An exponential bound on the number of non-isotopic commutative semifields

Göloğlu, Faruk; Kölsch, Lukas

doi:10.1090/tran/8785

An exponential bound on the number of non-isotopic commutative semifields
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by Faruk Göloğlu and Lukas Kölsch;

Trans. Amer. Math. Soc. 376 (2023), 1683-1716

DOI: https://doi.org/10.1090/tran/8785

Published electronically: December 16, 2022

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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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