Robust Sylvester-Gallai type theorem for quadratic polynomials

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 $δ>0$ such that for every $Q\in\mathcal{Q}$ there are at least $δm$ 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 $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/δ)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/δ)$ on the dimension (in the first work an upper bound of $O(1/δ^2)$ was given, which was improved to $O(1/δ)$ in the second work).

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