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arXiv · 2407.19317

Number of roots of the continuant over a finite local ring

Flavien Mabilat
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Abstract

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 $\frac{\mathbb{F}_{q}[X]}{\langle X^{m} \rangle}$. Besides, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for $\lambda$-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter's friezes) over the rings $\mathbb{Z}/p^{m}\mathbb{Z}$.

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Flavien Mabilat. 2024-07-27. Number of roots of the continuant over a finite local ring. https://arxiv.org/abs/2407.19317

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