Maximal arcs in Desarguesian planes of odd order do not exist (Q1375053)

The title of the paper is exactly stating the contents of the paper. A maximal arc \({\mathcal K}\) of degree \(d\) in a projective plane of order \(n\) is a set of points such that each line of the plane intersects the set in 0 or \(d\) points (hence \(|{\mathcal K}|=nd-n+d\)). If \(1<d<n\), then all the known examples have \(d\) even (note that \(d\) has to divide \(n\)). It is a long standing conjecture that no maximal arcs of odd degree \(d\), \(1<d<n\), exist in projective planes of odd order \(n\). The authors prove that this conjecture is true in case of a Desarguesian plane, no use to say that this is an outstanding result. The proof (which is rather complicated) is based on the so-called polynomial technique. A more simplified proof by the first two authors is going to appear.