CHAPTER 1 - Introduction

Abstract

Although the study of random algebraic polynomials is of independent theoretical interest in that their study leads to probabilistic generalizations of classical results on algebraic polynomials, many problems in the applied mathematical sciences lead to random algebraic polynomials. This chapter presents a number of examples to illustrate the origin of random algebraic polynomials. It discusses some concrete situations that lead to random algebraic polynomials. Some of these situations are: (a) a random algebraic polynomial will arise if the coefficients of an algebraic polynomial are subject to random error, (b) random algebraic polynomials arise in the study of difference and differential equations with random coefficients, (c) random algebraic polynomials arise in the spectral theory of random matrices with subsequent applications in many of the applied mathematical sciences that use matrix methods, (d) an interesting class of random algebraic polynomials arises in the study of approximate solution of operator equations. The chapter also presents a historical background of research on random algebraic polynomials. It further discusses other types of random polynomials, namely, trigonometric, orthogonal, and some random polynomials that arise in approximation theory.