2026/05/31 by Faruk Göloğlu, Lukas Kölsch
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Polynomial and algebraic computation #Advanced Topics in Algebra
paper · doi:10.1112/jlms.70635
Abstract The Menichetti–Kaplansky theorem ( J. Algebra 47(2) (1977)) 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 is equivalent to a Dembowski–Ostrom monomial. In this paper, we give a large class of semiquadratic homogeneous bijections of that are inequivalent to Dembowski–Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are nonisotopic 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 have been classified only recently by the first‐named author ( Finite Fields Appl . 81 (2022)) and Ding and Zieve ( Proc. Lond. Math. Soc . 127(2) (2023)) 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 .