Space-filling subsets of a normal rational curve
A normal rational curve in \(PG(n,q)\), \(2 \leq n \leq q-2\), is a set of \(q+1\) points projectively equivalent to \(\{(1,t,\ldots,t^n)||t \in GF(q)\} \cup \{(0,\ldots,0,1)\}\). It is the classical example of a \(k\)-arc in \(PG(n,q)\); that is, a set of \(k\) points of which every \(n+1\) are linearly independent. One of the main problems in the study of arcs is the problem of finding other examples of \(k\)-arcs. A method to obtain new examples of arcs is to construct \(k\)-arcs consisting of a large number of points of a normal rational curve \(L\), plus some extra points not belonging to \(L\). This article discusses three particular classes of arcs sharing approximately \(q/2\) points with a normal rational curve. They have size \(q/2+2\) in \(PG(n,q)\), \(q\) even, \(n \geq 4\), and sizes \((q+3)/2\) and \((q+5)/2\) in \(PG(n,q)\), \(q\) odd, \(n \geq 3\). For large values of \(q\) and \(4 \leq n \leq 0.09q\), it is known by results of Storme and Szönyi that these arcs have the largest possible intersection with a normal rational curve. Moreover, these arcs are not extendable to larger arcs. Here, the non-extendability of these arcs to larger arcs again is studied. It is shown that these three types of arcs are non-extendable to larger arcs for all dimensions \(4 \leq n \leq (q-3)/2 -\sqrt{7(q+1)\log q}\). This improves greatly the previous results by Storme and Szönyi on the non-extendability of these arcs. To obtain these results, first of all a detailed study of the bisecants of an affinely regular \(n\)-gon inscribed in an ellipse is made. This, together with previously obtained results by Korchmáros and Szönyi for the hyperbolas and parabolas, is combined with statistical arguments, in order to obtain the improvements.
scientific article; zbMATH DE number 1040404
scientific article