On Hilbert's irreducibility theorem

We obtain new quantitative forms of Hilbert’s irreducibility theorem. In particular, we show that if f(X,T1,…,Ts) is an irreducible polynomial with integer coefficients, having Galois group G over the function field Q(T1,…,Ts), and K is any subgroup of G, then there are at most Of,ε(Hs−1+∣∣G/K∣∣−1+ε) specialisations t∈Zs with |t|≤H such that the resulting polynomial f(X) has Galois group K over the rationals.