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Artinian reduction applied to the Largest Intersection Problem
Abstract
Suppose we have r hypersurfaces in ℙm of degree d, whose defining polynomials are linearly independent and whose intersection is zero-dimensional. What is the maximum number of points in their intersection? This problem is motivated by computing the generalized Hamming weights of projective Reed-Muller codes, and a conjectured formula for this maximum was given in [21]. We prove the conjectured formula for arbitrary m when binomm+dd−binomd+22<r≤binomm+dd. We use the notion of Artinian reduction to reformulate the problem in terms of graded Artinian algebras and then apply combinatorial tools to bound their Hilbert functions.
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BibTeX
@misc{lin2026artinian,
title = {{Artinian reduction applied to the Largest Intersection Problem}},
author = {Yuxin Lin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01273},
eprint = {2611.01273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01273},
}
RIS
TY - UNPB
TI - Artinian reduction applied to the Largest Intersection Problem
AU - Yuxin Lin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01273
UR - https://arxiv.org/abs/2611.01273
AB - Suppose we have r hypersurfaces in ℙ^m of degree d, whose defining polynomials are linearly independent and whose intersection is zero-dimensional. What is the maximum number of points in their intersection? This problem is motivated by computing the generalized Hamming weights of projective Reed-Muller codes, and a conjectured formula for this maximum was given in [21]. We prove the conjectured formula for arbitrary m when binomm+dd−binomd+22<r≤binomm+dd. We use the notion of Artinian reduction to reformulate the problem in terms of graded Artinian algebras and then apply combinatorial tools to bound their Hilbert functions.
ER -