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.     

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...     

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.     

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.     

Discrete Concavity and Zeros of Polynomials

Murota et al. have recently developed a theory of discrete convex analysis as a framework to solve combinatorial optimization problems using ideas from continuous optimization. This theory concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over the field of Puiseux series) with prescribed non-vanishing properties. We also provide a short proof of Speyer's “hive theorem” which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices. Due to limited space a more coherent treatment and proofs will appear elsewhere.     

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.     

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.     

微分多项式的零点及其相关的正规定则

对于平面区域D上的亚纯函数族F,F中的每个函数的极点重数至少为k,零点重数至少为s.设a,b为两个有限复数a≠0.若对于F中的每对函数f(z),g(z)∈F,f~((k))-af~3和g~((k))-ag~3分担b,则F在区域D内正规,其中k是正整数,k≥2.当k=2,有s=3;当k≥3时,有s=k.     

On Growth of Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n which does not vanish in |z| k, k ≥ 1.It is known that for each 0 ≤ s n and 1 ≤ R ≤ k,M (P~(s), R )≤( 1/(R~s+ k~s))[{d~((s)/dx(s))(1+x~n)}_(x=1)]((R+k)/(1+k))~nM(P,1).In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial P(z).     

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.     

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→∞.     

线性微分多项式的零点与极点

对Frank-Weissenborn不等式中导数f~((k))能否被替换成一般的线性微分多项式a_0f+a_1f′+…+a_kf~((k))进行了研究,并彻底解决了这一问题.作为此结果的应用,Hayman-Yang不等式等几个已有的定理也得到了推广.例子表明,本文所得到的几个不等式的条件是基本的.