ZEROS OF RANDOM POLYNOMIALS ON C m

By Thomas F. Bloom, Bernard Shiffman. 2006

86 citations recorded by the source corpus.

For a regular compact set $K$ in $\mathbb{C}^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $\mathcal{P}_N$ 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 $\mathcal{P}_N$ 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 $N\to\infty$.\end

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4c35694444e3049cee4477add6383c0bf01d856a

Citations in the corpus

• Equidistribution of zeros of random holomorphic sections (2013, unknown): “If a (n) j are standard complex Gaussian then it is classical that (see [26, 27, 7]) the identity (3.”