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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.19317 |
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| _version_ | 1866910544896720896 |
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| author | Mabilat, Flavien |
| author_facet | Mabilat, Flavien |
| contents | The aim of this article is to obtain a formula giving, for a positive integer $n$, the number of roots of the $n^{th}$ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the continuant over the local rings $\mathbb{F}_{q}$, $\mathbb{Z}/p^{m}\mathbb{Z}$ and $\frac{\mathbb{F}_{q}[X]}{<X^{m}>}$. To conclude, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for $λ$-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter's friezes) over the rings $\mathbb{Z}/p^{m}\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Number of roots of the continuant over a finite local ring Mabilat, Flavien Combinatorics The aim of this article is to obtain a formula giving, for a positive integer $n$, the number of roots of the $n^{th}$ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the continuant over the local rings $\mathbb{F}_{q}$, $\mathbb{Z}/p^{m}\mathbb{Z}$ and $\frac{\mathbb{F}_{q}[X]}{<X^{m}>}$. To conclude, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for $λ$-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter's friezes) over the rings $\mathbb{Z}/p^{m}\mathbb{Z}$. |
| title | Number of roots of the continuant over a finite local ring |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.19317 |