Space-filling subsets of a normal rational curve

✍ Scribed by G. Korchmáros; L. Storme; T. Szőnyi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 1009 KB
• Volume: 58
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(96)00063-8

✦ Synopsis

In this paper we show that the bisecants of an affinely regular n-gon inscribed in an ellipse of AG(2, q) cover all the points of AG(2, q) except the remaining points of the ellipse and possibly the center of the ellipse, if n >~ x/2q 3/4. This completes the previous investigations of Korehm~iros (1993) and Szrnyi (1987), who studied hyperbolas and parabolas. The results can be used to construct various complete plane arcs using the method of Segre (1962) and Lombardo-Radice (1956). It is also shown how these results can be used to prove the completeness of particular k-arcs in spaces PG(n, q) of n dimensions that share k-1 or k -2 points with a normal rational curve. The best results are obtained by combining algebraic techniques with probabilistic ideas.