CHAPTER 6 - The Variance of the Number of Real Zeros of Random Algebraic Polynomials

Abstract

This chapter discusses the variance of the number of real zeros of a random algebraic polynomial. It examines the theoretical estimates with the computer-generated numerical results. It gives an estimate of V{Nn(R, ω)} in the case where the random coefficients are real-valued dependent standard Gaussian random variables and states this result. The chapter presents some preliminary results, lemmas, and the proof of the main theorem. It also presents some computational results based on the theoretical estimates and computer-generated samples of random algebraic polynomials. As is well known, iterative and projective methods for the solution of many types of random operator equations lead to systems of random algebraic equations hence, knowledge of the statistical properties of the zeros of the random characteristic polynomials of the associated random matrices is of great importance for the numerical solution of random operator equations.