  The existence of the scaling limit and its universality, for correlations
between zeros of {\it Gaussian} random polynomials, or more generally, {\it
Gaussian} random sections of powers of a line bundle over a compact manifold
has been proved in a great generality in the works [BBL2], [Ha], [BD],
[BSZ1]-[BSZ4], and others. In the present work we prove the existence of the
scaling limit for a class of {\it non-Gaussian} random polynomials. Our main
result is that away from the origin the scaling limit exists and is universal,
so that it does not depend on the distribution of the coefficients. At the
origin the scaling limit is not universal, and we find a crossover from the
nonuniversal asymptotics of the density of the probability distribution of
zeros at the origin to the universal one away from the origin.
