Remarks on Tsfasman-Boguslavsky Conjecture and Higher Weights of Projective Reed-Muller Codes

TL;DR

This paper investigates the Tsfasman-Boguslavsky Conjecture regarding the maximum number of zeros of polynomial systems over finite fields, proving it for some cases, providing counterexamples for others, and exploring implications for projective Reed-Muller codes.

Contribution

It provides a self-contained proof confirming the conjecture for three polynomials, presents a counterexample with five quadrics, and connects these results to the weights of Reed-Muller codes.

Findings

Conjecture holds for systems of three homogeneous polynomials.

Counterexample found for five quadrics in 3D projective space.

Connections established between the conjecture and code weight determination.

Abstract

Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective space. We give a self-contained proof to show that this conjecture holds in the affirmative in the case of systems of three homogeneous polynomials, and also to show that the conjecture is false in the case of five quadrics in the 3-dimensional projective space over a finite field. Connections between the Tsfasman-Boguslavsky Conjecture and the determination of generalized Hamming weights of projective Reed-Muller codes are outlined and these are also exploited to show that this conjecture holds in the affirmative in the case of systems of a "large" number of three homogeneous polynomials, and to deduce the counterexample of 5 quadrics. An application to the nonexistence of lines in certain Veronese varieties over finite fields is also included.