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Abstract
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For a finite abelian group the Lind–Lehmer constant is the minimum positive
logarithmic Lind–Mahler measure for that group. Finding this is equivalent to
determining the minimal nontrivial group determinant when the matrix entries are
integers.
For a group of the form
with
we show that this
minimum is always
,
a case of sharpness in the trivial bound. For
with
the minimum
is
, and for
the minimum is
. Previously the minimum
was only known for
-
and
-groups of the form
or .
We also show that a congruence satisfied by the group determinant when
generalizes to other
abelian
-groups.
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Keywords
Lind–Lehmer constant, Mahler measure, group determinant
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Mathematical Subject Classification 2010
Primary: 11R06
Secondary: 11B83, 11C08, 11G50, 11T22, 43A40
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Milestones
Received: 21 June 2018
Accepted: 24 July 2018
Published: 20 May 2019
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