\'Etude des liens entre la taille et l'irr\'eductibilit\'e des solutions monomiales minimales dans $SL_{2}(\mathbb{Z}/N\mathbb{Z})$

TL;DR

This paper investigates the relationship between the size and irreducibility of minimal monomial solutions in modular matrix equations, providing bounds and conditions for irreducibility within the context of the modular group.

Contribution

It introduces new bounds on the size of irreducible solutions and establishes criteria linking size to irreducibility for minimal monomial solutions.

Findings

Upper bounds on the size of irreducible solutions

Certain sizes automatically imply irreducibility

Connections between solution size and modular group properties

Abstract

This article aims to study some $n$-tuples of elements belonging to a ring $\mathbb{Z}/N\mathbb{Z}$ related to the combinatorics of congruence subgroups of the modular group. More precisely, we will focus here on the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing during the study of Coxeter's friezes), modulo an integer $N$, all of whose components are identical and minimal for this property. Our objective here is to study the links between the size of minimal monomial solutions and a property of irreducibility which is central in the study of the combinatorics of the modular group. In particular, we will obtain an upper bound of the size of irreducible monomial solutions and we will prove that some sizes automatically lead to irreducibility.