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A bound on the number of points of a plane curve. (English) Zbl 1185.14017
Summary: A conjecture is formulated for an upper bound on the number of points in PG\((2,q)\) of a plane curve without linear components, defined over GF\((q)\). We prove a new bound which is half-way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity.

MSC:
14G15 Finite ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
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[1] Ball, S.; Blokhuis, A., On the incompleteness of \((k, n)\)-arcs in Desarguesian planes of order q where n divides q, Geom. dedicata, 74, 325-332, (1999) · Zbl 0923.51006
[2] Barlotti, A., Sui \(\{k, n \}\)-archi di un piano lineare finito, Boll. unione mat. ital., 11, 553-556, (1956) · Zbl 0072.38103
[3] Blokhuis, A., On multiple nuclei and a conjecture of lunelli and sce, A tribute to J.A. thas, Gent, 1994, Bull. belg. math. soc. Simon stevin, 1, 349-353, (1994) · Zbl 0803.51012
[4] Hadnagy, É.; Szőnyi, T., On the embedding of large \((k, n)\)-arcs and partial unitals, Ars combin., 65, 299-308, (2002) · Zbl 1073.51502
[5] Hirschfeld, J.W.P., Projective geometries over finite fields, (1979), Clarendon Oxford, 2nd ed., 1998, 555 pp · Zbl 0418.51002
[6] Lunelli, L.; Sce, M., Considerazioni aritmetiche e risultati sperimentali sui \(\{K, n \}_q\)-archi, Istit. lombardo accad. sci. lett. rend. A, 98, 3-52, (1964) · Zbl 0131.36802
[7] Thas, J.A., Elementary proofs of two fundamental theorems of B. Segre without using the hasse – weil theorem, J. combin. theory ser. A, 34, 381-384, (1983) · Zbl 0517.51014
[8] Weiner, Zs., On \((k, p^e)\)-arcs in Desarguesian planes, Finite fields appl., 10, 390-404, (2004) · Zbl 1050.51007
[9] Weil, A., Sur LES courbes algébrique et LES varietés qui s’en déduisent, Actualités scientifiques et industrielles, vol. 1041, (1948), Herman & Cie Paris · Zbl 0036.16001
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