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@GerryMyerson I would like to add that, if we can find a formula for Resultant of general trinomials, then the resilt will subsume this paper. ijpam.eu/contents/2012-74-1/5/5.pdf
and what if $x < log^2(d_L) $ is any such estimate known ?? Also I cant read Serre's survey because it is in French! Can you suggest me its English version.