Related papers: Solutions of the matrix equation $p(X)=A$, with po…

Let $K$ be an algebrically closed field and let $n\geq 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$, $P\neq 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of $\mathbb N^n$ such that $P=\sum_{i\in I(P)}a_iX^i$ with $a_i\in…

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$-modular and $11$-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a…

We describe all solutions of the matrix Hamburger moment problem in a general case (no conditions besides solvability are assumed). We use the fundamental results of A.V. Shtraus on the generalized resolvents of symmetric operators. All…

Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], we show how to compute the Hermite form H of A, and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in terms of…

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…

We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the…

Let $\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\ell$ with field of constants $\kappa$. Assume that there exists a prime $P_\infty$ of $F$ which has degree $1$, and let…

Solving a quadratic equation $P(x)=ax^2+bx+c=0$ with real coefficients is known to middle school students. Solving the equation over the quaternions is not straightforward. Huang and So \cite{Huang} give a complete set of formulas, breaking…

Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\ldots,p_n(x)$, $x\in\mathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline…

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The…

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…

Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{p_{i}}A_{i}=Q with p_{i}>0. Sufficient and necessary conditions for the existence of positive definite solutions to the equation with p_{i}>0 are derived. Two perturbation…

We solve direct and inverse problems for two-dimensional (quasi) canonical systems related to exponential polynomials of a specific but sufficiently general type. The approach to the inverse problem in this paper provides an interpretation…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either…

In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

In this paper we obtain a Nevanlinna-type formula for the matrix Hamburger moment problem in a general case. We only assume that the problem is solvable and has more that one solution. We express the matrix coefficients of the corresponding…

Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$…

We investigate special cases of the quadratic assignment problem (QAP) where one of the two underlying matrices carries a simple block structure. For the special case where the second underlying matrix is a monotone anti-Monge matrix, we…