On some extensions of the Ailon–Rudnick theorem
In this paper we present some extensions of the Ailon–Rudnick theorem, which says that if f , g ∈ C [ T ] , then gcd ( f n - 1 , g m - 1 ) is bounded for all n , m ≥ 1. More precisely, using a uniform bound for the number of torsion points on curves and results on the intersection of curves with alg...
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Published in: | Monatshefte für Mathematik 2016-10, Vol.181 (2), p.451-471 |
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Main Author: | |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | In this paper we present some extensions of the Ailon–Rudnick theorem, which says that if
f
,
g
∈
C
[
T
]
, then
gcd
(
f
n
-
1
,
g
m
-
1
)
is bounded for all
n
,
m
≥
1. More precisely, using a uniform bound for the number of torsion points on curves and results on the intersection of curves with algebraic subgroups of codimension at least 2, we present two such extensions in the univariate case. We also give two multivariate analogues of the Ailon–Rudnick theorem based on Hilbert’s irreducibility theorem and a result of Granville and Rudnick about torsion points on hypersurfaces. |
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ISSN: | 0026-9255 1436-5081 |
DOI: | 10.1007/s00605-016-0911-3 |