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2019 The Lind–Lehmer Constant for $\mathbb{Z}_2^r \times \mathbb{Z}_4^s$
Michael J. Mossinghoff, Vincent Pigno, Christopher Pinner
Mosc. J. Comb. Number Theory 8(2): 151-162 (2019). DOI: 10.2140/moscow.2019.8.151

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 n3 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.

Citation

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Michael J. Mossinghoff. Vincent Pigno. Christopher Pinner. "The Lind–Lehmer Constant for $\mathbb{Z}_2^r \times \mathbb{Z}_4^s$." Mosc. J. Comb. Number Theory 8 (2) 151 - 162, 2019. https://doi.org/10.2140/moscow.2019.8.151

Information

Received: 21 June 2018; Accepted: 24 July 2018; Published: 2019
First available in Project Euclid: 29 May 2019

zbMATH: 07063272
MathSciNet: MR3959883
Digital Object Identifier: 10.2140/moscow.2019.8.151

Subjects:
Primary: 11R06
Secondary: 11B83, 11C08, 11G50, 11T22, 43A40

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.8 • No. 2 • 2019
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