CHAPTER 4 - The Number and Expected Number of Real Zeros of Random Algebraic Polynomials

Abstract

This chapter presents a survey of available results on the estimates of the number of real zeros, the average number of real zeros, the average number of maxima, and the upper and lower bounds of the number of real zeros of random algebraic polynomials to illustrate the analytic techniques that are utilized in the theory of random algebraic polynomials and investigations of the statistical properties of the zeros. The results are for the cases in which the random coefficients are: (1) independent random variables with known mean and variance, (2) independent normal random variables, (3) dependent normal random variables, (4) independent Cauchy random variables, (5) independent stable random variables, and (6) independent uniform random variables. The chapter also presents a generalization of the Kac–Rice formula and its application for evaluating the estimates of the number of real zeros and a few results on the expected number of real zeros when the coefficients of the random algebraic polynomial are complex-valued random variables.