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.
| 1 | Introduction | 1 |
| 2 | Preliminaries | 3 |
| 3 | TBC for Systems of Three Polynomial Equations | 5 |
| 4 | Projective Reed-Muller codes and their higher weights | 8 |
| 5 | A Counterexample to the TBC | 11 |
| Acknowledgments | 12 | |
| References | 12 |
This paper is published in: Arithmetic, Geometry, Cryptography and Coding Theory (Luminy, France, May 2015), A. Bassa, A. Couvreur and D. Kohel Eds., Contemporary Mathematics, Vol. 686, American Mathematical Society, Providence, RI, 2017, pp. 157--169.
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