Related papers: A Simple Trigonometric Classification of Quartic R…

We raise a question on the existence of continuous roots of families of monic polynomials (by the root of a family of polynomials we mean a function of the coefficients of polynomials of a given family that maps each tuple of coefficients…

Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the…

A formula expressing a point of order 8 on an elliptic curve, in terms of the roots of the associated cubic polynomial, is given. Doubling such a point yields a point of order 4 distinct from the well-known points of order 4 given in…

Traditionally, the P3P problem is solved by firstly transforming its 3 quadratic equations into a quartic one, then by locating the roots of the resulting quartic equation and verifying whether a root does really correspond to a true…

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence…

Viewing a bivariate polynomial f in R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic…

This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…

In this study linear and nonlinear higher order singularly perturbed problems are examined by a numerical approach, the differential quadrature method. Here, the main idea is using Chebyshev polynomials to acquire the weighting coefficient…

We introduce a new algorithm denoted DSC2 to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f.…

This paper presents a Kharitonov-type algorithm for complex interval Hurwitz polynomials that determines whether all roots of a given interval polynomial lie within a prescribed angular sector of the complex plane. The method requires…

We start by constructing a new root system for rational triple singularities and determine the number of roots for each rational triple singularity. Then we show that, for each root, we obtain a linear free divisor. So we obtain a new…

We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…

This paper presents a primitive $n$th root of unity in $\mathbb C$. The approach is very elementary and avoids the following: the complex exponential function, trigonometry, and group theory. It also avoids differentiation, integration, and…

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the $[\Theta(\log n), \Theta(n)]$ region, in two settings. We present one…

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…

In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…

We provide a method for solving the roots of the general polynomial equation a[n]*x^n+a[n-1]*x^(n-1)+..+a1*x+a0=0. To do so, we express x as a powerseries of s, and calculate the first n-2 coefficients. We turn the polynomial equation into…

Let $K$ be a field, complete with respect to a discrete non-archimedian valuation and let $k$ be the residue field. Consider a system $F$ of $n$ polynomial equations in $K\vars$. Our first result is a reformulation of the classical Hensel's…

A solution to the problem of topological classification of real cubic fourfolds is presented. It is shown that the real locus of a real non-singular cubic fourfold is obtained from a projective 4-space either by adding several trivial one-…