  We show that for Gaussian random SU(m+1) polynomials of a large degree N the
probability that there are no zeros in the disk of radius r is less than
$e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$.
Enroute to this result, we also derive a more general result: probability
estimates for the event where the volume of the zero set of a random polynomial
of high degree deviates significantly from its mean.
