PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

Robust Radical Sylvester-Gallai Theorem for Quadratics

We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials, generalizing the result in [S'20]. More precisely, given a parameter $0 < δ\leq 1$ and a finite collection $\mathcal{F}$ of irreducible and pairwise independent polynomials of degree at most 2, we say that $\mathcal{F}$ is a $(δ, 2)$-radical Sylvester-Gallai configuration if for any polynomial $F_i \in \mathcal{F}$, there exist $δ(|\mathcal{F}| -1)$ polynomials $F_j$ such that $|\mathrm{rad}(F_i, F_j) \cap \mathcal{F}| \geq 3$, that is, the radical of $F_i, F_j$ contains a third polynomial in the set…

从Sylvester-Gallai检验到秩界：改进的深度3电路黑盒恒等式检验

From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-box Identity Test for Depth-3 Circuits

本文研究了k阶和d阶的深度3电路的恒等式测试问题。我们给出了这类恒等式的一个新的结构定理。我们的定理的直接应用改进了已知的有理数上的确定性d^{k^k}-时间黑盒恒等式测试（Kayal-Saraf，FOCS 2009），使其需要d^{k^2}-时间。我们的结构定理本质上说，在一个真正的深度3单位的独立变量的数量是非常小的。这个定理肯定地解决了Dvir-Shpilka（STOC 2005）和Kayal-Saraf（FOCS 2009）提出的更强的秩问题。我们的技术提供了一个统一的框架，实际上击败了所有已知的排名界限，因此为黑盒身份测试提供了最佳的运行时间（对于每个字段）。我们的主要定理（几乎最优）钉在一个非常透明的方式高维西尔维斯特-Gallai定理和深度3身份证的秩之间的关系。Dvir-Shpilka（STOC 2005）暗示了这一点的存在，但Kayal-Saraf（FOCS 2009）首先证明了这一点。我们引入了Sylvester-Gallai秩界的概念，并证明了这与深度3恒等秩界之间的密切联系。我们还证明了第一个定理高维Sylvester-Gallai配置在任何领域。我们的证明和技术是非常不同的，从以前的结果和设计一个非常有趣的组合数学和代数系综。后者的概念是理想的理论和涉及一个新的中国剩余定理。我们的证明方法解释了任何深度为3的身份C的结构：有一个核的C，形成一个低秩的身份，…

近似共线性的Sylvester-Gallai型定理

Sylvester-Gallai type theorems for approximate collinearity

我们研究的问题，在发病率几何点的精确位置是“模糊”（例如，由于噪音，不准确或错误）。因此，直线被细管代替，更一般的仿射子空间被它们的小邻域代替。我们证明了在d维复空间中的一组点中存在足够多的近似共线三元组意味着这些点接近于一个低维仿射子空间。这可以被看作是Sylvester-Gallai定理及其扩展的稳定变体。建立在最近发现的Sylvester-Gallai型定理和复杂的局部可纠正码（LCC）之间的连接，我们定义了新的概念，稳定的LCC，其中（本地）校正程序也可以处理小扰动的欧几里德度量。我们证明了这种稳定的代码与恒定的查询复杂度是不存在的。在任何此类当地环境中，2次以上查询均未获得已知的不可能结果。

Robust Sylvester-Gallai type theorem for quadratic polynomials

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if $\mathcal{Q}\subset \mathbb{C}[x_1.\ldots,x_n]$ is a finite set, $|\mathcal{Q}|=m$, of irreducible quadratic polynomials that satisfy the following condition: There is $δ>0$ such that for every $Q\in\mathcal{Q}$ there are at least $δm$ polynomials $P\in \mathcal{Q}$ such that whenever $Q$ and $P$ vanish then so does a third polynomial in $\mathcal{Q}\setminus\{Q,P\}$,…

Sylvester-Gallai type theorems for quadratic polynomials

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal Q$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in {\mathcal Q}$ there is a third polynomial $Q_3\in{\mathcal Q}$ so that whenever $Q_1$ and $Q_2$ vanish then also $Q_3$ vanishes, then the linear span of the polynomials in ${\mathcal Q}$ has dimension $O(1)$. We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there …

A generalized Sylvester-Gallai type theorem for quadratic polynomials

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $Σ^{[3]}ΠΣΠ^{[2]}$ circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials $\mathcal{Q}$ satisfy that for every two polynomials $Q_1,Q_2\in \mathcal{Q}$ there is a subset $\mathcal{K}\subset \mathcal{Q}$, such that $Q_1,Q_2 \notin \mathcal{K}$ and whenever $Q_1$ and $Q_2$ vanish then also $\prod_{i\in \mathcal{K}} Q_i$ vanishes, then the linear span of the polynom…