  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
$\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$ 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 $Q_1,Q_2 \notin \mathcal{K}$ 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 $|\mathcal{K}| = 1$.
  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.
