Commutative Semifields from bijections of the Desarguesian plane

TL;DR

This paper introduces new classes of semiquadratic homogeneous bijections of the projective plane over finite fields, leading to the construction of numerous non-Desarguesian commutative semifield planes.

Contribution

It provides a large class of semiquadratic bijections in $ ext{P}^2( ext{F}_q)$ that are inequivalent to known monomials, enabling the creation of many new non-isotopic commutative semifields.

Findings

Constructed a large family of non-isotopic commutative semifields.

Demonstrated that not all semiquadratic bijections are equivalent to Dembowski-Ostrom monomials.

Produced non-Desarguesian commutative semifield planes.

Abstract

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 $\mathbb{P}^2(\mathbb{F}_q)$.