On Hilbert's irreducibility theorem

In this paper we obtain new quantitative forms of Hilbert’s Irreducibility Theorem. In particular, we show that if $f(X,T_1,\dots ,T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb{Q}(T_1,\dots ,T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$specialisations$$\mathbf{t}$ \in $\mathbb{Z}$^s$with$|$\mathbf{t}$| \le H$suchthattheresultingpolynomial$f(X)$hasGaloisgroup$K$ over the rationals.