CHAPTER 7 - Distribution of the Zeros of Random Algebraic Polynomials

Abstract

An important and difficult problem in the study of random polynomials is the determination of the probability distribution or density of the zeros of a random polynomial—algebraic, trigonometric, and orthogonal—when the probability distribution or density of the random coefficients is given. This chapter presents the probability distribution or density of the zeros of random algebraic polynomials and discusses the problem of determining the distribution of zeros of random algebraic polynomials with complex coefficients. It explains the problem of determining the distribution function of the solutions of random linear and quadratic equations and presents some explicit results in these cases. It also presents a result that enables to determine the distribution of the zeros of a random algebraic polynomial with complex coefficients. The chapter discusses Hammersley's approach to the problem of determining the distribution of the zeros of a random algebraic equation based on the so-called conditional distribution of the zeros. It presents the distribution of the number of real zeros of random algebraic polynomials and some graphs based on computer-generated numerical results that illustrate the distribution of the real zeros and the distribution of the number of real zeros of random algebraic polynomials. Under certain conditions, a theorem for the distribution of the zeros of random algebraic polynomials can be obtained. The chapter presents a result on the limiting distribution of the zeros of random algebraic polynomials.