Common zeros of homogeneous polynomials and the weight hierarchy of projective Reed-Muller codes
Abstract
We determine the maximum number of common zeros in projective space of any set of linearly independent homogeneous polynomials of the same degree, when the degree is smaller than the cardinality of the finite field. This proves the conjecture of Beelen, Datta and Ghorpade and determines the complete weight hierarchy of the corresponding projective Reed-Muller codes. The proof uses Green's hyperplane restriction bound, combined with a footprint argument that reduces the problem to the affine case solved by Heijnen and Pellikaan. Wei duality then determines the hierarchy in sufficiently large degrees, recovering the complete hierarchy in every degree for the projective plane. Finally, we prove an upper bound valid in every degree, and conjecture that it is always sharp.
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BibTeX
@misc{sanjose2026common,
title = {{Common zeros of homogeneous polynomials and the weight hierarchy of projective Reed-Muller codes}},
author = {Rodrigo San-José},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08203},
eprint = {2611.08203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08203},
}
RIS
TY - UNPB
TI - Common zeros of homogeneous polynomials and the weight hierarchy of projective Reed-Muller codes
AU - Rodrigo San-José
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08203
UR - https://arxiv.org/abs/2611.08203
AB - We determine the maximum number of common zeros in projective space of any set of linearly independent homogeneous polynomials of the same degree, when the degree is smaller than the cardinality of the finite field. This proves the conjecture of Beelen, Datta and Ghorpade and determines the complete weight hierarchy of the corresponding projective Reed-Muller codes. The proof uses Green's hyperplane restriction bound, combined with a footprint argument that reduces the problem to the affine case solved by Heijnen and Pellikaan. Wei duality then determines the hierarchy in sufficiently large degrees, recovering the complete hierarchy in every degree for the projective plane. Finally, we prove an upper bound valid in every degree, and conjecture that it is always sharp.
ER -