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Published online by Cambridge University Press: 23 September 2009
Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among
for n ∈
. Fields of this type include the Shanks fields and their generalizations.