Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces

Yuxin Lin; Deepesh Singhal

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arXiv · 2507.05732

Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces

Yuxin Lin·Deepesh Singhal

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Abstract

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 $m=2$.

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Yuxin Lin, Deepesh Singhal. 2025-07-08. Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces. https://arxiv.org/abs/2507.05732

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