Robust Sylvester-Gallai type theorem for quadratic polynomials

Abstract

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 δm polynomials P ∈ Q such that whenever Q and P vanish then so does a third polynomial in Q \ {Q, P}, then dim(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).