On a Question of Davenport Original Research Article

Letkbe a number field and denote by imagekits ring of integers. Let image be a non-zero prime ideal of imagek. Denote byfthe polynomial derived fromfby reducing the coefficients modulo image. SetVimage(f)={f(u)miduset membership, variantimagek/image}. Davenport raised the following question (withkbeing the rationals). Supposefandgare polynomials in imagek[X] such thatVimage(f)=Vimage(g) for all but finitely many non-zero prime ideals of imagek. Does this implyf(X)=g(aX+b) for somea, bset membership, variantk? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counter-examples.