Related papers: The root distribution of polynomials with a three-…

In this paper we present a complete method for finding the roots of all polynomials of the form $\phi(z)=c_n z^n+c_{n-1} z^{n-1}+\dots+c_1 z+c_0$ over a given octonion division algebra. When $\phi(z)$ is monic we also consider the companion…

We propose and study the properties of a set of polynomials $M_{n\alpha, H\ }^{s}(z)$, $C_{n\alpha, H}^{s}(z)$ $W_{n\alpha, H}^{s}(z)$ with $n,s\in N$ $;\alpha =\pm 1;$and where $H$ stands for Hermite ; the ''root '' polynomial >.These…

We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible…

We study the repartition of the roots of a random p-adic polynomial in an algebraic closure of Qp.We prove that the mean number of roots generating a fixed finite extension K of Qp depends mostly on the discriminant of K, an extension…

The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…

In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…

Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the…

We introduce certain polynomials, so-called H.Weyl and H.Minkowski polynomials, which have a geometric origin. The location of roots of these polynomials is studied.

We give a new proof of the identity $\zeta(\{2,1\}^l)=\zeta(\{3\}^l)$ of the multiple zeta values, where $l=1,2,\dots$, using generating functions of the underlying generalized polylogarithms. In the course of study we arrive at…

We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We…

We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the…

We consider the following question: Which real sequences (a(n)) that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both (a(n)) and (-a(n)), if the…

We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…

In this article we apply a formula for the $n$-th power of a $3\times 3$ matrix (found previously by the authors) to investigate a procedure of Khovanskii's for finding the cube root of a positive integer. We show, for each positive integer…

The goal of this paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given given operator T=\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set r=max_i (deg…

The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this…

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…