CHAPTER 2 - Random Algebraic Polynomials: Basic Definitions and Properties

Abstract

This chapter presents an introduction to the concept of random algebraic polynomials and discusses the basic properties of this class of random functions. Random algebraic polynomials are defined as a specialization of random power series. Random power series are introduced and random algebraic polynomials are defined as a specialization of random power series. The chapter presents some other definitions of random algebraic polynomials, several of which are based on notions from probabilistic functional analysis. It discusses the measurability of the zeros of a random algebraic polynomial and the measurability of the number of zeros, respectively. The chapter reviews some additional properties of random algebraic polynomials—the continuity, separability, and measurability of random algebraic polynomials and shows that under certain conditions random algebraic polynomials are martingales.