Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finiteness of the number of irreducible $λ$-quiddities over a finite commutative and unitary ring

Flavien Mabilat

TL;DR

The paper proves that over any finite commutative unital ring $A$, there are only finitely many irreducible λ-quiddities, and it derives explicit upper bounds on their maximal size $\ell_A$ in terms of the ring’s characteristic and the size of $SL_2(A)$. The core technique combines a careful definition of λ-quiddities, an irreducibility criterion, and a pigeonhole-based argument on matrix products $M_n(a_1,\dots,a_n)$ and cosets in $SL_2(A)$ to force reducibility beyond a computable threshold. It also provides concrete bounds and refinements for specific finite rings (notably $\mathbb{Z}/N\mathbb{Z}$ and $\mathbb{F}_q$, including a sharp bound for $\mathbb{F}_9$), along with a discussion of how these results relate to Coxeter friezes and prior conjectures. The paper concludes with open problems on finiteness when restricting to finite submagma subsets of infinite rings, highlighting the limits of the current approach and outlining directions for future work.

Abstract

A $λ$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion of irreducibility. The main objective of this text is to demonstrate that there is a finite number of irreducible $λ$-quiddities over a finite unitary commutative ring and to obtain in this case an upper bound for their maximal size.

Finiteness of the number of irreducible $λ$-quiddities over a finite commutative and unitary ring

TL;DR

The paper proves that over any finite commutative unital ring , there are only finitely many irreducible λ-quiddities, and it derives explicit upper bounds on their maximal size in terms of the ring’s characteristic and the size of . The core technique combines a careful definition of λ-quiddities, an irreducibility criterion, and a pigeonhole-based argument on matrix products and cosets in to force reducibility beyond a computable threshold. It also provides concrete bounds and refinements for specific finite rings (notably and , including a sharp bound for ), along with a discussion of how these results relate to Coxeter friezes and prior conjectures. The paper concludes with open problems on finiteness when restricting to finite submagma subsets of infinite rings, highlighting the limits of the current approach and outlining directions for future work.

Abstract

A -quiddity of size is an -tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion of irreducibility. The main objective of this text is to demonstrate that there is a finite number of irreducible -quiddities over a finite unitary commutative ring and to obtain in this case an upper bound for their maximal size.
Paper Structure (10 sections, 16 theorems, 28 equations)

This paper contains 10 sections, 16 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Quiddities
  4. Some classification results
  5. Preliminary results
  6. Lower and upper bounds for $\ell_{A}$
  7. Proof ot Theorem \ref{['prin1']}
  8. Applications
  9. A refinement of Theorem \ref{['prin1']} in the case $A=\mathbb{F}_{9}$
  10. A still open problem

Key Result

Theorem 1.1

Let $(A,+,\times)$ be a finite commutative and unitary ring (different from $\{0_{A}\}$) and $R \subset A$ a submagma of $(A,+)$, that is to say a subpart of $A$ closed under $+$. i) There is a finite number of irreducible $\lambda$-quiddities over $R$. ii) Let $\ell_{A}$ be the maximum size of an i

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: C, Definition 2.2
  • Definition 2.2: C, Lemma 2.7
  • Definition 2.3: C, Definition 2.5
  • Definition 2.4: C, Definition 2.9
  • Example
  • Remark
  • Theorem 2.5: C, Theorems 3.1 and 3.2
  • Theorem 2.6
  • ...and 21 more