Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields

Beelen, Peter; Datta, Mrinmoy; Ghorpade, Sudhir

doi:10.1090/proc/13863

Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields
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by Peter Beelen, Mrinmoy Datta and Sudhir R. Ghorpade PDF

Proc. Amer. Math. Soc. 146 (2018), 1451-1468 Request permission

Abstract:

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can have in the corresponding $m$-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise. Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. In this paper, we prove that this new conjecture holds true for several values of $r$. In particular, this settles the new conjecture completely when $d=3$. Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of $d=q-1$ and of $d=q$. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes.

References

M. Boguslavsky, On the number of solutions of polynomial systems, Finite Fields Appl. 3 (1997), no. 4, 287–299. MR 1478830, DOI 10.1006/ffta.1997.0186
Alain Couvreur, An upper bound on the number of rational points of arbitrary projective varieties over finite fields, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3671–3685. MR 3513530, DOI 10.1090/proc/13015
Mrinmoy Datta and 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 (2015), no. 4, 715–725. MR 3438829, DOI 10.17323/1609-4514-2015-15-4-715-725
Mrinmoy Datta and Sudhir R. Ghorpade, Number of solutions of systems of homogeneous polynomial equations over finite fields, Proc. Amer. Math. Soc. 145 (2017), no. 2, 525–541. MR 3577858, DOI 10.1090/proc/13239
Mrinmoy Datta and Sudhir R. Ghorpade, Remarks on the Tsfasman-Boguslavsky Conjecture and higher weights of projective Reed-Muller codes, Arithmetic, geometry, cryptography and coding theory, Contemp. Math., vol. 686, Amer. Math. Soc., Providence, RI, 2017, pp. 157–169. MR 3630614, DOI 10.1090/conm/686
Petra Heijnen and Ruud Pellikaan, Generalized Hamming weights of $q$-ary Reed-Muller codes, IEEE Trans. Inform. Theory 44 (1998), no. 1, 181–196. MR 1486657, DOI 10.1109/18.651015
Masaaki Homma and Seon Jeong Kim, An elementary bound for the number of points of a hypersurface over a finite field, Finite Fields Appl. 20 (2013), 76–83. MR 3015353, DOI 10.1016/j.ffa.2012.11.002
Gilles Lachaud and Robert Rolland, On the number of points of algebraic sets over finite fields, J. Pure Appl. Algebra 219 (2015), no. 11, 5117–5136 (English, with English and French summaries). MR 3351576, DOI 10.1016/j.jpaa.2015.05.008
Jean-Pierre Serre, Lettre à M. Tsfasman, Astérisque 198-200 (1991), 11, 351–353 (1992) (French, with English summary). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144337
Anders Bjært Sørensen, Projective Reed-Muller codes, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1567–1576. MR 1134296, DOI 10.1109/18.104317
Corrado Zanella, Linear sections of the finite Veronese varieties and authentication systems defined using geometry, Des. Codes Cryptogr. 13 (1998), no. 2, 199–212. MR 1600208, DOI 10.1023/A:1008286614783