Zeros of random polynomials on ℂm

Abstract

For a regular compact set K in ℂm and a measure μ on K satisfying the Bernstein-Markov inequality, we consider the ensemble P N of polynomials of degree N, endowed with the Gaussian probability measure induced by L2(μ). We show that for large N, the simultaneous zeros of m polynomials in PN tend to concentrate around the Silov boundary of K; more precisely, their expected distribution is asymptotic to Nmμed where μeq is the equilibrium measure of K. For the case where K is the unit ball, we give scaling asymptotics for the expected distribution of zeros as N → ∞. © International Press 2007.