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The second largest number of points on plane curves over finite fields. (English) Zbl 1411.11059
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 \(d\geq q+1\).

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
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[1] Hirschfeld, J. W.P.; Korchm√°ros, G.; Torres, F., Algebraic curves over a finite field, (2008), 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, Contemp. Math., vol. 518, (2010), 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
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