Related papers: Images of polynomials with involution on $2\times …

In this paper, we study the partial bi-free $S$-transform of a pair $(a,b)$ of random variables, and the $S$-transform of the $2\times 2$ matrix-valued random variable $\left(\begin{matrix}a&0\\0&b\end{matrix}\right)$ associated with…

This paper studies locally linear involutions on S^4. Our main theorem shows that any such involution with a 1-dimensional fixed-point set is necessarily linear, provided the fixed-point set admits an equivariant tubular neighborhood. The…

Let $M=(E,\mathcal B)$ be an $\mathbb F_q$-linear matroid; denote by ${\mathcal B}$ the family of its bases, $s(M;\alpha)=\sum_{B\in\mathcal B}\prod_{e \in B} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$. According to the Kontsevich…

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…

Let $F$ be a field of characteristic zero. We prove that if a group grading on $UT_m(F)$ admits a graded involution then this grading is a coarsening of a $\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor}$-grading on $UT_m(F)$ and the graded…

Let $\Rx$ denote the ring of polynomials in $g$ freely non-commuting variables $x=(x_1,...,x_g)$. There is a natural involution * on $\Rx$ determined by $x_j^*=x_j$ and $(pq)^*=q^* p^*$ and a free polynomial $p\in\Rx$ is symmetric if it is…

In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded…

Optimizing and certifying the positivity of polynomials are fundamental primitives across mathematics and engineering applications, from dynamical systems to operations research. However, solving these problems in practice requires large…

We will study evolution algebras $A$ which are free modules of dimension $2$ over domains. Furthermore, we will assume that these algebras are perfect, that is $A^2=A$. We start by making some general considerations about algebras over…

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this…

Multidimensional imaging, capturing image data in more than two dimensions, has been an emerging field with diverse applications. Due to the limitation of two-dimensional detectors in obtaining the high-dimensional image data, computational…

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with…

We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees…

We construct a set $M_d$ whose points parametrize families of Meixner polynomials in $d$ variables. There is a natural bispectral involution $b$ on $M_d$ which corresponds to a symmetry between the variables and the degree indices of the…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a…

We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due…

For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the…

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2…