On a Question of Davenport

AbstractLetkbe a number field and denote by okits ring of integers. Let p be a non-zero prime ideal of ok. Denote byfthe polynomial derived fromfby reducing the coefficients modulo p. SetVp(f)={f(u)∣u∈ok/p}. Davenport raised the following question (withkbeing the rationals). Supposefandgare polynomials in ok[X] such thatVp(f)=Vp(g) for all but finitely many non-zero prime ideals of ok. Does this implyf(X)=g(aX+b) for somea, b∈k? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counter-examples