Related papers: Generic skew-symmetric matrix polynomials with fix…

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties…

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix…

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a skew-symmetric bilinear form with values in a line bundle L and that \Lambda^2 V^* \otimes L is ample. Suppose that the maximum…

Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to…

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

We define a regular polynomial skew product $(p(z),q(z,w))$ of $\mathbb{C}^2$ of degree $d\geq 2$ to be special if it is triangularly conjugate to a map of the form $(p(z),q(w))$, where $p$ and $q$ are power maps or $\pm$Chebyshev maps, or…

This work is a thorough investigation of skew-orthogonal polynomials with respect to a quartic Freud weight. We provide an explicit method to evaluate skew-orthogonal polynomials of any degree as linear combinations of orthogonal…

We show that every principal submatrix of a square matrix $A$ with only entries in $\{0,1,-1\}$ has even rank if and only if $A$ is skew-symmetric up to multiplication of rows/columns by $-1$.

For a fixed integer $e \geqslant 3$ and $n$ large enough, we show that the number of congruence classes modulo $2^e$ of characteristic polynomials of $n \times n$ symmetric $\{\pm 1\}$-matrices with constant diagonal is equal to…

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the…

In this paper we introduce and investigate a one-parameter family of polynomials. They are semisymmetric, i.e. symmetric in the variables with odd and even index separately. In fact, the family forms a basis of the space of semisymmetric…

A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

Particular class of skew orthogonal polynomials are introduced and investigated, which possess Laurent symmetry. They are also shown to appear as eigenfunctions of symplectic generalized eigenvalue problems. The modification of these…

Here we study the typical rank for real bivariate homogeneous polynomials of degree $d\ge 6$ (the case $d\le 5$ being settled by P. Comon and G. Ottaviani). We prove that $d-1$ is a typical rank and that if $d$ is odd, then $(d+3)/2$ is a…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\operatorname{Rep}(D^\omega(\mathbb Z_7\rtimes…

Exact integral expressions of the skew orthogonal polynomials involved in Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are obtained: the (even rank) skew orthogonal polynomials are average characteristic polynomials…