Strong Algebras and Radical Sylvester-Gallai Configurations

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let K be an algebraically closed field of characteristic 0 and F = tF 1 , ¨¨¨, F m u Ă K[x 1 , ¨¨¨, x N ] be a set of irreducible homogeneous polynomials of degree at most d such that F i is not a scalar multiple of F j for i ‰ j. Suppose that for any two distinct F i , F j P F, there is k ‰ i, j such that F k P rad(F i , F j ). We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function λ : N Ñ N, independent of K, N and m, such that any such configuration F must satisfy dim(span K tFu) ď λ(d).

Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and generalizes the quadratic and cubic Sylvester-Gallai theorems of [Shp20, OS22]. Our result takes us one step closer towards the first deterministic polynomial time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4 circuits of bounded top and bottom fanins. Our result, when combined with the Stillman uniformity type results of [AH20a, DLL19, ESS21], yields uniform bounds for several algebraic invariants such as projective dimension, Betti numbers and Castelnuovo-Mumford regularity of ideals generated by radical SG configurations.