Complete arcs containing \((q+1)/2\) points of a conic in Galois fields of odd order
Given a \(k\)-arc \(H\) in a projective plane, a point of the plane is said to be \(H\)-free if no secant of \(H\) (chord of \(H\)) passes through the point. The author investigates the following problem. In a Desarguesian plane, \(PG(2,q)\), \(q\) odd, let \(\Gamma\) be a non-degenerate conic and let \(H\) be a set of \((q+1)/2\) points of \(\Gamma\) such that there exist at least two \(H\)-free interior points to \(\Gamma\) on an exterior line of \(\Gamma\). Determine the orders of complete arcs other than \(\Gamma\) which contain \(H\). Letting \(A\) and \(B\) denote the hypothesized points and \(O\) the pole (with respect to \(\Gamma\)) of the line containing them, the author shows that when \(q>13\) an \(H\) can be constructed so that, \(H\cup\{A,B\}\) is a complete arc, and if \(q=4t+1\), \(H\cup\{O\}\) is also a complete arc. The technique involves introducing coordinates drawn from the ring of residues modulo \((q+1)\), and using some number-theoretic properties of certain sum-free sets in that ring. This paper is a continuation of earlier work of the author and there is a heavy reliance on these previous papers.
scientific article; zbMATH DE number 490577
scientific article