Statistics of real roots of random polynomials

In this thesis, we present two approaches in order to study the expected number of real zeros of random univariate polynomials. Namely, the Kac-Rice method and Edelman-Kostlan's geometric approach. We derive a remarkable result called the Kac-Rice formula concerning the expected number of real zeros and apply this result to certain random polynomial ensembles. We also report some basic facts from potential theory in the complex plane and its connection to complex random polynomials. In addition, we consider certain random orthogonal polynomials associated to suitable weight functions supported in the complex plane, and we present some known results in this directio