In 1893, Sylvester asked a basic question in combinatorial geometry: given a finite set of distinct points v1,…,vm∈ℝN such that the line defined by any pair of distinct points vi,vj contains a third point vk in the set, must all points in the set be collinear?
Generalizations of Sylvester’s problem, which are known as Sylvester-Gallai type problems, have found applications in algebraic complexity theory (in Polynomial Identity Testing—PIT) and coding theory (Locally Correctable Codes). The underlying theme in all these types of questions is the following:
Are Sylvester-Gallai type configurations always low-dimensional?
In 2014, Gupta, motivated by such applications in algebraic complexity theory, proposed wide-ranging non-linear generalizations of Sylvester’s question, with applications on the PIT problem.
In this talk, we will discuss these non-linear generalizations of Sylvester’s conjecture, their intrinsic relation to algebraic computation, and a recent theorem proving that radical Sylvester-Gallai configurations for cubic polynomials must have small dimension. And time permitting a little bit more.