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

Abstract

We show that the number of non-isotopic commutative semifields of odd order p n p^{n} is exponential in n n when n = 4 t n = 4t and t t is not a power of 2 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 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].