Event Type:

Analysis Seminar

Date/Time:

Monday, April 29, 2013 - 05:00

Location:

Kidder 280

Guest Speaker:

Institution:

University of Oregon

Abstract:

The generalized Mahler measure of a polynomial is a measure of complexity formed by integrating the logarithm of the absolute value of the polynomial against \omega_K, the logarithmic equilibrium distribution of a set K (in the case of the classical Mahler measure, K is the unit disk). The roots of a complex degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a determinantal point process on the complex plane. According to the large deviation principle, when N is large, it is exponentially likely to find the roots near the outer boundary of K distributed as \omega_K. Determinantal nature of this point process means that the local statistics of the roots can be expressed via determinants of a single kernel function. We investigate the asymptotics of this kernel in different regions of the complex plane.