  This article is concerned with random holomorphic polynomials and their
generalizations to algebraic and symplectic geometry. A natural
algebro-geometric generalization studied in our prior work involves random
holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L
\to M$ over any complex manifold. Our main interest is in the statistics of
zeros of $k$ independent sections (generalized polynomials) of degree $N$ as
$N\to\infty$. We fix a point $P$ and focus on the ball of radius $1/\sqrt{N}$
about $P$. Under a microscope magnifying the ball by the factor $\sqrt{N}$, the
statistics of the configurations of simultaneous zeros of random $k$-tuples of
sections tends to a universal limit independent of $P,M,L$. We review this
result and generalize it further to the case of pre-quantum line bundles over
almost-complex symplectic manifolds $(M,J,\omega)$. Following [SZ2], we replace
$H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic' sections
defined by Boutet de Monvel-Guillemin and (from another point of view) by
Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of
the Kac-Rice formula for zero correlations together with the results of [SZ2],
we prove that the scaling limits of the correlation functions for zeros of
random $k$-tuples of asymptotically holomorphic sections belong to the same
universality class as in the complex case.
