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

Flavien Mabilat

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.

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

Abstract

This article aims to study some -tuples of elements belonging to a ring 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 , 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.

Paper Structure

This paper contains 10 sections, 18 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Définitions et résultats principaux
  3. Majoration de la taille des solutions monomiales minimales irréductibles
  4. Résultats préliminaires
  5. Démonstration du théorème principal
  6. Le cas des entiers de la forme $3m$ avec $m$ premier avec 6
  7. Solutions monomiales minimales de taille impaire
  8. Étude de certaines tailles dépendant de $N$
  9. Solutions monomiales minimales de \ref{['p']} taille $N$
  10. Quelques résultats supplémentaires

Key Result

Proposition \oldthetheorem

Soit $N=p_{1}^{\alpha_{1}}\ldots p_{r}^{\alpha_{r}}$, avec, pour tout $i$, $p_{i}$ premier, $\alpha_{i} \geq 1$ et $p_{i} \neq p_{j}$ si $i \neq j$. Soient $\overline{k} \in \mathbb{Z}/N\mathbb{Z}$ et $l_{i}$ la taille de la solution $(k+p_{i}^{\alpha_{i}}\mathbb{Z})$-monomiale minimale de $(E_{p_{i

Theorems & Definitions (41)

  • proof
  • proof
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • proof
  • Proposition \oldthetheorem: M3 proposition 4.1
  • proof
  • ...and 31 more