One formulation of the famous SylvesterGallai 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 SylvesterGallai theorem was used to device
deterministic algorithms for testing depth3 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
depth4 identities.
This is joint work with Amir Shpilka.
