An oval partition of the central units of certain semifield planes Original Research Article

Let oxy denote a fixed autotopism triangle of a finite commutative semifield plane of even order qN, with middle nucleus GF(q). A point I ∉ {x, y, o} is called a central unit of the plane, relative to oxy, if coordinatising the plane by a semifield, in the standard way with I chosen as unit point and ox, oy as axes, yields a commutative semifield DI. It is shown that the set of all central units, relative to a fixed oxy, is partitioned by a set of q − 1 translation (hyper)ovals, any two of which share only the origin o as a common point. The full autotopism group acts transitively on these q − 1 translation ovals, and the ovals, together with the lines ox and oy, define a rational Desarguesian net of degree q.