Sylvester-Gallai type theorems for quadratic polynomials

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection Q, of irreducible polynomials of degree at most 2, satisfy that for every two polynomials Q₁,Q₂ ∈ Q there is a third polynomial Q₃ ∈ Q so that whenever Q₁ and Q₂ vanish then also Q3 vanishes, then the linear span of the polynomials in 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 [Gup14] 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.