1Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 S Morgan St Chicago, IL 60607, U.S.A. 2Department of Mathematics Royal Holloway University of London TW20 0EX Egham, United Kingdom
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.
Keywords:
obtain quantitative forms hilbert irreducibility theorem particular ldots irreducible polynomial integer coefficients having galois group function field mathbb ldots subgroup there varepsilon s varepsilon specialisations mathbf mathbb mathbf resulting polynomial has galois group rationals
Affiliations des auteurs :
Abel Castillo
1
;
Rainer Dietmann
2
1
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 S Morgan St Chicago, IL 60607, U.S.A.
2
Department of Mathematics Royal Holloway University of London TW20 0EX Egham, United Kingdom
@article{10_4064_aa8380_2_2017,
author = {Abel Castillo and Rainer Dietmann},
title = {On {Hilbert{\textquoteright}s} irreducibility theorem},
journal = {Acta Arithmetica},
pages = {1--14},
year = {2017},
volume = {180},
number = {1},
doi = {10.4064/aa8380-2-2017},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8380-2-2017/}
}
TY - JOUR
AU - Abel Castillo
AU - Rainer Dietmann
TI - On Hilbert’s irreducibility theorem
JO - Acta Arithmetica
PY - 2017
SP - 1
EP - 14
VL - 180
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4064/aa8380-2-2017/
DO - 10.4064/aa8380-2-2017
LA - en
ID - 10_4064_aa8380_2_2017
ER -
