Commutative Semifields from bijections of the Desarguesian plane

The Menichetti-Kaplansky theorem states that a finite semifield that is three-dimensional over its center is either a field or a twisted field of Albert.

This implies that a quadratic homogeneous bijection of $\mathbb{P}^2(\mathbb{F}_q)$ is equivalent to a Dembowski-Ostrom monomial.

In this paper, we give a large class of semiquadratic homogeneous bijections of $\mathbb{P}^2(\mathbb{F}_q)$ that are inequivalent to Dembowski-Ostrom monomials.

Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes.

Semiquadratic homogeneous bijections of $\mathbb{P}^1(\mathbb{F}_q)$ have been classified only recently by the first-named author, and Ding and Zieve with the result that all such bijections are either equivalent to Dembowski-Ostrom monomials or degenerate.

We demonstrate that this is not the case for $\mathbb{P}^2(\mathbb{F}_q)$.