Homma, Masaaki; Kim, Seon Jeong The second largest number of points on plane curves over finite fields. (English) Zbl 1411.11059 Finite Fields Appl. 49, 80-93 (2018). Summary: A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree \(d\) over the finite field of \(q\) elements is also given for \(d\geq q+1\). Cited in 2 Documents MSC: 11G20 Curves over finite and local fields 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 14G15 Finite ground fields in algebraic geometry 14N05 Projective techniques in algebraic geometry Keywords:finite field; basis of the ideal; plane curve PDFBibTeX XMLCite \textit{M. Homma} and \textit{S. J. Kim}, Finite Fields Appl. 49, 80--93 (2018; Zbl 1411.11059) Full Text: DOI arXiv References: [1] Hirschfeld, J. W.P.; Korchmáros, G.; Torres, F., Algebraic Curves over a Finite Field (2008), Princeton Univ. Press: Princeton Univ. Press Princeton and Oxford · Zbl 1200.11042 [2] 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 [3] Homma, M.; Kim, S. J., Sziklai’s conjecture on the number of points of a plane curve over a finite field II, (McGuire, G.; Mullen, G. L.; Panario, D.; Shparlinski, I. E., Finite Fields: Theory and Applications. Finite Fields: Theory and Applications, Contemp. Math., vol. 518 (2010), AMS: AMS Providence), 225-234, An update is available at · Zbl 1211.14037 [4] Homma, M.; Kim, S. J., Sziklai’s conjecture on the number of points of a plane curve over a finite field III, Finite Fields Appl., 16, 315-319 (2010) · Zbl 1196.14030 [5] Homma, M.; Kim, S. J., Toward determination of optimal plane curves with a fixed degree over a finite field, Finite Fields Appl., 17, 240-253 (2011) · Zbl 1215.14033 [6] Homma, M.; Kim, S. J., The characterization of Hermitian surfaces by the number of points, J. Geom., 107, 509-521 (2016) · Zbl 1410.11081 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.