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 Σ[3]ΠΣΠ[2] circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials Q satisfies that for every two polynomials Q1, Q2 ∈ Q there is a subset κ ⊂ Q such that Q1, Q2 ∈ κ and whenever Q1 and Q2 vanish, then Πi∈κ vanishes, then the linear span of the polynomials in Q has dimension Q(1). This extends the earlier result [21] that holds for the case |κ| = 1.