Robust Radical Sylvester-Gallai Theorem for Quadratics

03/10/2022

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by   Abhibhav Garg, et al.

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We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials, generalizing the result in [S'20]. More precisely, given a parameter 0 < δ≤ 1 and a finite collection ℱ of irreducible and pairwise independent polynomials of degree at most 2, we say that ℱ is a (δ, 2)-radical Sylvester-Gallai configuration if for any polynomial F_i ∈ℱ, there exist δ(|ℱ| -1) polynomials F_j such that |rad(F_i, F_j) ∩ℱ| ≥ 3, that is, the radical of F_i, F_j contains a third polynomial in the set. In this work, we prove that any (δ, 2)-radical Sylvester-Gallai configuration ℱ must be of low dimension: that is span(ℱ) = poly(1/δ).