On Hilbert's irreducibility theorem

TL;DR

This paper develops new quantitative bounds for Hilbert's Irreducibility Theorem, estimating the number of specializations that yield polynomials with a given Galois group over the rationals.

Contribution

It introduces novel bounds on the count of specializations with prescribed Galois groups, extending Hilbert's Irreducibility Theorem quantitatively.

Findings

Bound on the number of specializations with a specific Galois group

Quantitative estimates depending on polynomial degree and Galois group index

Extension of Hilbert's theorem to more precise algebraic settings

Abstract

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb{Q}(T_1, \ldots, T_s)$, 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 $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.