Classification of 8-dimensional rank two commutative semifields

TL;DR

This paper classifies 8-dimensional rank two commutative semifields over finite fields using computational methods linked to linear sets, and extends the classification to 10-dimensional cases over , with implications for various geometric structures.

Contribution

It introduces a computational approach to classify rank two commutative semifields in specific dimensions and extends the classification to new cases, connecting algebraic and geometric structures.

Findings

Classified 8-dimensional rank two commutative semifields.

Extended classification to 10-dimensional cases over .

Connected semifield classifications to geometric structures like ovoids and eggs.

Abstract

We classify the rank two commutative semifields which are 8-dimensional over their center $\mathbb{F}_{q}$. This is done using computational methods utilizing the connection to linear sets in $\mathrm{PG}(2,q^{4})$. We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over $\mathbb{F}_{3}$. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.