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

TL;DR

This work analyzes λ-quiddities, the size-$n$ solutions of $M_n(a_1, abla\tdots,a_n)=\pm\mathrm{Id}$, and their role in Coxeter friezes. It develops explicit counting formulas over finite fields and composite rings, starting with detailed recursion-based counts $u_{n,q}^{B}$ for matrices in $\text{SL}_2(\mathbb{F}_q)$ and then extending to $\mathbb{Z}/N\mathbb{Z}$ via the Chinese Remainder Theorem, yielding closed forms for odd and even $n$ and products over prime-power factors. The paper also analyzes irreducible solutions, proving asymptotic growth properties of the number of equivalence classes $v_N$ and providing numerical data for small $N$, highlighting non-monotonic behavior. Together, these results deepen understanding of friezes, their modular representations, and the combinatorics of λ-quiddities, with potential implications for counting friezes and representations of $\mathrm{SL}_2$ modulo $N$.

Abstract

The $λ$-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 $λ$-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 $λ$-quiddities verifying an irreducibility condition over $\mathbb{Z}/N\mathbb{Z}$ when $N$ goes to the infinity.

Figures (1)

• Figure 1: À gauche une triangulation et sa quiddité, à droite la séquence de parité de cette triangulation

Theorems & Definitions (16)

• Proposition 2.5
• Proposition 2.11
• Proposition 2.14: M0, remarque 5.4
• Proposition 2.15
• proof
• proof
• proof
• Proposition 2.18
• proof
• proof : Démonstration du théorème \ref{['thm412']}