Robust Radical Sylvester-Gallai Theorem for Quadratics

TL;DR

This paper extends the Sylvester-Gallai theorem to quadratic polynomials, showing that configurations with a certain radical intersection property are necessarily low-dimensional, which has implications for algebraic geometry and combinatorics.

Contribution

It introduces a robust generalization of the Sylvester-Gallai theorem for quadratic polynomials, establishing low-dimensionality for configurations with radical intersection properties.

Findings

Configurations are of low dimension, polynomial in 1/δ.

Generalizes Sylvester-Gallai theorem to quadratic polynomials.

Provides bounds on the dimension of radical Sylvester-Gallai configurations.

Abstract

We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials, generalizing the result in [S'20]. More precisely, given a parameter $0 < \delta \leq 1$ and a finite collection $\mathcal{F}$ of irreducible and pairwise independent polynomials of degree at most 2, we say that $\mathcal{F}$ is a $(\delta, 2)$-radical Sylvester-Gallai configuration if for any polynomial $F_i \in \mathcal{F}$, there exist $\delta(|\mathcal{F}| -1)$ polynomials $F_j$ such that $|\mathrm{rad}(F_i, F_j) \cap \mathcal{F}| \geq 3$, that is, the radical of $F_i, F_j$ contains a third polynomial in the set. In this work, we prove that any $(\delta, 2)$-radical Sylvester-Gallai configuration $\mathcal{F}$ must be of low dimension: that is $$\dim \mathrm{span}(\mathcal{F}) = \mathrm{poly}(1/\delta).$$