Number of roots of the continuant over a finite local ring

TL;DR

This paper derives formulas for counting roots of the $n$th continuant polynomial over finite local rings, including applications to $ ext{lambda}$-quiddities and Coxeter friezes.

Contribution

It provides new explicit counting formulas for roots of continuant polynomials over various finite local rings and offers a novel proof for $ ext{lambda}$-quiddities over $ ext{Z}/p^m ext{Z}$.

Findings

Formulas for roots over finite fields and rings.

Counting methods for $ ext{lambda}$-quiddities.

Simplified proof for solutions of matrix equations.

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