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

TL;DR

This paper provides a concise proof of a known inequality related to the maximum number of points on hypersurfaces over finite fields, explores a conjectural extension, and discusses applications to coding theory.

Contribution

It offers a short proof of Serre's inequality, examines the validity of a conjectural extension, and connects these results to coding theory applications.

Findings

The inequality by Serre is reaffirmed with a short proof.

The proposed extension conjecture is false in general but true in specific cases.

Applications to projective Reed-Muller codes are discussed.

Abstract

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on 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.