Moscow Journal of Combinatorics and Number Theory

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

Michael J. Mossinghoff, Vincent Pigno, and Christopher Pinner

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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 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.

Article information

Mosc. J. Comb. Number Theory, Volume 8, Number 2 (2019), 151-162.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11B83: Special sequences and polynomials 11C08: Polynomials [See also 13F20] 11G50: Heights [See also 14G40, 37P30] 11T22: Cyclotomy 43A40: Character groups and dual objects

Lind–Lehmer constant Mahler measure group determinant


Mossinghoff, Michael J.; Pigno, Vincent; Pinner, Christopher. The Lind–Lehmer Constant for $\mathbb{Z}_2^r \times \mathbb{Z}_4^s$. Mosc. J. Comb. Number Theory 8 (2019), no. 2, 151--162. doi:10.2140/moscow.2019.8.151.

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