Polynomials with All Zeros Real and in a Prescribed Interval

We provide a characterization of the real-valued univariate polynomials that have only real zeros, all in a prescribed interval [a,b]. The conditions are stated in terms of positive semidefiniteness of related Hankel matrices.     

Zeros of Polynomials on Banach Spaces: The Real Story

Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Further Investigations Involving Rook Polynomials With Only Real Zeros

We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham . Another involves a version of the ‘hit polynomial’ of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is analogous to a theorem of the author, K. Ono, and D. G. Wagner, namely that for Ferrers boards the hit polynomial has only real zeros. We also show that for each of these families there is a general conjecture involving arrays of numbers satisfying inequalities which contains these theorems as special cases. We provide evidence for the truth of these conjectures by proving other special cases and discussing computational experiments.     

关于微分多项式零点的注记

The value distribution of differential polynomials is studied.The re- suits in this paper improve and generalize some previous theorems given by Yang Chungchun(On deficiencies of differential polynomials,Math.Z.,116(1970),197- 204),H.S.Gopalakrishna and S.S.Bhoosnurmath(On distribution of values of differential polynomials,Indian J.Pure Appl.Math.,17(1986),367-372),I.Lahiri (A note on distribution of nonhomogeneous differential polynomials,Hokkaido Math. J.,31(2002),453-458)and Yi Hongxun(On zeros of differential polynomials,Adv. in Math.,18(1989),335-351)et al.Examples show that the results in this paper are sharp.     

On the Zeros of Orthogonal Polynomials: The Elliptic Case

Constructive Approximation - Let E = [–1, α] \cup [β, 1], –1 &;lt; α &;lt; β &;lt; 1, and let (pn) be orthogonal on E with respect to the weight function...     

On the Behaviour of Zeros of Jacobi Polynomials

Denote by x_n,k(α,β) and x_n,k(λ)=x_n,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P^(α,β)_n(x) and of the ultraspherical (Gegenbauer) polynomial C^λ_n(x), respectively. The monotonicity of x_n,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P^(a,b)_n(x) are smaller (greater) than the zeros of P^(α,β)_n(x) are provided. A. Markov proved that x_n,k(a,b)<x_n,k(α,β) (x_n,k(a,b)>x_n,k(α,β)) for every n and each k, 1kn if a>α and b<β (a<α and b>β). We prove the converse statement of Markov's theorem. The question of how large the function f_n(λ) could be such that the products f_n(λ)x_n,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that f_n(λ)=(λ+(2n²+1)/(4n+2))^1/2 obeys this property. We establish the sharpness of their result.     

Zeros of Two-Sided Quadratic Quaternion Polynomials

The structure and computation are given for the zeros of a two-sided quadratic quaternion polynomial. In particular, the computation formulae and the sharp bound of essential number are established for the zeros of a two-sided quaternion polynomial of the form \({u^2 + puq + r (p, q, r \in \mathbb{H})}\) . As applications, some known results are improved and a conjecture presented by Janovská and Opfer is answered.     

On the Zeros of Polynomials of Best Approximation

Given a function f, uniform limit of analytic polynomials on a compact, regular set E^N, we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate L^q norms. These results give multivariable versions of one-variable results due to Blatt–Saff, Pleniak and Wójcik.     

Zeros of Difference Polynomials of Meromorphic Functions

This research is a continuation of a recent paper, due to Liu and Laine, dealing with difference polynomials of entire function. In this paper, we investigate the value distribution of difference polynomials of meromorphic functions and prove some difference analogues to some classical results for differential polynomials.     

Asymptotic Properties of Zeros of Hypergeometric Polynomials

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |z^k(1−z)|=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.     

Asymptotic Formulas for the Zeros of the Meixner Polynomials

The zeros of the Meixner polynomialm_n(x; β, c) are real, distinct, and lie in (0, ∞). Letα_{n, s}denote thesth zero ofm_n(nα; β, c), counted from the right; and letα_{n, s}denote thesth zero ofm_n(nα; β, c), counted from the left. For each fixeds, asymptotic formulas are obtained for bothα_{n, s}andα_{n, s}, asn→∞.     

The Krzyz Problem and Polynomials with Zeros on the Unit Circle

Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman.     

四元数体上两类多项式的零点

本文研究四元数体 Q上多项式的零点 ,特别对于其中两类多项式——系数两两可换的多项式和二次多项式建立了系统而完善的零点理论 .     

Asymptotic Behavior of the Pollaczek Polynomials and Their Zeros

In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials P_n(x; a, b) when , where t > 0 is fixed. He divided the positive t-axis into two regions, 0 < t < (a + b)^1/2 and t > (a + b)^1/2, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two-term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ε ≤ t ≤ M < ∞, and show that Ismail's result is incorrect.     

Self-Inversive Polynomials with All Zeros on the Unit Circle

Conditions are given in the coefficients of a self-inversive polynomial under which all its zeros are on the unit circle.To my friend, Jean-Louis Nicolas at the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—30C15     

Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.     

基于Jacobi多项式零点的Grünwald插值算子

本文考虑基于一般Jacobi多项式J_n~(α,β)(x)(—1<α,β<1)零点的Grnwald插值多项式G_n(f,x);主要证明了G_n(f,x)在(—1,1)内几乎一致收敛于连续函数f(x),并给出了点态逼近估计;拓广和完善了文献[1,2]的结果。