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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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An exponential bound on the number of non-isotopic commutative semifields
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by Faruk Göloğlu and Lukas Kölsch PDF
Trans. Amer. Math. Soc. 376 (2023), 1683-1716 Request permission

Abstract:

We show that the number of non-isotopic commutative semifields of odd order $p^{n}$ is exponential in $n$ when $n = 4t$ and $t$ is not a power of $2$. We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in $n$ and given by Zhou and Pott [Adv. Math. 234 (2013), pp. 43–60]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003), pp. 96–114] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004), pp. 895–938].
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Additional Information
  • Faruk Göloğlu
  • Affiliation: Faculty of Mathematics and Physics, Charles University in Prague, Czechia
  • ORCID: 0000-0002-1223-3093
  • Email: Faruk.Gologlu@mff.cuni.cz
  • Lukas Kölsch
  • Affiliation: University of Rostock, Germany; and Department of Mathematics and Statistics, University of South Florida
  • ORCID: 0000-0002-2966-0710
  • Email: lukas.koelsch.math@gmail.com
  • Received by editor(s): August 22, 2021
  • Received by editor(s) in revised form: June 19, 2022, and July 19, 2022
  • Published electronically: December 16, 2022
  • Additional Notes: The first author was supported by the GAČR Grant 18-19087S - 301-13/201843. The second author was supported by the National Science Foundation under grant No. 2127742.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 1683-1716
  • MSC (2020): Primary 12K10, 17A35; Secondary 51A35, 51A40
  • DOI: https://doi.org/10.1090/tran/8785
  • MathSciNet review: 4549689