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Abstract
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Let
be a number field
and let
be any subgroup
of the units of .
If
,
Lehmer’s conjecture predicts that the height of any nontorsion element of
is bounded below by an absolute positive constant. If
),
Zimmert proved a lower bound on the regulator of
which grows
exponentially with
.
By sharpening a 1997 conjecture of Daniel Bertrand’s, Fernando Rodriguez
Villegas “interpolated” between these two extremes of rank with a new
higher-dimensional version of Lehmer’s conjecture. Here we prove a high-rank
case of the Bertrand–Rodriguez Villegas conjecture. Namely, it holds if
contains a
subfield
for
which
and
contains the kernel of
the norm map from
to
.
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Keywords
Lehmer's conjecture, Mahler measure, units
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Mathematical Subject Classification
Primary: 11R06
Secondary: 11R27
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Milestones
Received: 11 September 2020
Revised: 4 August 2022
Accepted: 25 October 2022
Published: 3 March 2023
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