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Clément de Seguins Pazzis

Publications and source records attributed to Clément de Seguins Pazzis.

At least 19 recordsLinked to original sources
arXiv2026

Affine subspaces of units in simple algebras

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the spaces that have the greatest possible dimension. This is equivalent to the problem of determining the greatest possible dimension for an $F$-linear subspace $S$ of $A$ in which $x-1_A$ is a unit for all $x \in S$, and we elucidate the structure of these linear subspaces up to conjugation when their dimension reaches the greatest possible one. These classifications involve the associative composition algebras over $F$. Over fields of characteristic other than $2$, the first problem is essentially reduced to the classification of nonisotropic quadratic forms over $F$ and of nonisotropic Hermitian forms over quadratic and quaternionic extensions of $F$. These results are intimately connected with the problem of intransitive operator spaces between finite-dimensional vector spaces over division rings, which we study in depth: in particular, we generalize a dual version of Atkinson's theorem on primitive spaces of bounded rank matrices.

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arXiv2026

Spaces of triangularizable matrices (III): Perfect non-quadratically closed fields with characteristic 2

Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than $n-1$ elements. In this article, we prove that the equality $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension $t_n(\mathbb{F})$.

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arXiv2026

Range-compatible homomorphisms on Hermitian matrices

Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathcal{M}_n(\mathbb{D})$. Here, we give a complete solution to the following problem: Determine all group homomorphisms from $\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ (respectively, from $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ unless $(-)^\star$ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when $(-)^\star$ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.

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arXiv2026

The Free Hamilton Algebra

Over an arbitrary field $\mathbb{F}$, let $p$ and $q$ be monic polynomials with degree $2$ in $\mathbb{F}[t]$. The free Hamilton algebra of the pair $(p,q)$ is the free noncommutative algebra in two generators $a$ and $b$ subject only to the relations $p(a)=0=q(b)$. Free Hamilton algebras are models of free products of two $2$-dimensional algebras over $\mathbb{F}$. They can be viewed as the most elementary nontrivial noncommutative algebras over fields. It has been recently observed that the free Hamilton algebra has surprising connections with quaternion algebras. Here, we exploit these connections to investigate its zero divisors, group of units, maximal ideals, finite-dimensional subalgebras, and its automorphism group.

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arXiv2026

Large affine spaces of symplectic forms

Let F be a field and V be a 2n-dimensional vector space over F. In a previous article, we have proved that if F has more than 2n-2 elements then the greatest possible dimension for an affine space of symplectic forms on V is n(n-1). Here, under the same cardinality assumption we study the spaces that have the critical dimension n(n-1). In particular, if the characteristic of F is not 2 the classification of these spaces up to congruence is reduced to: (1) the classification of nonisotropic quadratic forms over F, up to equivalence and multiplication with a nonzero scalar; (2) the classification of nonisotropic Hermitian forms over all quadratic extensions of F, up to equivalence and multiplication by $-1$. In particular, for quadratically closed fields it is shown that there is exactly one solution up to congruence.

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arXiv2026

Sums of two nilpotent quaternionic matrices

Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de la Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices. Here, we give a simple characterization of the square matrices with entries in $\mathcal{Q}$ that are the sum of two nilpotent ones. When $n \geq 3$, the special cases involve the scalar matrices and their perturbations by rank $1$ matrices, as well as the very special case of $3$-by-$3$ unispectral diagonalisable matrices.

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arXiv2026

Spaces of matrices with few eigenvalues (II)

Let $F$ be a field, and $\mathcal{M}$ be a linear subspace of $n$-by-$n$ matrices with entries in $F$ that have at most two eigenvalues in $F$ (respectively, at most one non-zero eigenvalue in $F$). In a previous article, we have determined the greatest possible dimension for $\mathcal{M}$ when the characteristic of $F$ is not $2$. In this article and its sequel, we solve this problem for all fields with characteristic $2$.

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arXiv2025

Spaces of triangularizable matrices (II): Finite fields with odd characteristic

Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed. The structure of the spaces of maximal dimension was also elucidated provided $\mathbb{F}$ is infinite and not quadratically closed. In this sequel, we extend this result to finite fields with odd characteristic. More specifically, we prove that if $\mathbb{F}$ is finite with odd characteristic, then the space of all upper-triangular $n$-by-$n$ matrices is, up to conjugation, the sole vector space of $n$-by-$n$ matrices that has dimension $\frac{n(n+1)}{2}$ and consists only of triangularizable matrices.

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arXiv2025

Spaces of triangularizable matrices

Let F be a field. We investigate the greatest possible dimension t_n(F) for a vector space of n-by-n matrices with entries in F and in which every element is triangularizable over the ground field F. It is obvious that t_n(F) is greater than or equal to n(n+1)/2, and we prove that equality holds if and only if F is not quadratically closed or n=1, excluding finite fields with characteristic 2. If F is infinite and not quadratically closed, we give an explicit description of the solutions with the critical dimension t_n(F), reducing the problem to the one of deciding for which integers k between 2 and n all k-by-k symmetric matrices over F are triangularizable.

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arXiv2025

A Convex-Analytical Proof of the Fundamental Theorem of Algebra

A weak version of Birkhoff's generalization of the Perron-Frobenius theorem states that every endomorphism of a finite-dimensional real vector that leaves invariant a non-degenerate closed convex cone has an eigenvector in that cone. Here, we show that this theorem, whose proof relies only upon basic convex analysis, yields very short proofs of both the spectral theorem for selfadjoint operators of Euclidean spaces and the Fundamental Theorem of Algebra.

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arXiv2024

Products of involutions in symplectic groups over general fields (II)

Let $s$ be an $n$-dimensional symplectic form over a field $\mathbb{F}$ of characteristic other than $2$, with $n>2$. In a previous article, we have proved that if $\mathbb{F}$ is infinite then every element of the symplectic group $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$ and of five involutions otherwise. Here, we adapt this result to all finite fields with characteristic not $2$, with the sole exception of the very special situation where $n=4$ and $|\mathbb{F}|=3$, a special case which we study extensively.

math.GR↗
arXiv2024

On affine spaces of rectangular matrices with constant rank

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices with entries in $\mathbb{F}$ in which all the elements have rank $r$. In this note, we generalize her result to an arbitrary field with more than $r+1$ elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of $\mathrm{M}_s(\mathbb{F})$ with dimension $\dbinom{s}{2}$. The latter is known to be connected to the classification of nonisotropic quadratic forms over $\mathbb{F}$ up to congruence.

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arXiv2023

Products of involutions in symplectic groups over general fields (I)

Let $s$ be an $n$-dimensional symplectic form over an arbitrary field with characteristic not $2$, with $n>2$. The simplicity of the group $\mathrm{Sp}(s)/\{\pm \mathrm{id}\}$ and the existence of a non-trivial involution in $\mathrm{Sp}(s)$ yield that every element of $\mathrm{Sp}(s)$ is a product of involutions. Extending and improving recent results of Awa, de La Cruz, Ellers and Villa with the help of a completely new method, we prove that if the underlying field is infinite, every element of $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$, and of five involutions otherwise. The first part of this result is shown to be optimal for all multiples of $4$ and all fields, and is shown to fail for the fields with three elements and for $n=4$. Whether the second part of the result is optimal remains an open question. Finite fields will be tackled in a subsequent article.

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arXiv2023

On affine spaces of alternating matrices with constant rank

Let $\mathbb{F}$ be a field, and $n \geq r>0$ be integers, with $r$ even. Denote by $\mathrm{A}_n(\mathbb{F})$ the space of all $n$-by-$n$ alternating matrices with entries in $\mathbb{F}$. We consider the problem of determining the greatest possible dimension for an affine subspace of $\mathrm{A}_n(\mathbb{F})$ in which every matrix has rank equal to $r$ (or rank at least $r$). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that $n \geq r+3$ and $|\mathbb{F}|\geq \min\bigl(r-1,\frac{r}{2}+2\bigr)$, we also determine the affine subspaces of rank $r$ matrices in $\mathrm{A}_n(\mathbb{F})$ that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case $n\leq r+2$.

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arXiv2023

Products of unipotent elements of index $2$ in orthogonal and symplectic groups

An automorphism $u$ of a vector space is called unipotent of index $2$ whenever $(u-\mathrm{id})^2=0$. Let $b$ be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space $V$ over a field $\mathbb{F}$ of characteristic different from $2$. Here, we characterize the elements of the isometry group of $b$ that are the product of two unipotent isometries of index $2$. In particular, if $b$ is symplectic we prove that an element of the symplectic group of $b$ is the product of two unipotent isometries of index $2$ if and only if it has no Jordan cell of odd size for the eigenvalue $-1$. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index $2$ (and no less in general). For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article.

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arXiv2023

The quadratic sum problem for symplectic pairs

Let $(b,u)$ be a pair consisting of a symplectic form $b$ on a finite-dimensional vector space $V$ over a field $\mathbb{F}$, and of a $b$-alternating endomorphism $u$ of $V$ (i.e. $b(x,u(x))=0$ for all $x$ in $V$). Let $p$ and $q$ be arbitrary polynomials of degree $2$ with coefficients in $\mathbb{F}$. We characterize, in terms of the invariant factors of $u$, the condition that $u$ splits into $u_1+u_2$ for some pair $(u_1,u_2)$ of $b$-alternating endomorphisms such that $p(u_1)=q(u_2)=0$.

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arXiv2023

A note on bireflectional elements of an algebra

A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field $\mathbb{F}$ is algebraically closed with characteristic other than $2$. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over $\mathbb{F}$.

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