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
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Additional Information
- Peter Beelen
- Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK 2800, Kgs. Lyngby, Denmark
- MR Author ID: 672478
- Email: pabe@dtu.dk
- Mrinmoy Datta
- Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK 2800, Kgs. Lyngby, Denmark
- MR Author ID: 1120609
- ORCID: 0000-0003-1138-0953
- Email: mrinmoy.dat@gmail.com
- Sudhir R. Ghorpade
- Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
- MR Author ID: 306883
- ORCID: 0000-0002-6516-3623
- Email: srg@math.iitb.ac.in
- Received by editor(s): August 12, 2016
- Received by editor(s) in revised form: May 29, 2017
- Published electronically: December 7, 2017
- Communicated by: Mathew A. Papanikolas
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1451-1468
- MSC (2010): Primary 14G15, 11T06, 11G25, 14G05; Secondary 51E20, 05B25
- DOI: https://doi.org/10.1090/proc/13863
- MathSciNet review: 3754333