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Mathematics of Computation

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The Lind-Lehmer constant for certain $ p$-groups


Authors: Dilum De Silva, Michael J. Mossinghoff, Vincent Pigno and Christopher Pinner
Journal: Math. Comp.
MSC (2010): Primary 11R06; Secondary 11B83, 11C08, 11G50, 11R09, 11T22, 43A40
DOI: https://doi.org/10.1090/mcom/3350
Published electronically: May 18, 2018
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Abstract: We establish some new congruences satisfied by the Lind Mahler measure on $ p$-groups, and use them to determine the Lind-Lehmer constant for many finite groups. First, we determine the minimal nontrivial measure of $ p$-groups where one component has particularly high order. Second, we describe an algorithm that determines a small set of possible values for the minimal nontrivial measure of a $ p$-group of the form $ \mathbb{Z}_p\times \mathbb{Z}_{p^k}$ with $ k\geq 2$. This algorithm is remarkably effective: applying it to more than 600000 groups the minimum was determined in all but six cases. Finally, we employ the results of our calculations to compute the Lind-Lehmer constant for nearly $ 8$ million additional $ p$-groups.


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Additional Information

Dilum De Silva
Affiliation: Department of Mathematics, Bowling Green State University, Firelands, Huron, Ohio 44839
Email: dilumd@bgsu.edu

Michael J. Mossinghoff
Affiliation: Department of Mathematics and Computer Science, Davidson College, Davidson, North Carolina 28035-6996
Email: mimossinghoff@davidson.edu

Vincent Pigno
Affiliation: Department of Mathematics and Statistics, California State University, Sacramento, California 95819
Email: vincent.pigno@csus.edu

Christopher Pinner
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: pinner@math.ksu.edu

DOI: https://doi.org/10.1090/mcom/3350
Keywords: Lind-Lehmer constant, Mahler measure, group determinants
Received by editor(s): September 7, 2017
Received by editor(s) in revised form: December 1, 2017
Published electronically: May 18, 2018
Additional Notes: The second author was supported by a grant from the Simons Foundation (#426694).
Article copyright: © Copyright 2018 American Mathematical Society

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