On Hilbert's irreducibility theorem

Abstract

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 $O_{f,\epsilon }(H^{s-1+|G/K{|}^{-1}+\epsilon })$ specialisations $\mathbf{t}\in {\mathbb{Z}}^s$ with $|\mathbf{t}|\le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.

arXiv comment12 pages