Sylvester-Gallai type theorems for quadratic polynomials
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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 Q_1,Q_2∈ Q there is a third polynomial Q_3∈ Q so that whenever Q_1 and Q_2 vanish then also Q_3 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 [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).
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