  We show that the Newton polytope of a polynomial has a strong impact on the
distribution of its mass and zeros. The basic theme is that Newton polytopes
determine allowed and forbidden regions for these distributions. We equip the
space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables
with its usual $SU(m + 1)$-invariant Gaussian probability measure and then
consider the conditional measure induced on the subspace of polynomials with
fixed Newton polytope $P$. We then determine the asymptotics of the conditional
expectation $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ polynomials with
Newton polytope $NP$ as $N \to \infty$. When $P = \Sigma$, the unit simplex,
then it is obvious and well-known that the expected zero distribution
$E_{|N\Sigma}(Z^k)$ is uniform relative to the Fubini-Study form. For a convex
polytope $P\subset p\Sigma$, we show that there is an allowed region on which
$N^{-k}E_{|N P}(Z^k)$ is asymptotically uniform as the scaling factor
$N\to\infty$. However, the zeros have an exotic distribution in the
complementary forbidden region, and when $k=m$, the expected percentage of
simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$,
yielding a quantitative version of the Kouchnirenko-Bernstein theorem.
