Here is a code to generate high resolution video efficiently with a few more graphics details:
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ClearAll["`*"]SeedRandom[7];coef[n_, k_] := coef[n, k] = (RandomReal[{-1, 1}] + RandomReal[{-1, 1}] I);nframes = 240;solve[theta_, n_] := NSolve[ Sum[coef[n, k] z^k, {k, 1, n}] + 0.2 E^(I theta) + 0.9 E^(I 2 theta) z^3 If[n >= 3, 1, 0] + 0.9 E^(I 3 theta) z^6 If[n >= 6, 1, 0] == 0, z]ClearAll[disks];disks[theta_, n_] := Disk @@@ Flatten[Table[ {{Re[z], Im[z]}, Max[0.3/n, 0.001]} /. solve[theta, n], {n, 4, 200}], 1];Graphics[{Black,Opacity[.85],disks[0,n]},PlotRange->1.3,ImageSize->1080]
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frames=ParallelTable[Graphics[{Black,Opacity[.85],disks[theta,n]},PlotRange->1.3,ImageSize->1080],{theta,0,2Pi(1-1/nframes),2Pi/nframes}];//AbsoluteTiming
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{92.3478,Null}
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Parallelize@FrameListVideo[frames,RasterSize->1080,FrameRate->30]
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