An oval partition of the central units of certain semifield planes
Let the translation plane \(\pi\) be coordinatised by the commutative semified \((D,+, \circ)\) of even order \(q^N\) and assume that the middle nucleus of \(D\) is isomorphic to \(GF(q)\). An affine point \(I\) of \(\pi\) not on the coordinate axes is called a central unit if the semifield obtained from \(D\) by recoordinatising \(\pi\) with \(I\) as unit point is also commutative. The authors show that all nonzero points on the translation oval \(\{(x,x\circ x)\mid x\in D\}\) are central units and that all other central units are obtained by applying middle nucleus homologies to this translation oval. It follows that the set of central units is partitioned into \(q-1\) translation ovals minus the origin.
scientific article; zbMATH DE number 932532
scientific article