary ring and $\mathbb{K}$ is a commutative field. Moreover, the stated results will also make it possible to treat easily many situations in which $A$ is infinite. Finally, we shall give a complete answer to the initial question for all rings of formal power series.