A generalized Sylvester-Gallai type theorem for quadratic polynomials

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of ${\mathrm{\Sigma }}^{[3]}\mathrm{\Pi }\mathrm{\Sigma }{\mathrm{\Pi }}^{[2]}$ circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials $\mathcal{Q}$ satisfy that for every two polynomials $Q_1,Q_2\in \mathcal{Q}$ there is a subset $\mathcal{K}\subset \mathcal{Q}$, such that $Q_1,Q_2\mathrm{\notin }\mathcal{K}$ and whenever $Q_1$ and $Q_2$ vanish then also ${\prod }_{i\in \mathcal{K}}{Q}_i$ vanishes, then the linear span of the polynomials in $\mathcal{Q}$ has dimension $O(1)$. This extends the earlier result [Shpilka19] that showed a similar conclusion when $\mathrm{\mid }\mathcal{K}\mathrm{\mid }=1$. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generates by the two quadratics. This step extends a result from [Shpilka19]that studied the case when one quadratic polynomial is in the radical of two other quadratics.