arXiv subjects
Publications and source records attributed to Flavien Mabilat.
In this note, we give a complete characterization of square matrices of size $n$ over a commutative field which can be expressed as a product of a nilpotent matrix and an involutive matrix and matrices which can be expressed as a product of a nilpotent matrix and an idempotent matrix, in terms of invariant factors.
In this note, we prove that a square matrix of size $n$ over a field containing at least $2n$ elements can be expressed as the product of two matrices similar to companion matrices, that is to say matrices with the same minimal and characteristic polynomial, if and only if the rank of $A$ is greater than $n-2$, using only classical facts. We will also give some partial results valid over smaller fields.
We will show in this text that, for all non-negative integers $n$ and $l$, the following equality is verified: \[\sum_{i=0}^{l} {n-i \choose i}{l+i \choose 2i+1}=\sum_{i=0}^{l} {n-i \choose i-1}{l+i \choose 2i}.\] We will first address the case where $l \leq n$, for which both sums contain only classical binomial coefficients. Then, we will consider the general framework using generalized binomial coefficients.
The study of the combinatorics of the modular group and of Coxeter's friezes naturally leads to the investigation of a matrix equation, sometimes referred to as the Conway-Coxeter equation. The solutions of size $n$ of this equation, called $\lambda$-quiddities, are $n$-tuples of elements of a given ring $B$. A detailled understanding of these objects relies on the notion of irreducible solutions, from which all $\lambda$-quiddities can be reconstructed. One of the central questions that naturally arises in this context is whether the irreducible $\lambda$-quiddities over $B$ have bounded size, and, if so, how to determine such a bound. In this paper, we aim to list results that address this question in the case of polynomial rings $A[X]$ and $\mathbb{K}[X]$, where $A$ is a finite commutative unitary ring and $\mathbb{K}$ is a commutative field. Moreover, the stated results will also make it possible to treat easily many situations in which $A$ is infinite. Finally, we shall give a complete answer to the initial question for all rings of formal power series.
$\lambda$-quiddities of size $n$ are $n$-tuples of elements from a fixed set that are solutions to a matrix equation which is fundamental in the study of the combinatorics of the modular group and Coxeter's friezes. To gain further insight into these objects, we use a notion of irreducibility, which allows restricting the study to a limited number of elements that must be determined for each set. Our goal here is to define several families of $\lambda$-quiddities over finite fields and to study their irreducibility properties, with the specific aim of establishing lower bounds on the maximal size of irreducible elements over $\mathbb{F}_{q}$.
The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_{1} & -1 \\[4pt] 1 & 0 \end{pmatrix}=\pm M$ when $M$ is equal to the generators of the modular group $S=\begin{pmatrix} 0 & -1 \\[4pt] 1 & 0 \end{pmatrix}$ and $T=\begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}$. To count these elements, we will study the $\lambda$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed.
The aim of this article is to obtain a formula giving, for a positive integer $n$, the number of roots of the $n^{th}$ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the continuant over the local rings $\mathbb{F}_{q}$, $\mathbb{Z}/p^{m}\mathbb{Z}$ and $\frac{\mathbb{F}_{q}[X]}{\langle X^{m} \rangle}$. Besides, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for $\lambda$-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter's friezes) over the rings $\mathbb{Z}/p^{m}\mathbb{Z}$.
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.
A $\lambda$-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 $\lambda$-quiddities over a finite unitary commutative ring and to obtain in this case an upper bound for their maximal size.
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. Their number and their properties are closely linked to the structure and the cardinality of the chosen set. The main objective of this text is to obtain an explicit formula giving the number of $\lambda$-quiddities of odd size, and a lower and upper bound for the number of $\lambda$-quiddities of even size, over the rings $\mathbb{Z}/2^{m}\mathbb{Z}$ ($m \geq 2$). We also give explicit formulas concerning the number of $\lambda$-quiddities of size $n$ over $\mathbb{Z}/8\mathbb{Z}$.
The aim of this article is to continue the study of the notion of $λ$-quiddity over a ring, which appeared during the study of Coxeter's friezes. For this, we will focus here on situations where the ring used can be seen as a direct product of unitary commutative rings. In particular, we will consider the cases of direct products of rings containing at least two rings of characteristic 0 and we will also consider some products of the type $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$.
In this article, we study the combinatorics of congruence subgroups of the modular group. More precisely, we consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes), modulo an integer $N$, whose components are identical and minimal for this property. The objective here is to characterize the solutions of this type verifying a property of irreducibility modulo a prime power.
In this note, we consider matrices similar to $X$-form matrices, which are the matrices for which only the diagonal and the anti-diagonal elements can be different from zero. First, we give a characterization of these matrices using the minimal polynomial. Then, we prove that the set of matrices similar to $X$-form matrices over $\mathbb{R}$ and $\mathbb{C}$ are dense and we give a characterization of the interior of this set.
In this article, we study the classification of some natural numbers 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 in the study of Coxeter friezes), modulo an integer $N$, whose components are identical and minimal for this property. Our aim here is to study the integers $N$ for which the minimal monomial solutions satisfying some fixed conditions have an irreducibility property. In particular, we will classify the monomially irreducible integers which are the integers for which all the nonzero minimal monomial solutions are irreducible.
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.
During his work devoted to Coxeter's friezes, M. Cuntz initiated the study of the notion of $\lambda$-quiddity and raised the problem of the study of this over some subsets of $\mathbb C$. More specifically, $\lambda$-quiddities are the solutions to a matrix equation, related to various mathematical objects, which we seek to solve over different sets. The aim of this text is to provide some new insights into the problem raised by M. Cuntz in the case of some cyclic subgroups of ($\mathbb{C},+$) generated by an algebraic number. In particular, we will study the cases of subgroups generated by $a+b\sqrt{k}$.
In this paper, we study combinatorics of congruence subgroups of the modular group. More precisely, we consider the matrix equation that naturally arises in the theory of Coxeter friezes and investigate its irreducible solutions. We give new properties for minimal monomial solutions. Furthermore, we introduce the notion of minimal dynomial solutions and study their irreducibility.
During the study of Coxeter's friezes, M. Cuntz defined the concept of $\lambda$-quiddities and gave the problem of studying them over some subsets of $\mathbb{C}$. The objective of this text is to carry out this study in the case of some cyclic subgroups of ($\mathbb{C},+$). In particular we will study the case of the cyclic subgroups generated by $\sqrt{k}$ and $i\sqrt{k}$, with $k \in \mathbb{N}$.