Solution to an isotopism question concerning rank 2 semifields

TL;DR

This paper resolves an open problem by determining the isotopism classes of certain rank 2 semifields constructed from Dembowski-Ostrom polynomials, establishing the novelty of two classes for dimensions greater than three.

Contribution

It proves that two classes of semifields are new for n>3, while one class is isotopic to known structures, clarifying their classification.

Findings

Classes D_A and D_AB are new for n>3.

Family D_B semifields are isotopic to Generalized Twisted Fields.

The paper completes the classification of these semifields.

Abstract

In [U. Dempwolff: More Translation Planes and Semifields from Dembowski-Ostrom Polynomials, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81-103], the author gives a construction of three classes of rank two semifields of order $q^{2n}$, with $q$ and $n$ odd, using Dembowski-Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for $n>3$, in particular we prove that two of these classes, labeled $D_{A}$ and $D_{AB}$, are new for $n>3$, whereas presemifields in family $D_{B}$ are isotopic to Generalized Twisted Fields for each $n\geq 3$.