Roots Of Polynomial Research Articles

The On power integral bases of certain pure number fields defined by $x^{120}-m $

Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Exponential concentration for the number of roots of random trigonometric polynomials

Exponential concentration for the number of roots of random trigonometric polynomials

An Optimal Choice of Characteristic Polynomial Roots for Pole Placement Control Design

The problem of finding the arrangement of closed-loop control system poles that minimizes an objective function is considered. The system optimality criterion is the value of the H∞ norm of the frequency transfer function relative to the disturbance with constraints imposed on the system pole placement and the values of the H∞ norm of the sensitivity function and the transfer function from measurement noise to control. An optimization problem is formulated as follows: the vector of variables consists of the characteristic polynomial roots of the closed loop system with the admissible values restricted to a given pole placement region; in addition to the optimality criterion, the objective function includes penalty elements for other constraints. It is proposed to use a logarithmic scale for the moduli of the characteristic polynomial roots as elements of the vector of variables. The multi-extremality problem of the objective function is solved using the multiple start procedure. A coordinate descent modification with a pair of coordinates varied simultaneously is used for search.

Asynchronous progressive iterative approximation method for least squares fitting

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014), in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective.

On the Number of Rational Roots of a Rational Polynomial

We propose an extension of Gauss’s lemma and Schönemann-Eisenstein’s irreducibility criterion to determine the impossibility of decomposing an integer polynomial into several polynomials with rational coefficients. This extension allows us to obtain some limitations on the number of rational roots and on the degrees of the factors of a rational polynomial.

Expected Energy of Zeros of Elliptic Polynomials

AbstractIn 2011, Armentano, Beltrán and Shub obtained a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.

Computing a compact local Smith–McMillan form

We define a compact local Smith–McMillan form of a rational matrix R ( λ ) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of R ( λ ) . We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of R ( λ ) around an arbitrary complex point λ 0 , allows us to compute a compact local Smith-McMillan form of that rational matrix R ( λ ) at the point λ 0 , provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point λ 0 . Numerical tests illustrate the promising performance of the resulting algorithm.

Discriminant and integral basis of number fields defined by exponential Taylor polynomials

AbstractLet $K_n=\mathbb{Q}(\alpha_n)$ be a family of algebraic number fields where $\alpha_n\in \mathbb{C}$ is a root of the nth exponential Taylor polynomial $\frac{x^n}{n!}+ \frac{x^{n-1}}{(n-1)!}+ \cdots +\frac{x^2}{2!}+\frac{x}{1!}+1$, $n\in \mathbb{N}$. In this paper, we give a formula for the exact power of any prime p dividing the discriminant of Kn in terms of the p-adic expansion of n. An explicit p-integral basis of Kn is also given for each prime p. These p-integral bases quickly lead to the construction of an integral basis of Kn.

The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation

Abstract We extend the free convolution of Brown measures of $R$-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Abstract Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m<n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

In search of Newton-type inequalities

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.

The inverse kinematics of lobster arms

The roots of the closure polynomial associated with a given mechanism determine its assembly modes. In the case of 6R closed-loop mechanisms, these polynomials are usually expressed in the half-angle tangent of one of its joints. In this paper, we derive closure polynomials of 6R robots in terms of distances, not angles. The use of a distance-based formulation provides, in general, a fundamental advantage since it leads to closure conditions without requiring neither variable eliminations nor variable substitutions. We restrict our attention, though, to robots with coplanar consecutive joint axes, i.e., robots whose consecutive axes intersect at either proper or improper points. We show that this particular arrangement of joints does not result in a reduction in the maximum number of the inverse kinematic solutions with respect to the general case. Moreover, this family of robots include broadly used offset-wrist arms. For instance, in this paper, we obtain closure polynomials for robots such as the FANUC CRX-10iA/L, the UR10e, and the KUKA LBR iiwa R800 robot in generic form (i.e., as a function of their end-effector locations).

On Variance and Average Moduli of Zeros and Critical Points of Polynomials

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial.

Planar polynomial of the graphs

In this paper, the planar polynomial of a graph is introduced and some of its properties are discussed. The planar polynomial of the graph G is real-rooted if and only if G is planar. This polynomial is not EE-invariant. Some useful results about the roots of planar polynomial of complete graphs are presented. Moreover, all the graphs whose planar polynomial is of degree five are characterized.

Analytical Formula for the Roots of the General Complex Cubic Polynomial

Analytical Formula for the Roots of the General Complex Cubic Polynomial

Inverse design of arbitrary non-reciprocal acoustic dispersion in non-local phononic crystals and metamaterials

We present inverse methods to obtain arbitrary non-reciprocal dispersion relations in periodic, multi-degree-of-freedom lattices with non-local interactions. Just as phase-delayed interactions such as local resonance govern the frequency-dependence of a lattice’s dispersion relation, spatially non-local interactions govern the wavenumber dependence of the coefficients of the lattice’s characteristic equation. Thus, the use of fully general non-local interactions can, in principle, specify every branch of the dispersion diagram at every wavenumber. However, calculating a viable set of non-local interactions for a given set of desired dispersion curves is challenging, as there are multiple possible sets of coefficients that yield the same polynomial roots. To solve this problem, we first present a general method to calculate the dispersion relation of multi-degree-of-freedom non-local lattices, showing that non-local interactions create a Fourier series expansion that governs the wavenumber dependence of each matrix element. Next, we present numerical techniques to solve the inverse problem—i.e., given the desired dispersion relation, calculate the required non-local interactions—and discuss the associated computational challenges. Finally, we discuss practical methods to realize non-local interactions leveraging piezoelectric sensors and actuators, and we highlight how non-local interactions can be introduced in unit-cell based finite-element dispersion calculations.

Numerical methods for finding the roots of polynomials with real and complex coefficients

Numerical methods for finding the roots of polynomials with real and complex coefficients

Some preconditioning techniques for a class of double saddle point problems

SummaryIn this paper, we describe and analyze the spectral properties of several exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the (F)GMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.

Integral Representations over Finite Limits for Quantum Amplitudes

We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,∞]. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|=x12−2x1x2cosθ+x22 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [0,1] are numerically stable, as was the prior version, having integrals running over the interval [0,∞], and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [0,1] than over [0,∞]. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (ax2+bx+c)/x, for which a single tabled integral exists for the integrals from running over the interval [0,∞] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [0,1]. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [0,1].