Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909679148335104 |
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| author | Singhal, Deepesh Lin, Yuxin |
| author_facet | Singhal, Deepesh Lin, Yuxin |
| contents | Suppose we have $r$ hypersurfaces in $\mathbb{P}^m$ of degree $d$, whose defining polynomials are linearly independent, and their intersection has dimension $0$. Then what is the largest possible intersection of the $r$ hypersurfaces? We conjecture an exact formula for this problem and prove it when $m=2$. We show that this can be used to compute the generalized hamming weights of the projective Reed-Muller code $\operatorname{PRM}_q(d,2)$ and hence settle a conjecture of Beelen, Datta and Ghorpade for $m=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces Singhal, Deepesh Lin, Yuxin Algebraic Geometry Combinatorics Number Theory Suppose we have $r$ hypersurfaces in $\mathbb{P}^m$ of degree $d$, whose defining polynomials are linearly independent, and their intersection has dimension $0$. Then what is the largest possible intersection of the $r$ hypersurfaces? We conjecture an exact formula for this problem and prove it when $m=2$. We show that this can be used to compute the generalized hamming weights of the projective Reed-Muller code $\operatorname{PRM}_q(d,2)$ and hence settle a conjecture of Beelen, Datta and Ghorpade for $m=2$. |
| title | Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces |
| topic | Algebraic Geometry Combinatorics Number Theory |
| url | https://arxiv.org/abs/2507.05732 |