Related papers: Zeros and factorizations of quaternion polynomials…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of…

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition…

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial…

In this paper we give new methods to construct zero divisors in A_n =R^(2^n) the Cayley_Dickson algebras over the real numbers, for n larger than 4, and we also relate the set of zero divisors in A_{n+1} with the Stiefel Manifold V_{2^n…

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.

The paper presents analogues of some Vieta formulas for quaternionic polynomials of the form $R_n(w)=\sum\limits_{m=0}^{n}A_mx^m$. A criterion for the sphericity of the root (zero) of a polynomial $R_n(w)$ is established.

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this…

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…

We construct many irreducible polynomials within semigroups generated by sets of the form $S=\{x^2+c_1,\dots,x^2+c_s\}$ under composition.

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…

The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…

As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes…

We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear…

Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited…

This paper discusses the left and right ranks of quaternion matrices with Hankel structure. While they are in general different for arbitrary quaternion matrices, we show that the left and right ranks of quaternion Hankel matrices are…