Arcs in Desarguesian nets


Journal article


A. Beato, G. Faina, M. Giulietti
Contributions Discret. Math., 2008

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APA   Click to copy
Beato, A., Faina, G., & Giulietti, M. (2008). Arcs in Desarguesian nets. Contributions Discret. Math.


Chicago/Turabian   Click to copy
Beato, A., G. Faina, and M. Giulietti. “Arcs in Desarguesian Nets.” Contributions Discret. Math. (2008).


MLA   Click to copy
Beato, A., et al. “Arcs in Desarguesian Nets.” Contributions Discret. Math., 2008.


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@article{a2008a,
  title = {Arcs in Desarguesian nets},
  year = {2008},
  journal = {Contributions Discret. Math.},
  author = {Beato, A. and Faina, G. and Giulietti, M.}
}

Abstract

A trivial upper bound on the size k of an arc in an r-net is $k \leq r + 1$. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case $k = r + 1$ cannot occur, and $k \geq r - 1$ implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference $r - k$ does not exceed $\sqrt{k/18}$. Moreover, it is proved that the same assumption ensures that the arc can be extended to an oval of the net.


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