Sylvester-Gallai type theorems for quadratic polynomials

Abstract

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal{Q}$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in \mathcal{Q}$ there is a third polynomial $Q_3\in \mathcal{Q}$ so that whenever $Q_1$ and $Q_2$ vanish then also $Q_3$ vanishes, then the linear span of the polynomials in $\mathcal{Q}$ has dimension $O(1)$. We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there is a polynomial in the third set satisfying the same vanishing condition then all polynomials are contained in an $O(1)$-dimensional space. This answers affirmatively two conjectures of Gupta [ECCC 2014] that were raised in the context of solving certain depth-$4$ polynomial identities. To obtain our main theorems we prove a new result classifying the possible ways that a quadratic polynomial $Q$ can vanish when two other quadratic polynomials vanish. Our proofs also require robust versions of a theorem of Edelstein and Kelly (that extends the Sylvester-Gallai theorem to colored sets).