Abstract :

How many real roots does a polynomial of degree n with random coefficients have? For independent random signs (+1/-1) and independent standard Gaussians, the answer is ~(2/\pi)\log n, as is known for about 75+ years now. It is possible to change the nature of randomness and get different answers. In this talk we survey various results including recent ones, and give an idea of some of the techniques involved. The talk is intended to be accessible to graduate students with basic knowledge of analysis.