Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.
Serre,Topics in Galois Theory, 2nd ed., Research Notes in Mathematics, vol
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Componentwise height bounds for polynomial value-set lifting
Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.