This is great! Thanks a lot. Do you happen to have any other suggestions?

Thanks a lot for your great answer! Your sketch of the proof has been very helpful indeed!

@AlexandreEremenko, thank you so much for your response. You have been enormously helpful. I have been out of action for the last few days so I have not been able to study your answer in any detail yet. However, I can at least quickly answer your question about Kac-polynomials. Terry Tao has a nice discussion on his blog. I also recently came across a question about them here on mathoverflow.

Yes, sorry, I am being rather vague because I would be happy with various statements. In the case where $p$ and $q$ are assumed to be random polynomials with iid coefficients, then I am interested in the distribution of zeroes in the complex plane. Presumably in this case one would need to assume that the degree $n$ of $p$ and $q$ is large and then take the large $n$ limit in order to make any kind of sharp statement. For low degree polynomials, I would be happy even with a simple condition for the existence of zeroes.