Robust Radical Sylvester-Gallai Theorem for Quadratics

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 $0<\delta \le 1$ 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 $|\mathrm{rad}(F_i,F_j)\cap \mathcal{F}|\ge 3$, 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 $$\dim \mathrm{span}(\mathcal{F})=\mathrm{poly}(1/\delta ).$$

41 pages