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

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

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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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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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
4. S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322
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