I could imagine that one uses the derivative here. If x+iy and x-iy are complex roots, and y is really small, then the derivative at x should be small as well (

I could imagine that one uses the derivative here. If x+iy and x-iy are complex roots, and y is really small, then the derivative at x should be small as well (if y = 0, you have a double zero, so p'(x)=0). So if p'(x) is far away from zero, you have a single real zero. And I guess double zeroes don't really happen with random coefficients.