Let P_n be a polynomial of degree n whose coefficients are independent (but not necessarily identical) real random variables. How do its real roots distribute ?
This problem has been studied for more than 70 years,
started with fundamental results of Littlewood-Offord and Kac in the 1940s. On the other hand, many basic
problems stay open. For instance, little was known about the local maxima and minima of Rademacher polynomial,
whose coefficients are +-1 with probability 1/2.
In this talk, we are going to give a brief survey about some basic problems, and then report a recent
progress which provides complete answers in the case the coefficients have polynomial growth. In particular,
we get precise answers about roots of all derivatives of Rademacher polynomials. Our results also improve upon
an old estimate by Erdos-Offord from 1956.
Joint work with Y. Do (Univ of Virginia) and V. Vu (Yale).