Comptage des quiddit{\'e}s sur les corps finis et sur quelques anneaux $\mathbb{Z}/N\mathbb{Z}$

TL;DR

This paper provides explicit formulas for counting $ ext{lambda}$-quiddities over finite fields and certain rings, analyzing their asymptotic behavior and solutions related to Coxeter's friezes.

Contribution

It introduces explicit formulas for enumerating $ ext{lambda}$-quiddities over finite fields and rings $ ext{Z}/N ext{Z}$ with specific conditions, extending previous combinatorial results.

Findings

Derived formulas for counting $ ext{lambda}$-quiddities over finite fields.

Established enumeration formulas for rings $ ext{Z}/N ext{Z}$ with $N=4m$ and $m$ square free.

Analyzed asymptotic behavior of irreducible $ ext{lambda}$-quiddities as $N$ approaches infinity.

Abstract

The $\lambda$-quiddities of size $n$ are $n$-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter's friezes. These can be considered on various sets with very different structures from one set to another. The main objective of this text is to obtain explicit formulas giving the number of $\lambda$-quiddities of size $n$ over finite fields and over the rings $\mathbb{Z}/N\mathbb{Z}$ with $N=4m$ and $m$ square free. We will also give some elements about the asymptotic behavior of the number of $\lambda$-quiddities verifying an irreducibility condition over $\mathbb{Z}/N\mathbb{Z}$ when $N$ goes to the infinity.