Arcs and ovals in infinite K-clan geometry

Abstract

For a finite field GF(q); to a q-clan of matrices there are associated generalized quadrangles, flocks of quadratic cones in PG(3, q), translation planes and, for q even, ovals in PG(2, q). The connections with generalized quadrangles, flocks and translation planes have recently been extended to the case of an infinite field K, under certain extra assumptions. In this note we extend the theory of ovals in PG(2, q) associated with q-clans, q even, to ovals in PG(2,K) associated with K-clans for (infinite) fields K of characteristic 2. Again, certain extra assumptions on the field K are made.