Zeros of univariate interval polynomials

Polynomials with perturbed coefficients, which can be regarded as interval polynomials, are very common in the area of scientific computing due to floating point operations in a computer environment. In this paper, the zeros of interval polynomials are investigated. We show that, for a degree n interval polynomial, the number of interval zeros is at most n and the number of complex block zeros is exactly n if multiplicities are counted. The boundaries of complex block zeros on a complex plane are analyzed. Numeric algorithms to bound interval zeros and complex block zeros are presented.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies

Using the Schwarz function of an ellipse, it was recently shown that galaxies with density constant on confocal ellipses can produce at most four “bright” images of a single source. The more physically interesting example of an isothermal galaxy has density that is constant on homothetic ellipses. In that case bright images can be seen to correspond to zeros of a certain transcendental harmonic mapping. We use complex dynamics to give an upper bound on the total number of such zeros.     

Weierstrass method for quaternionic polynomial root‐finding

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass‐like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.     

Zeros of the Derivative of a Rational Function and Coinvariant Subspaces for the Shift Operator on the Bergman Space

If all n (n > 1) zeros of a rational function r with simple poles are in a half-plane, then the derivative of r has at least one zero in the same half-plane. This result is used to prove that the number of zeros of a linear combination of n Bergman kernels in the unit disk may range from 0 to 2n-3. Bibliography: 7 titles.     

On Oscillation of Functions with Bounded Spectral Band

In this paper, we consider extremal oscillatory properties of functions with bounded spectrum, i.e., with bounded support (in the sense of distributions) of the Fourier transform. For such functions f, we give criteria of extendability of }f} from the real axis to a function F on the complex plane with derivatives F ^(m) having no real zeros and without enlarging the width of spectrum. In particular, we give examples of functions $f$ from the real Paley–Wiener space such that every function f ^(m), m=0, 1,..., has a finite number of real zeros.     

Computing zeros of analytic mappings: A logarithmic residue approach

LetD be a polydisk in ℂ ⁿ and a mapping that is analytic in and has no zeros on the boundary ofD. Thenf has only a finite number of zeros inD and these zeros are all isolated. We consider the problem of computing these zeros. A multidimensional generalization of the classical logarithmic residue formula from the theory of functions of one complex variable will be our means of obtaining information about the location of these zeros. This integral formula involves the integral of a differential form, which we will transform into a sum ofn Riemann integrals of dimension 2n−1. We will show how the zeros and their multiplicities can be computed from these integrals by solving a generalized eigenvalue problem that has Hankel structure, andn Vandermonde systems. Numerical examples are included. The first author was supported by a grant from the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT). This work is part of the project “Counting and computing all isolated solutions of systems of nonlinear equations”, funded by the Fund for Scientific Research, Flanders.     

An extention of the Kac-Rice formula for the average number of zeros of random algebraic polynomials

We extend the Kac-Rice formula for the expected number of real zeros of random algebraic polynomials on R¹ with R¹-valued random coefficients to complex zeros of random algebraic polynomials on C¹ with C¹-valued random coefficients. Our method directly extends to multivariable cases     

Circular arithmetic and the determination of polynomial zeros

Summary Suppose all zeros of a polynomialp but one are known to lie in specified circular regions, and the value of the logarithmic derivativepp ^–1 is known at a point. What can be said about the location of the remaining zero? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newton's method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.     

On the zeros of complex Van Vleck polynomials

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials, have been studied extensively in the case of real number parameters. In the complex case, relatively little is known. Numerical investigations of the location of the zeros of the Stieltjes and Van Vleck polynomials in special cases reveal intriguing patterns in the complex case, suggestive of a deeper structure. In this article we report on these investigations, with the main result being a proof of a theorem confirming that the zeros of the Van Vleck polynomials lie on special line segments in the case of the complex generalized Lamé equation having three free parameters. Furthermore, as a result of this proposition, we are able to obtain in this case a strengthening of a classical result of Heine on the number of possible Van Vleck polynomials associated with a given Stieltjes polynomial.     

Constraints on the angular distribution of the zeros of a polynomial of low complexity

LetP(z) be a polynomial of degreen with complex coefficients. Theorem.There exists a constant C such that if P has at most k terms then the number of zeros of P in any open sector of aperture π/n at the origin is no more than C ^k2. The main point of this bound is that it is independent both ofn and the coefficients ofP. The proof is a simple application of Khovanskii's real Bezout theorem to the system of real equations ReP=ImP=0. We also describe a measure of additive complexity for sets of integers and use it to estimate the angular distribution more finely.     

On the Distribution of the Derivative Zeros of a Complex Polynomial

We consider the class S(n) of all complex polynomials of degree n > 1 having all their zeros in the closed unit disk ē. By S(n,β) we denote the subclass of p ? S(n) vanishing in the prescribed point β ? ē. For an arbitrary point α ? C and p ? S(n,β) let |p| _α be the distance of α and the set of zeros of P'. Then there exists some P ? S(n,β) with maximal |P|α. We give an estimation for the number of zeros of P on |z| = 1$ resp. P' on $ |z-α| = |P| _α .     

Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities

Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L ² for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

On some diophantine results related to Euler polynomials

In this paper we prove that there is at most one complex number b for which the shifted Euler polynomial E _n (x) + b has at most two zeros of odd multiplicity. Supported in part, by Grants T48791 and F68872 form HNFSR, and by the Hungarian Academy of Sciences.     

On Zeros of Certain Analytic Functions

Given a function s which is analytic and bounded by one in modulus in the open unit disk \mathbb D{{\mathbb D}} and given a finite Blaschke product J{\vartheta} of degree k, we relate the number of zeros of the function s-J{s-\vartheta} inside \mathbb D{{\mathbb D}} to the number of boundary zeros of special type of the same function.     

On Siegel zeros of Hecke-Landau zeta-functions

The primary concern of this paper is to deal with Siegel zeros of Hecke-Landau zeta-functions in an algebraic number field of finite degree over the rationals. As in the rational case with DirichletL-functions, the location of such zeros is closely connected with lower bounds for the corresponding zeta-functions at the points=1. This will be the theme in the first part of the paper. In this second part we first derive a form of the Brun-Titchmarsh theorem in the setting of a number field which is appropriate in our context. Then we turn our attention to the fact that an improvement of the constant in this inequality would lead to the nonexistence of Siegel zeros. The procedure is based on a weighted algebraic form of Selberg's upper bound sieve.     

On Locating Clusters of Zeros of Analytic Functions

Given an analytic function f and a Jordan curve that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside , together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along . If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.     

On a modification of the Koenig theorem

We present a modified Koenig theorem for the simultaneous determination of all distinct poles ofG(z)/F(z), whereG(z) is an analytic function inside a simple smooth closed contourC, F(z) is an analytic function inside and onC, with a known numbern of simple zeros insideC, andF(z), G(z) have no common zeros insideC. It turns out that complex and interval iterations of higher order can be constructed, and several algorithms are available for doing this. Some of them are well known and discussed in many papers.The author is grateful to the referees for their valuable comments and suggestions. Also, she would like to thank Andrey Andreev and Nikolay Kjurkchiev for their helpful discussions.     

Non-real zeros of linear differential polynomials

Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f″ + ω f, where ω is a positive real constant.