Abstract
We study the rationality of the intersection points of certain lines and smooth plane quartics C defined over \(\mathbb{F}_q\). For q ≥ 127, we prove the existence of a line such that the intersection points with C are all rational. Using another approach, we further prove the existence of a tangent line with the same property as soon as char\(\mathbb{F}_q \neq 2\) and q ≥ 662 + 1. Finally, we study the probability of the existence of a rational flex on C and exhibit a curious behavior when char \(\mathbb{F}_q=3\).
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Oyono, R., Ritzenthaler, C. (2010). On Rationality of the Intersection Points of a Line with a Plane Quartic. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_16
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