Commutative Semifields from bijections of the Desarguesian plane

@misc{gologlu2026commutative,
  title = {{Commutative Semifields from bijections of the Desarguesian plane}},
  author = {Faruk Göloğlu and Lukas Kölsch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.14009},
  eprint = {2605.14009},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.14009},
  note = {CC0},
}

TY  - UNPB
TI  - Commutative Semifields from bijections of the Desarguesian plane
AU  - Faruk Göloğlu
AU  - Lukas Kölsch
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2605.14009
UR  - https://arxiv.org/abs/2605.14009
AB  - 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 ℙ^2(𝔽_q) is equivalent to a Dembowski-Ostrom monomial. In this paper, we give a large class of semiquadratic homogeneous bijections of ℙ^2(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 ℙ^1(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 ℙ^2(𝔽_q).
ER  -