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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
##### Keywords:
finite field; basis of the ideal; plane curve
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##### References:
 [1] Hirschfeld, J. W.P.; Korchmáros, 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 [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
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