Bounds for Hilbert's irreducibility theorem

In the context of Hilbert's irreducibility theorem, it is an open question whether there exists a bound for the least hilbertian specialization in N that is polynomial in the degree d and the logarithmic height log(H) of the polynomial P(T, Y) in question. A positive answer would be useful. notably for algorithmic applications. We obtain a polynomial bound in log(H) and d(Hi(P)) where Hi(P) - the Hilbert index of P - is a pure group-theoretical invariant, we define and which we show to be absolutely bounded for many classes of polynomials. We also discuss further questions related to effectiveness in Hilbert's irreducibility theorem.