Number Theory Seminar

Tuesday, Nov. 18
Andrew Ledoan, U of R
The complex zeros of random polynomials

ABSTRACT:

n this talk I will prove a 1995 result of L. A. Shepp and R. J. Vanderbei on the expectation of the number, $\nu_n(\Omega)$, of complex zeros of a random polynomial, $P_n(z) = \sum_{j=0}^{n-1} \eta_j z^j$, where $\eta_0, \dots, \eta_{n-1}$ are independent standard normal random variables, in any measurable subset $\Omega$ of $\mathbb{C}$ whose boundary intersects the real axis at most finitely many times. This result is an extension of a theorem of M. Kac where $\Omega$ is any measurable subset of the reals.