On the zeros of random polynomials

In this dissertation we will prove several results pertaining to the properties of zeros of random polynomials. We will begin with a discussion of the expected number of real zeros for random polynomials with dependent standard normal coefficients. With certain restrictions imposed on the spectral density of the coefficients' covariance function, we will show that similar behavior to the independent case can be expected. Specifically, the value of the expected number of real zeros grows asymptotically like 2p log n, as n → infinity. After studying the real zeros, we will next consider the number of K-level crossings. Again imposing certain restrictions on the spectral density, an asymptotic value will be derived for the expected number of K-level crossings of random polynomials with dependent standard normal coefficients.;The next problem that will be considered is a study of the distribution of the complex zeros. Once more imposing restrictions on the spectral density, we will show that the complex zeros of random polynomials with dependent standard normal coefficients converge to the unit circle. Additionally, we will derive expressions approximating how fast this convergence happens. By then adapting the techniques used in the aforementioned problem, we will study the behavior of random polynomials which have applications to the GSM (Global System for Mobile Communications)/EDGE (Enhanced Data Rates for GSM Evolution) standard for mobile phones.;The last part of our work will consider the zeros of random sums of orthogonal polynomials. For a random sum of the Chebyshev polynomials of the first kind, orthogonalized over the interval [-1, 1], we will show that the distribution of zeros converges to the corresponding equilibrium measure for this set. This result will lay the foundation for some further work in the area of random sums of orthogonal polynomials.