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arXiv · 1705.10185

Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields

Peter Beelen·Mrinmoy Datta·Sudhir R. Ghorpade
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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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Peter Beelen, Mrinmoy Datta, Sudhir R. Ghorpade. 2017-05-29. Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields. https://doi.org/10.1090/proc%2F13863

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