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

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.