2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127712We 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.This paper generalizes one which was previously posted by the authorComplex VariablesProbability30B20; 30C15; 60G60; 82B10text