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An oval partition of the central units of certain semifield planes

✍ Scribed by V. Jha; G. Wene

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 343 KB
Volume: 155
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(94)00376-t

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✦ Synopsis

Let oxy denote a fixed autotopism triangle of a finite commutative semifield plane of even order qjV, with middle nucleus GF(q). A point 1 f[ {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/91. 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.

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