Vol. 8, No. 2, 2019

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The Lind–Lehmer Constant for $\mathbb{Z}_2^r \times \mathbb{Z}_4^s$

Michael J. Mossinghoff, Vincent Pigno and Christopher Pinner

Vol. 8 (2019), No. 2, 151–162
Abstract

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 G = 2r × 4s with |G| 4 we show that this minimum is always |G| 1, a case of sharpness in the trivial bound. For G = 2 × 2n with n 3 the minimum is 9, and for G = 3 × 3n the minimum is 8. Previously the minimum was only known for 2- and 3-groups of the form G = pk or pk. We also show that a congruence satisfied by the group determinant when G = pr generalizes to other abelian p-groups.

Keywords
Lind–Lehmer constant, Mahler measure, group determinant
Mathematical Subject Classification 2010
Primary: 11R06
Secondary: 11B83, 11C08, 11G50, 11T22, 43A40
Milestones
Received: 21 June 2018
Accepted: 24 July 2018
Published: 20 May 2019
Authors
Michael J. Mossinghoff
Department of Mathematics & Computer Science
Davidson College
Davidson, NC
United States
Vincent Pigno
Department of Mathematics & Statistics
California State University
Sacramento, CA
United States
Christopher Pinner
Department of Mathematics
Kansas State University
Manhattan, KS
United States