On (k,pe)-arcs in Desarguesian planes

In this paper we improve Sz nyiʹs embeddability result on (k,p)-arcs, in PG(2,q), q=ph, p prime. Sz nyi proved that for k>qp−q+p− , , a (k,p)-arc can be embedded in a maximal arc. Our main theorem is that this result can be extended for , furthermore it can be generalized to (k,pe)-arcs, . This and the result of Ball, Blokhuis and Mazzocca on the non-existence of maximal arcs for p>2, yields an upper bound on the size of a (k,pe)-arc. In the particular case p=2, Segre showed that when , any k-arc can be extended to a hyperoval. This result is sharp, since there are complete arcs of size . A new proof for Segreʹs theorem is also presented.