Related papers: Computing Multiplicative Relations between Roots o…

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$…

In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more…

The matching polynomial of a graph encodes rich combinatorial information through its roots. We determine the maximum multiplicity of a non-zero matching polynomial root and characterize all graphs attaining the bound. We also generalize…

Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the…

This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…

For a $t$-nomial $f(x) = \sum_{i = 1}^t c_i x^{a_i} \in \mathbb{F}_q[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2 (q-1)^{1-\varepsilon} C^\varepsilon$, where $\varepsilon = 1/(t-1)$ and $C$ is the…

Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral…

General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, ${\cal L}\psi=0$, where ${\cal L}=\sum_l p_l(z) d_z^l$, $p_l(z)$ are polynomials, $z$ is a…

In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…

We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(\alpha)\mid\alpha\in\mathbb{F}_{q}\}$ and denote the…

We investigate the spectral properties of rooted trees with the intention of improving the currently existing results that deal with this matter. The concept of an assigned rational function is recursively defined for each vertex of a…

We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an $n$-variate degree-$d$ polynomial $f$ which can be written as $$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$ where each $c_i\in…

We completely characterize the faces of the root polytope $\tilde Q_G = \text{conv}\{\mathbf 0, \mathbf e_i - \mathbf e_j\: (i,j) \in E(G)\}$ combinatorially. Our results specialize to state of the art results in a straightforward way.

We construct parametric families of (monic) reducible polynomials having two roots very close to each other.

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…

We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these…

We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new…

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Math\'ematiques, (2), vol 5 (1881), pp.393-395], some results of D.A.…