Seminar in Real and Complex Geometry

Thursday, April 11, 2019, 16:00-18:00, Schreiber building, room 209




Shir Peleg (Tel Aviv University)

Sylvester-Gallai type theorem for quadratic polynomials


Abstract
             

One formulation of the famous Sylvester-Gallai theorem states that if a finite set of linear forms L satisfy that for every two forms, L1, L2 in L there is a subset I of L, such that L1, L2 not in I and whenever L1 and L2 vanish then also the product of Li vanishes, then the linear span ofL has dimension at most 3. This version of the Sylvester-Gallai theorem was used to device deterministic algorithms for testing depth-3 polynomial identities. In this work we prove a similar statement in which the functions under consideration are quadratic polynomials. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials Q1,Q2 in Q there is a subset I of Q, such that Q1,Q2 not in I and whenever Q1 and Q2 vanish then also the product of Qi vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends an earlier result by [Shp18]. This result brings us one step closer towards obtaining a deterministic polynomial time algorithm for the polynomial identity question of certain depth-4 identities. This is joint work with Amir Shpilka.