Zeros of random polynomials on C^m

Abstract

For a regular compact set $K$ in $C^m$ 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 $L^2(μ)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $P_N$ tend to concentrate around the Silov boundary of $K$; more precisely, their expected distribution is asymptotic to $N^m μ_{eq}$, where $μ_{eq}$ 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 $N\to\infty$.