Abstract
The nonsingular Hermitian surface of degree \({\sqrt{q} +1}\) is characterized by its number of \({\mathbb{F}_q}\) -points among the surfaces over \({\mathbb{F}_q}\) of degree \({\sqrt{q} +1}\) in the projective 3-space without \({\mathbb{F}_q}\) -plane components.
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Bose R.C., Chakravarti I.M.: Hermitian varieties in a finite projective space PG(N, q 2). Can. J. Math. 18, 1161–1182 (1966)
Deligne P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)
Hirschfeld J.W.P., Storme L., Thas J.A., Voloch J.F.A.: A characterization of Hermitian curves. J. Geom. 41, 72–78 (1991)
Homma, M.: A bound on the number of points of a curve in a projective space over a finite field. In: Lavrauw, M., Mullen, G.L., Nikova, S., Panario, D., Storme, L. (eds.) Theory and Applications of Finite Fields, pp. 103–110. Contemporary Mathematics, vol. 579. AMS, Providence (2012)
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)
Homma, M., Kim, S.J.: Sziklai’s conjecture on the number of points of a plane curve over a finite field II. In: McGuire, G., Mullen, G.L., Panario, D., Shparlinski, I.E. (eds.) Finite Fields: Theory and Applications, pp. 225–234. Contemporary Mathematics, vol. 518. AMS, Providence (2010) (An update is available at arXiv:0907.1325v2)
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)
Homma M., Kim S.J.: An elementary bound for the number of points of a hypersurface over a finite field. Finite Fields Appl. 20, 76–83 (2013)
Homma, M., Kim, S.J.: Numbers of points of surfaces in the projective 3-space over finite fields. Finite Fields Appl. 35, 52–60 (2015) (A previous version is available at arXiv:1409.5842v1)
Rück H.-G., Stichtenoth H.: A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math. 457, 185–188 (1994)
Segre B.: Le geometrie di Galois. Ann. Mat. Pura Appl. (4) 48, 1–96 (1959)
Sørensen, A.B. : On the number of rational points on codimension-1 algebraic sets in P n(F q ). Discrete Math. 135, 321–334 (1994)
Sziklai, P.: A bound on the number of points of a plane curve. Finite Fields Appl. 14, 41–43 (2008)
Tironi, A.L.: Hypersurfaces achieving the Homma–Kim bound (2014, preprint). Available at arXiv:1410.7320v2
Weil, A.: Numbers of solutions of equations in finite fields. Bull. Am. Math. Soc. 55, 497–508 (1949)
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M. Homma was partially supported by Grant-in-Aid for Scientific Research (24540056), JSPS.
S. J. Kim was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042228).
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Homma, M., Kim, S.J. The characterization of Hermitian surfaces by the number of points. J. Geom. 107, 509–521 (2016). https://doi.org/10.1007/s00022-015-0283-1
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DOI: https://doi.org/10.1007/s00022-015-0283-1