Georgia Institute of Technology College of Sciences

The area was essentially originated by the general question: How many zeros of a random polynomials are real? Kac showed that the expected number of real zeros for a polynomial with i.i.d. Gaussian coefficients is logarithmic in terms of the degree. Later, it was found that most of zeros of random polynomials are asymptotically uniformly distributed near the unit circumference (with probability one) under mild assumptions on the coefficients. Thus two main directions of research are related to the almost sure limits of the zero counting measures, and to the quantitative results on the expected number of zeros in various sets. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by various bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.