A generalized Sylvester–Gallai-type theorem for quadratic polynomials

Abstract In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining deterministic polynomial time algorithm testing zeroness $\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$ circuits. Specifically, that, if finite set irreducible ${\mathcal {Q}}$ satisfies every two $Q_1,Q_2\in {\mathcal there is subset {K}}\subset such $Q_1,Q_2 \notin {K}}$ and whenever $Q_1$ $Q_2$ vanish, then $\prod _{i\in {K}}} Q_i$ vanishes, linear span in has dimension $O(1)$ . This extends earlier result [21] holds case $|{\mathcal {K}}| = 1$