Robust Sylvester-Gallai type theorem for quadratic polynomials

02/10/2022

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by   Shir Peleg, et al.

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In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if 𝒬⊂ℂ[x_1.…,x_n] is a finite set, |𝒬|=m, of irreducible quadratic polynomials that satisfy the following condition: There is δ>0 such that for every Q∈𝒬 there are at least δ m polynomials P∈𝒬 such that whenever Q and P vanish then so does a third polynomial in 𝒬∖{Q,P}, then (span(𝒬))=poly(1/δ). The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ^2) was given, which was improved to O(1/δ) in the second work).