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

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)].