### Abstract

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≥√2q^{3/4}. This completes the previous investigations of Korchmáros (1993) and Szonyi (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.

Original language | English |
---|---|

Pages (from-to) | 93-110 |

Number of pages | 18 |

Journal | Journal of Statistical Planning and Inference |

Volume | 58 |

Issue number | 1 |

Publication status | Published - Mar 1 1997 |

### Fingerprint

### Keywords

- Affine regular polygons
- Arcs in n dimensions
- Complete arcs
- Normal rational curves

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Journal of Statistical Planning and Inference*,

*58*(1), 93-110.

**Space-filling subsets of a normal rational curve.** / Korchmáros, G.; Storme, L.; Szőnyi, T.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol. 58, no. 1, pp. 93-110.

}

TY - JOUR

T1 - Space-filling subsets of a normal rational curve

AU - Korchmáros, G.

AU - Storme, L.

AU - Szőnyi, T.

PY - 1997/3/1

Y1 - 1997/3/1

N2 - 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≥√2q3/4. This completes the previous investigations of Korchmáros (1993) and Szonyi (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.

AB - 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≥√2q3/4. This completes the previous investigations of Korchmáros (1993) and Szonyi (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.

KW - Affine regular polygons

KW - Arcs in n dimensions

KW - Complete arcs

KW - Normal rational curves

UR - http://www.scopus.com/inward/record.url?scp=0031093569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031093569&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031093569

VL - 58

SP - 93

EP - 110

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 1

ER -