Zeros of random polynomials on C^m

Details Zeros of random polynomials on C^m Zeros of random polynomials on C^m

2006-05-30

arxiv.org/abs/math/0605739v1

Math. Res. Lett. 14 (2007), 469-479

Mathematics

Complex Variables

10 pages

Scientific paper

For a regular compact set $K$ in $C^m$ and a measure $\mu$ 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 $L^2(\mu)$. 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 $N^m \mu_{eq}$, where $\mu_{eq}$ 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 $N\to\infty$.

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