Robust Sylvester-Gallai type theorem for quadratic polynomials

TL;DR

This paper extends the robust Sylvester-Gallai theorem to quadratic polynomials, showing that under certain conditions, the span of such polynomials has a dimension bounded polynomially in the inverse of a parameter delta.

Contribution

It generalizes the robust Sylvester-Gallai theorem from linear to quadratic polynomials, establishing a polynomial bound on the dimension of their span.

Findings

Dimension of quadratic polynomial set is polynomial in 1/delta.

Conditions ensure a structured dependency among polynomials.

Extends previous linear polynomial results to quadratic case.

Abstract

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if $\mathcal{Q}\subset \mathbb{C}[x_1.\ldots,x_n]$ is a finite set, $|\mathcal{Q}|=m$, of irreducible quadratic polynomials that satisfy the following condition: There is $\delta>0$ such that for every $Q\in\mathcal{Q}$ there are at least $\delta m$ polynomials $P\in \mathcal{Q}$ such that whenever $Q$ and $P$ vanish then so does a third polynomial in $\mathcal{Q}\setminus\{Q,P\}$, then $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/\delta)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/\delta)$ on the dimension (in the first work an upper bound of $O(1/\delta^2)$ was given, which was improved to $O(1/\delta)$ in the second work).