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 $\mathcal{F}=\{F_1,\cdots,F_m\} \subset \mathbb{K}[x_1,\cdots,x_N]$ 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 $i\neq j$. Suppose that for any two distinct $F_i,F_j\in \mathcal{F}$, there is $k\neq i,j$ such that $F_k\in \mathrm{rad}(F_i,F_j)$. We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function $\lambda : \mathbb{N} \to \mathbb{N}$, 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.

翻译：本文证明了经典Sylvester-Gallai定理的如下非线性推广。设$\mathbb{K}$为特征$0$的代数闭域，$\mathcal{F}=\{F_1,\cdots,F_m\} \subset \mathbb{K}[x_1,\cdots,x_N]$为一组次数不超过$d$的不可约齐次多项式，且当$i\neq j$时$F_i$不是$F_j$的标量倍。假设对任意两个不同的$F_i,F_j\in \mathcal{F}$，存在$k\neq i,j$使得$F_k\in \mathrm{rad}(F_i,F_j)$。我们证明此类根式SG构型必然具有低维性质。更精确地，我们证明存在一个与$\mathbb{K},N,m$均无关的函数$\lambda : \mathbb{N} \to \mathbb{N}$，使得任何此类构型$\mathcal{F}$满足$$ \dim (\mathrm{span}_{\mathbb{K}}{\mathcal{F}}) \leq \lambda(d). $$ 该结果证实了Gupta的一个猜想[Gup14, 猜想2]，并推广了[S20,OS22]中的二次与三次Sylvester-Gallai定理。我们的结果使我们在解决有界顶部和底部扇入的深度4电路的多项式恒等测试（PIT）问题方面，向首个确定性多项式时间算法迈进一步。该结果结合[AH20a,DLL19,ESS21]中Stillman一致型结果，可导出由根式SG构型生成的理想的若干代数不变量（如投射维数、Betti数和Castelnuovo-Mumford正则性）的一致上界。