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

TL;DR

This paper establishes an exponential lower bound on the number of non-isotopic commutative semifields of odd order, introducing new families and methods to improve previous quadratic bounds.

Contribution

The authors introduce a new family of commutative semifields and a novel method for isotopy classification, leading to an exponential bound on their count.

Findings

Number of non-isotopic commutative semifields is exponential in n for certain n.

New family of commutative semifields constructed.

Method for proving isotopy results developed.

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)]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003)] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004)].