How many real roots does a random polynomial have?

Van Vu (Yale)

Abstract:
Finding real roots of a polynomial is one of the oldest and most basic problems in mathematics. For polynomials with high degrees, even estimating the number of real roots is a very difficult. Classical estimates, such as Descartes’s rule of signs, are often sharp for certain classes of polynomials, but far from being optimal for some others.

What if our polynomial is chosen randomly ? Typically how many real roots are there? This question was already studied by Waring and Sylvester, with breakthrough results obtained by Littlewood-Offord and Kac in the early 1940s, which have led to the development of the theory of random functions.

We are going to discuss the study of this fascinating question through the last 7 decades, ending with some recent results which give surprisingly sharp answers and a few exciting open questions. Behind the scene is the development of a new approach to attack universality problems for random point processes, which I will also discuss if time permits.

Wednesday, Sept. 24, 4:30 – 5:30 p.m.
Mathematics 520
Tea will be served at 4:00 p.m.

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