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Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields


Authors: Peter Beelen, Mrinmoy Datta and Sudhir R. Ghorpade
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
Published electronically: December 7, 2017
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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.


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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
Email: pabe@dtu.dk

Mrinmoy Datta
Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK 2800, Kgs. Lyngby, Denmark
Email: mrinmoy.dat@gmail.com

Sudhir R. Ghorpade
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email: srg@math.iitb.ac.in

DOI: https://doi.org/10.1090/proc/13863
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
Article copyright: © Copyright 2017 American Mathematical Society

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