The classical Sylvester-Gallai theorem says that if a finite set of points in the Euclidean plane has the property that the line joining any two points contains a third point from the set, then all the points must be collinear. More generally, a Sylvester-Gallai type configuration is a finite set of geometric objects with certain "local" dependencies. A remarkable phenomenon is that the local constraints give rise to global dimension bounds for linear SG-type configurations, and such results have found far reaching applications to complexity theory and coding theory.

In this talk we will discuss non-linear generalizations of SG-type configurations which consist of polynomials. We will discuss how the commutative-algebraic principle of Stillman uniformity can shed light on low dimensionality of SG-configurations. I’ll talk about recent progress showing that these non-linear SG-type configurations are indeed low-dimensional as conjectured by Gupta. This is based on joint work with R. Oliveira.