# Moscow Mathematical Journal

Volume 15, Issue 4, October–December 2015 pp. 715–725.

On a Conjecture of Tsfasman and an Inequality of Serre for the Number of Points of Hypersurfaces over Finite Fields

**Authors**:
Mrinmoy Datta and Sudhir R. Ghorpade

**Author institution:**Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

**Summary: **

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.

2010 Math. Subj. Class. Primary: 14G15, 11G25, 14G05; Secondary: 11T27, 94B27, 51E20.

**Keywords:**Hypersurface, rational point, finite field, Veronese variety, Reed–Muller code, generalized Hamming weight.

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