Commutative Semifields from bijections of the Desarguesian plane
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913126071402496 |
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| author | Göloğlu, Faruk Kölsch, Lukas |
| author_facet | Göloğlu, Faruk Kölsch, Lukas |
| contents | 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 $\mathbb{P}^2(\mathbb{F}_q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commutative Semifields from bijections of the Desarguesian plane Göloğlu, Faruk Kölsch, Lukas Combinatorics Commutative Algebra 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 $\mathbb{P}^2(\mathbb{F}_q)$. |
| title | Commutative Semifields from bijections of the Desarguesian plane |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2605.14009 |