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
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
This is joint work with Amir Shpilka.