  The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong
impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of
simultaneous zeros of $m$ polynomials with given Newton polytopes. In this
article, we show that $P_f$ also has a strong impact on the distribution of
zeros of one or several polynomials. We equip the space of (holomorphic)
polynomials of degree $\leq N$ in $m$ complex variables with its usual
$SU(m+1)$-invariant Gaussian measure and then consider the conditional measures
$\gamma_{|NP}$ induced on the subspace of polynomials whose Newton polytope
$P_f\subset NP$. When $P=\Sigma$, the unit simplex, then it is obvious and
well-known that the expected distribution of zeros $Z_{f_1,...,f_k}$ (regarded
as a current) of $k$ polynomials $f_1,...,f_k$ of degree $N$ is uniform
relative to the Fubini-Study form. Our main results concern the conditional
expectation $E_{|NP} (Z_{f_1,...,f_k})$ of zeros of $k$ polynomials with Newton
polytope $NP\subset Np\Sigma$ (where $p=\deg P$); these results are asymptotic
as the scaling factor $N\to\infty$. We show that $E_{|NP} (Z_{f_1,...,f_k})$ is
asymptotically uniform on the inverse image $A_P$ of the open scaled polytope
$p^{-1}P^\circ$ via the moment map $\mu:CP^m\to\Sigma$ for projective space.
However, the zeros have an exotic distribution outside of $A_P$ and when $k=m$
(the case of the Kouchnirenko-Bernstein theorem) the percentage of zeros
outside $A_P$ approaches 0 as $N\to\infty$.
