# Roots of random polynomials: Universality and Number of Real roots

# Roots of random polynomials: Universality and Number of Real roots

Estimating the number of real roots of a polynomial is among the oldest and most basic questions in mathematics.The answer to this question depends very strongly on the structure of the coefficients, of course. What happens if we choose the coefficients randomly? In this case, the number of real roots become a random variable, whose value is between 0 and the degree of the polynomial. Can one understand this random variable? What is its mean, variance, and limiting distribution? Random polynomials were first studied by Waring in the 18th century. In the 1930s, Littlewood and Offord started their famous studied which led to the surprising fact that in general a random polynomials with iid coefficients have (with high probability), order log n real roots. Their investigation opened a whole new area of random polynomials and random functions, with deep contributions from several leading mathematicians, including Erdos, Turan, Kac, Kahane, Ibragimov etc. An exciting turn occurred in the 1990s, when physicists Bogomolny, Bohigas and Leboeuf established a link between roots of random polynomials and quantum chaotic dynamics. Since then, random polynomials and random series are also studied intensively by researchers from analysis, probability, and mathematical physics. In these studies, the complex roots become also important. One views the set of all roots as a random point process, and a fundamental problem, among others, is to understand the correlation between nearby points in a small region. (This correlation is often use to model the interaction between particles in certain physics problems.) In this talk, I will try to survey this fascinating area, presenting some of its main questions, results, and ideas. The talk is self-contained, and requires only basic knowledge in analysis and probability.