Effective Hilbert's irreducibility theorem for global fields

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field K. More precisely, we give effective bounds for the number of specializations t is an element of O-K that do not preserve the irreducibility or the Galois group of a given irreducible polynomial F(T, Y) is an element of K[T, Y]. The bounds are explicit in the height and degree of the polynomial F(T, Y), and are optimal in terms of the size of the parameter t is an element of O-K. Our proofs deal with the function field and number field cases in a unified way.