Zeros Of Polynomials Research Articles

Erratum to “Relationships Between the Zeros, Weights, and Weight Functions of Orthogonal Polynomials: Derivative Rule Conjecture (the DRC) Applied to Stieltjes and Spectral Imaging”

Presents corrections to the above named paper.

Some further q-shift difference results on Hayman conjecture

In this paper, we investigate the zero distributions of q-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential polynomials to q-shift difference. We also investigate the uniqueness problem of q-shift difference-differential polynomials sharing a non-zero polynomial with finite weight.

On the growth and zeros of polynomials attached to arithmetic functions

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0<h(n) le h(n+1). We put P_0^{g,h}(x)=1 and Png,h(x):=xh(n)∑k=1ng(k)Pn-kg,h(x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} P_n^{g,h}(x) := \\frac{x}{h(n)} \\sum _{k=1}^{n} g(k) \\, P_{n-k}^{g,h}(x). \\end{aligned}$$\\end{document}As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind eta -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: Bounds, asymptotics and zeros

This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with respect to these parameters. Criteria for knowing when the zeros of the (Sobolev) orthogonal polynomials and also the zeros of their derivatives stay within the unit disk have also been addressed. By numerical experiments some further information on the parameters is also found so that the zeros remain within the unit disk.

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

Cryptanalysis of Modular Exponentiation Outsourcing Protocols

Abstract Public-key cryptographic primitives are time consuming for resource-constrained devices. A classical problem is to securely offload group exponentiations from a (comparatively) weak device—the client—to an untrusted more powerful device—the server. A delegation protocol must usually meet two security objectives: privacy—the exponent or the base should not be revealed to a passive adversary—and verifiability—a malicious server should not be able to make the client accept an invalid value as the result of the delegated computation. Most proposed protocols relies on a secret splitting of the exponent and the base, and a considerable amount of literature has been devoted to their analysis. Recently, Su et al. (Su, Q., Zhang, R. and Xue, R. (2020) Secure outsourcing algorithms for composite modular exponentiation based on single untrusted cloud. Comput. J., 63, 1271.) and Rangasamy and Kuppusamy (Rangasamy, J. and Kuppusamy, L. (2018) Revisiting Single-Server Algorithms for Outsourcing Modular Exponentiation. In Chakraborty, D. and Iwata, T. (eds), Progress in Cryptology - INDOCRYPT 2018: 19th International Conference in Cryptology in India, New Delhi, India, December 912, Vol. 11356, Lecture Notes in Computer Science. Springer, Heidelberg, Germany, pp. 320. proposed outsourcing protocols for modular exponentiations. They claim that their protocols achieve security (privacy and verifiability). We show that these claims are flawed and that their schemes are broken beyond repair. They remain insecure even if one increases significantly the proposed parameters (and consequently the protocols computational and communication complexities). Our attacks rely on standard lattice-based cryptanalytic techniques, namely the Coppersmith methods to find small integer zeroes of modular multivariate polynomials and simultaneous Diophantine approximation methods for the so-called approximate greatest common divisor problem.

COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW

The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial.

Multiplicity of zeros of polynomials

Sharp bounds are given for the highest multiplicity of zeros of polynomials in terms of their norm on Jordan curves and arcs. The results extend a theorem of Erdős and Turán and solve a problem of them from 1940.

Relationships Between the Zeros, Weights, and Weight Functions of Orthogonal Polynomials: Derivative Rule Conjecture (the DRC) Applied to Stieltjes and Spectral Imaging

Zeros, x[ k, N ], k = 1 ⋯N,x[k,N], k=1...N, of orthogonal polynomials are useful for numerical applications, e.g., Gaussian quadrature were combined with quadrature weights, w[ k,N ],w[k,N], integrals with weight function ρ(x)ρ(x) are performed to high precision. For broad classes of OPs, the ratios w[ k,N ]/ρ(x[ k,N ]) w[k,N]/ρ(x[k,N])are independent of the OPs at hand if x[ k,N ] x[k,N]lie within a smooth part of ρ(x)ρ(x). This universality as N → ∞ N→∞suggests that these ratios may be evaluated as x'[ k,N ] = d(x[ k,N ])/dkx'[k,N]=d(x[k,N])/dk. This Derivative Rule Conjecture, or DRC, then gives, even for small NN, ρ(x[ K,N ])=w[ k,N ]/x'[ k,N ],ρ(x[K,N])=w[k,N]/x'[k,N], and Stieltjes imaging, or inversion, with exponential convergence. Similar methods apply to the numerics of Schrödinger resolvents. The example chosen to illustrate this latter indicates that the assumption of universality, while suggesting and validating the DRC, is not necessary.

Zeros of slice functions and polynomials over dual quaternions

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over an alternative $*$-algebra $A$ was defined by Ghiloni and Perotti in 2011, extending the class of slice regular functions introduced by Gentili and Struppa in 2006. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb{DH}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general alternative $*$-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb{DH}$, introduced by Heged\"us, Schicho, and Schr\"ocker in 2013 for their relevance in mechanism science.

Asymptotics of Polynomial Interpolation and the Bernstein Constants

It is well known that the interpolation error for left| xright| ^{alpha },alpha >0 in L_{infty }left[ -1,1right] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when alpha tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to left| xright| ^{alpha },alpha >0 in the L_{infty } norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

Selected Extensions on Eneström-kakeya Theorem

The theorem of Eneström-Kakeya is important within the hypothesis of dissemination of zeros ofpolynomials. In the literature, it can be found so many extensions on Eneström-Kakeya theorem by giving various relations between the coefficients of polynomial like increasing, decreasing, irregular order etc.This paper mainly deals with some extensions on the Abdul Aziz and B AZargar theorembygiving some relaxations to the hypothesis that the coefficients are real, positive and alternative coefficients must be in increasing order.

Markov’s Theorem for Weight Functions on the Unit Circle

The aim of this paper is to prove that Markov’s theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle.

Zeros of complex random polynomials spanned by Bergman polynomials

We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n/3$. Generalizing our basis functions $\{p_k(z)\}$ to be regular in the sense of Ullman--Stahl--Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n/3$.

AN ITERATIVE FORMULA FOR SIMULTANEOUS LOCATION OF THE ZEROS OF A POLYNOMIAL

Needless to say that the search for efficient algorithms for determining zeros of polynomials has been continually raised in many applications. In this paper we give a cubic iteration method for determining simultaneously all the zeros of a polynomial – assumed distinct – starting with ‘reasonably close’ initial approximations – also assumed distinct. The polynomial – in question – is expressed in its Taylor series expansion in terms of the initial approximations and their correction terms. A formula with cubic rate of convergence – based on retaining terms up to 2ndorder of the expansion in the correction terms – is derived.

A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form Gn=∑i=0nξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog⁡(1+|ξ0|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when pi(z)=zi. We also show that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0|>en)=o(n−1).Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in Lp(τ), where p∈(0,∞]. In particular, sequences of Lp-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

On signless Laplacian spectrum of the zero divisor graphs of the ring $\mathbb{Z}_{n}$

For a finite commutative ring $ R $ with identity $ 1\neq 0 $, the zero divisor graph $ \Gamma(R) $ is a simple connected graph having vertex set as the set of nonzero zero divisors of $ R $, where two vertices $ x $ and $ y $ are adjacent if and only if $ xy=0$. We find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for various values of $ n$. Also, we find signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^z , z\geq 2 $, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise $ n $ for which zero divisor graph $ \Gamma(\mathbb{Z}_{n}) $ are signless Laplacian integral.

The estimation of the zeros of some counting polynomials

In this work we study the zeros of the Eulerian and Bell polynomials and their generalizations. More concretely, lower estimates for the leftmost zeros of these polynomials will be given, complementing earlier results where upper estimations were presented.

Nonreal Zeros of Polynomials in a Polynomial Sequence Satisfying a Three-Term Recurrence Relation

This paper discusses the location of zeros of polynomials in a polynomial sequence $$\{P_{n}(z)\}_{n=1}^{\infty}$$ generated by a three-term recurrence relation of the form $$P_{n}(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0$$ with $$k>2$$ and the standard initial conditions $$P_{0}(z)=1$$ , $$P_{-1}(z)=P_{-k+1}(z)=0,$$ where $$A(z)$$ and $$B(z)$$ are arbitrary coprime real polynomials. We show that there always exist polynomials in $$\{P_{n}(z)\}_{n=1}^{\infty}$$ with nonreal zeros.

Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.