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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2604.12547 |
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| _version_ | 1866908962646917120 |
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| author | Mabilat, Flavien |
| author_facet | Mabilat, Flavien |
| contents | 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 $λ$-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 $λ$-quiddities can be reconstructed. One of the central questions that naturally arises in this context is whether the irreducible $λ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12547 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal size of irreducible $λ$-quiddities over polynomial and formal power series rings Mabilat, Flavien Combinatorics 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 $λ$-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 $λ$-quiddities can be reconstructed. One of the central questions that naturally arises in this context is whether the irreducible $λ$-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. |
| title | Maximal size of irreducible $λ$-quiddities over polynomial and formal power series rings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.12547 |