Seminar on low-dimensional mathematics April 21, 2006: Dmitry Zaporozhets (PDMI). On the roots of a random polynomial. Let $G_n=\xi_0+\xi_1 z+\dots+\xi_n z^n$ denote the random polynomial with i.i.d. coefficients. There are 2 questions exist: what is the expected number of real zeros of $G_n$, and what asymptotic behavior do complex roots exhibit? It was shown in the last century that for a wide class distribution of the coefficients the average number of the real roots of $G_n$ is approximate to $\log n$. Concerning complex case, it was shown that for a lot of different coefficients distributions complex roots concentrate on the unit circle. The random polynomial with quite different behavior of the roots (both real and complex) will be describe. Also, the necessary and sufficient conditions of the roots concentration on the unit circle a.s. will be presented. ------------------------------------------------ http://www.pdmi.ras.ru/~lowdimma