Homma, Masaaki; Kim, Seon Jeong Sziklai’s conjecture on the number of points of a plane curve over a finite field. III. (English) Zbl 1196.14030 Finite Fields Appl. 16, No. 5, 315-319 (2010). Summary: We manage an upper bound for the number of rational points of a Frobenius nonclassical plane curve over a finite field. Together with previous results, the modified Sziklai conjecture is settled affirmatively.For part II, cf. [Contemp. Math. 518, 225–234 (2010; Zbl 1211.14037)]. Cited in 5 ReviewsCited in 15 Documents MSC: 14H50 Plane and space curves 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14N10 Enumerative problems (combinatorial problems) in algebraic geometry Keywords:plane curve; finite field; rational point; Frobenius nonclassical curve PDF BibTeX XML Cite \textit{M. Homma} and \textit{S. J. Kim}, Finite Fields Appl. 16, No. 5, 315--319 (2010; Zbl 1196.14030) Full Text: DOI References: [1] Hefez, A.; Voloch, J.F., Frobenius nonclassical curves, Arch. math. (basel), Arch. math. (basel), 57, 416-273, (1991), correction: · Zbl 0758.14015 [2] Hirschfeld, J.W.P.; Korchmáros, G., On the number of solutions of an equation over a finite field, Bull. lond. math. soc., 33, 16-24, (2001) · Zbl 1047.11061 [3] Hirschfeld, J.W.P.; Korchmáros, G.; Torres, F., Algebraic curves over a finite field, (2008), Princeton Univ. Press Princeton, NJ · Zbl 1200.11042 [4] Homma, M.; Kim, S.J., Around Sziklai’s conjecture on the number of points of a plane curve over a finite field, Finite fields appl., 15, 468-474, (2009) · Zbl 1194.14031 [5] M. Homma, S.J. Kim, Sziklai’s conjecture on the number of points of a plane curve over a finite field II, in: G. McGuire, G.L. Mullen, D. Panario, I.E. Shparlinski (Eds.), Finite Fields: Theory and Applications, in: Contemp. Math., vol. 518, AMS, Providence, RI, 2010, in press; an earlier version is available at arXiv:0907.1325. · Zbl 1211.14037 [6] M. Homma, S.J. Kim, Toward determination of optimal plane curves with a fixed degree over a finite field, preprint, 2010. [7] Stöhr, K.-O.; Voloch, J.F., Weierstrass points and curves over finite fields, Proc. lond. math. soc. (3), 52, 1-19, (1986) · Zbl 0593.14020 [8] Sziklai, P., A bound on the number of points of a plane curve, Finite fields appl., 14, 41-43, (2008) · Zbl 1185.14017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.