Strong Algebras and Radical Sylvester-Gallai Configurations

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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={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. Suppose that for any two distinct Fi,Fj∈F, there is k≠i,j such that Fk∈rad(Fi,Fj). We prove that such radical SG 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

dim(spanKF)≤λ(d).

Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,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.