Zero distribution of random polynomials

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that under mild assumptions on the coefficients, their zeros are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane and quantify this convergence. In our results, random coefficients may be dependent and need not have identical distributions.     

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     

An improved lower bound for the expected number of real zeros of a random algebraic polynomials

In this note we estimate the lower bound of the average number of real zeros of a random algebraic polynomials when the random coefficients are standard normal random variables     

Computing orthogonal polynomials in Sobolev spaces

Summary. Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other - a generalization of "Stieltjes's procedure" - expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Received July 13, 1994 / Revised version received September 26, 1994     

On the Evaluation of Polynomial Coefficients

The evaluation of the coefficients of a polynomial from its zeros is considered. We show that when the evaluation is carried out by the standard algorithm in finite precision arithmetic, the accuracy of the computed coefficients depends on the order in which the zeros are introduced. An ordering that enhances the accuracy for many polynomials is presented.     

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.     

Weierstrass formula and zero-finding methods

Summary. Classical Weierstrass' formula [29] has been often the subject of investigation of many authors. In this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions. We are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros. Conditions for the safe convergence of Weierstrass' method, depending only on initial approximations, are given. In particular, we study polynomials with interval coefficients. Using an interval version of Weierstrass' method enclosures in the form of disks for the complex-valued set containing all zeros of a polynomial with varying coefficients are obtained. We also present Weierstrass-like algorithm for approximating, simultaneously, all zeros of a class of analytic functions in a given closed region. To demonstrate the proposed algorithms, three numerical examples are included. Received September 13, 1993     

ON THE ZEROS OF POLYNOMIALS OVER QUATERNIONS

ABSTRACT

We review and expand some results on the number of zeros of polynomials over Hamilton's quaternions, with particular emphasis on those polynomials with coefficients in a degree two subfield.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Interlacing properties of zeros of multiple orthogonal polynomials

It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result.     

Some properties of a class of sparse polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.     

Orthogonal polynomials and differential equations in neutron-transport and radiative-transfer theories

There is a set of orthogonal polynomials {g_n(x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuous in the interval [−1, + 1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g., the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular, neither the second-order differential equation nor the density of zeros (i.e., the number of zeros per unit of interval) of the polynomial g_n(x) have been found. Here we obtain the second-order differential equation in the case that these polynomials are hypergeometric, so leaving open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of g_n(x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.     

Number of zeros of interval polynomials

In this paper, we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval, without computing the roots of any polynomials. The result generalizes Sturm’s Theorem for counting the roots of univariate polynomials to univariate interval polynomials.     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Polynomial evaluation and associated polynomials

We show a connection between the Clenshaw algorithm for evaluating a polynomial , expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial given in monomial (!) representation can be evaluated for using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients. Received January 2, 1995 / Revised version received April 9, 1997     

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.     

Extremal polynomials for obtaining bounds for spherical codes and designs

We investigate two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs, respectively. We consider two basic properties of the solutions of these problems. Namely, we estimate from below the number of double zeros and find zero Gegenbauer coefficients of extremal polynomials. Our results allow us to search effectively for such solutions using a computer. The best polynomials we have obtained give substantial improvements in some cases on the previously known bounds for spherical codes and designs. Some examples are given in Section 6. This research was partially supported by the Bulgarian NSF under Contract I-35/1994.     

Accurate computation of the zeros of the generalized Bessel polynomials

Summary. A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper. Received July 25, 1997 / Revised version received May 19, 1999 / Published online June 8, 2000     

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

Turán inequalities for three-term recurrences with monotonic coefficients

We establish some new Turán type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. We deduce as a corollary asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.