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Mosc. Math. J., 2015, Volume 15, Number 4, Pages 715–725 (Mi mmj582)  

This article is cited in 5 scientific papers (total in 5 papers)

On a conjecture of Tsfasman and an inequality of Serre for the number of points of hypersurfaces over finite fields

Mrinmoy Datta, Sudhir R. Ghorpade

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

Abstract: We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points of hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality to an explicit formula for the maximum number of common solutions of a system of linearly independent multivariate homogeneous polynomials of the same degree with coefficients in a finite field. This conjecture is shown to be false, in general, but is also shown to hold in the affirmative in a special case. Applications to generalized Hamming weights of projective Reed–Muller codes are outlined and a comparison with an older conjecture of Lachaud and a recent result of Couvreur is given.

Key words and phrases: hypersurface, rational point, finite field, Veronese variety, Reed–Muller code, generalized Hamming weight.

Funding Agency Grant Number
Russian Foundation for Basic Research INT/RFBR/P-114
IITB 12IRAWD009
NBHM
The first named author was supported in part by a doctoral fellowship from the National Board for Higher Mathematics, a division of the Department of Atomic Energy, Govt. of India. The second named author was supported in part by Indo-Russian project INT/RFBR/P-114 from the Department of Science & Technology, Govt. of India and IRCC Award grant 12IRAWD009 from IIT Bombay.


DOI: https://doi.org/10.17323/1609-4514-2015-15-4-715-725

Full text: http://www.mathjournals.org/.../2015-015-004-008.html
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Bibliographic databases:

MSC: Primary 14G15, 11G25, 14G05; Secondary 11T27, 94B27, 51E20
Received: April 4, 2015; in revised form September 28, 2015
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Citation: Mrinmoy Datta, Sudhir R. Ghorpade, “On a conjecture of Tsfasman and an inequality of Serre for the number of points of hypersurfaces over finite fields”, Mosc. Math. J., 15:4 (2015), 715–725

Citation in format AMSBIB
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\by Mrinmoy~Datta, Sudhir~R.~Ghorpade
\paper On a~conjecture of Tsfasman and an inequality of Serre for the number of points of hypersurfaces over finite fields
\jour Mosc. Math.~J.
\yr 2015
\vol 15
\issue 4
\pages 715--725
\mathnet{http://mi.mathnet.ru/mmj582}
\crossref{https://doi.org/10.17323/1609-4514-2015-15-4-715-725}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3438829}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Couvreur, “An upper bound on the number of rational points of arbitrary projective varieties over finite fields”, Proc. Amer. Math. Soc., 144:9 (2016), 3671–3685  crossref  mathscinet  zmath  isi  scopus
    2. M. Datta, S. R. Ghorpade, “Number of solutions of systems of homogeneous polynomial equations over finite fields”, Proc. Amer. Math. Soc., 145:2 (2017), 525–541  crossref  mathscinet  zmath  isi  scopus
    3. M. Datta, S. R. Ghorpade, “Remarks on the Tsfasman–Boguslavsky conjecture and higher weights of projective Reed–Muller codes”, Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, 686, eds. A. Bassa, A. Couvreur, D. Kohel, Amer. Math. Soc., 2017, 157–169  crossref  mathscinet  zmath  isi  scopus
    4. S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. P. Beelen, M. Datta, S. R. Ghorpade, “Maximum number of common zeros of homogeneous polynomials over finite fields”, Proc. Amer. Math. Soc., 146:4 (2018), 1451–1468  crossref  mathscinet  zmath  isi  scopus
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