Moscow Journal of Combinatorics and Number Theory

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

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 $G=ℤ2r×ℤ4s$ with $|G|≥4$ we show that this minimum is always $|G|−1$, a case of sharpness in the trivial bound. For $G=ℤ2×ℤ2n$ with $n≥3$ the minimum is $9$, and for $G=ℤ3×ℤ3n$ the minimum is $8$. Previously the minimum was only known for $2$- and $3$-groups of the form $G=ℤpk$ or $ℤpk$. We also show that a congruence satisfied by the group determinant when $G=ℤpr$ generalizes to other abelian $p$-groups.

Article information

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

Dates
Accepted: 24 July 2018
First available in Project Euclid: 29 May 2019

https://projecteuclid.org/euclid.moscow/1559095390

Digital Object Identifier
doi:10.2140/moscow.2019.8.151

Mathematical Reviews number (MathSciNet)
MR3959883

Zentralblatt MATH identifier
07063272

Citation

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. https://projecteuclid.org/euclid.moscow/1559095390

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