On Hilbert's irreducibility theorem

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if \(f(X, T_1, \ldots, T_s)\) 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...

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Bibliographic Details
Published in:arXiv.org 2016-01
Main Authors: Castillo, Abel, Dietmann, Rainer
Format: Article
Language:English
Subjects:
Polynomials
Subgroups
Theorems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422