Related papers: Polynomials that preserve nonnegative monomial mat…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…
We establish an analogue of the fundamental theorem of algebra for polynomial matrix equations, in which the matrices-coefficients and unknown matrix are assumed to be circulant matrices.
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting…
In~\cite{algorithmic} was given an algorithm that computes arithmetical structures on matrices. We use some of the ideas contained there to get an algorithm that computes arithmetical structures over dominated polynomials. A dominated…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate…
A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…
Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a…
Following the classical approach of P\'olya-Schur theory we initiate in this paper the study of linear operators acting on $\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gr\"obner bases, factorization or sub-resultant computations.
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting…
In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several…
We formulate and prove a necessary condition for a sequence of analytic trigonometric polynomials with real non-negative coefficients to be flat a.e.
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory.…