Related papers: Supplement to the article "irreducible polynomials…

We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices,…

We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula…

In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random $\pm 1$ symmetric matrix is singular.

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial…

Random correlation matrices are studied for both theoretical interestingness and importance for applications. The author of [6] is interested in their interpretation as covariance matrices of purely random signals, the authors of [16]…

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length…

We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

We prove that every real nonnegative ternary quartic whose complex zero set is smooth can be represented as the determinant of a symmetric matrix with quadratic entries which is everywhere positive semidefinite. We show that the…

We investigate the limiting behaviour of the path of random bridges treated as random sets in $\mathbb{R}^{d}$ with the Euclidean metric and the dimension $d$ increasing to infinity. The main result states that, in the square integrable…

We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…

Bordered and framed Toeplitz/Hankel determinants have the same structure as Toeplitz/Hankel determinants except in small number of matrix rows and/or columns. We review these structured determinants and their connections to orthogonal…

One of the most significant challenges in Computing Determinant of Rectangular Matrices is high time complexity of its algorithm. Among all definitions of determinant of rectangular matrices, used definition has special features which make…

In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank…

In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded…

We derive upper and lower bounds on the determinant of an exponential matrix. They can be transformed into corresponding bounds for the determinant of a univariate Gaussian matrix.