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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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.

Article information

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

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

Permanent link to this document
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

Subjects
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

Keywords
Lind–Lehmer constant Mahler measure group determinant

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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References

  • T. M. Apostol, “Resultants of cyclotomic polynomials”, Proc. Amer. Math. Soc. 24 (1970), 457–462.
  • K. Conrad, “The origin of representation theory”, Enseign. Math. $(2)$ 44:3-4 (1998), 361–392.
  • R. Dedekind, Gesammelte mathematische Werke, II, Chelsea, New York, 1968.
  • D. DeSilva and C. Pinner, “The Lind Lehmer constant for $\mathbb{Z}_p^n$”, Proc. Amer. Math. Soc. 142:6 (2014), 1935–1941.
  • N. Kaiblinger, “On the Lehmer constant of finite cyclic groups”, Acta Arith. 142:1 (2010), 79–84.
  • S. Lang, Cyclotomic fields, Graduate Texts in Mathematics 59, Springer, 1978.
  • E. T. Lehmer, “A numerical function applied to cyclotomy”, Bull. Amer. Math. Soc. 36:4 (1930), 291–298.
  • D. H. Lehmer, “Factorization of certain cyclotomic functions”, Ann. of Math. $(2)$ 34:3 (1933), 461–479.
  • D. Lind, “Lehmer's problem for compact abelian groups”, Proc. Amer. Math. Soc. 133:5 (2005), 1411–1416.
  • V. Pigno and C. Pinner, “The Lind–Lehmer constant for cyclic groups of order less than 892,371,480”, Ramanujan J. 33:2 (2014), 295–300.
  • W. Vipismakul, The stabilizer of the group determinant and bounds for Lehmer's conjecture on finite abelian groups, Ph.D. thesis, University of Texas at Austin, 2013, http://hdl.handle.net/2152/21685.