Strong Algebras and Radical Sylvester-Gallai Configurations

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}={F1,\cdots,Fm} \subset \mathbb{K}[x1,\cdots,xN]$ be a set of irreducible homogeneous polynomials of degree at most $d$ such that $Fi$ is not a scalar multiple of $Fj$ for $i\neq j$. Suppose that for any two distinct $Fi,Fj\in \mathcal{F}$, there is $k\neq i,j$ such that $Fk\in \mathrm{rad}(Fi,Fj)$. We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function $\lambda : \mathbb{N} \to \mathbb{N}$, independent of $\mathbb{K},N$ and $m$, such that any such configuration $\mathcal{F}$ must satisfy $$ \dim (\mathrm{span}{\mathbb{K}}{\mathcal{F}}) \leq \lambda(d). $$ 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.