Strong Algebras and Radical Sylvester-Gallai Configurations

In this paper, we study the following non-linear generalization of the classical Sylvester-Gallai configuration. Let K be an algebraically closed field of characteristic 0 and F=F1,…,Fm ⊂ K[x1,…,xN] be a set of irreducible homogeneous polynomials of degree at most d such that Fi is not a scalar multiple of Fj for i ≠ j. We say that F is a radical Sylvester-Gallai configuration if for any two distinct Fi,Fj ∈ F, there is k ≠ i,j such that Fk ∈ rad(Fi,Fj). We prove that such radical Sylvester-Gallai configurations must be low dimensional. More precisely, we show that there exists a function λ : ℕ → ℕ, independent of K,N, and m, such that any such configuration F must satisfy