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

TL;DR

This paper investigates the maximum zero-dimensional intersection of multiple hypersurfaces in projective space, proves a conjecture for the case when the dimension is two, and applies results to coding theory, specifically projective Reed-Muller codes.

Contribution

It proposes a conjecture for the maximum intersection of hypersurfaces and proves it for the case when the ambient space dimension is two, linking algebraic geometry to coding theory.

Findings

Conjectured exact formula for maximum intersection dimension.

Proved the conjecture for the case m=2.

Computed generalized Hamming weights of PRM codes for m=2.

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 $m=2$.