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 $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$ 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 polynomials in $\mathcal{Q}$ has dimension $O(1)$. This extends the earlier result [Shpilka19] that showed a similar conclusion when $|\mathcal{K}| = 1$. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generates by the two quadratics. This step extends a result from [Shpilka19]that studied the case when one quadratic polynomial is in the radical of two other quadratics.

中文翻译：

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二次多项式的广义 Sylvester-Gallai 型定理

在这项工作中，我们证明了二次多项式的 Sylvester-Gallai 定理的一个版本，它使我们更接近于获得用于测试 $\Sigma^{[3]}\Pi\Sigma\Pi^{ 零性的确定性多项式时间算法[2]}$电路。具体来说，我们证明如果一个有限的不可约二次多项式 $\mathcal{Q}$ 满足对于每两个多项式 $Q_1,Q_2\in \mathcal{Q}$ 有一个子集 $\mathcal{K}\subset \mathcal{Q}$，使得 $Q_1,Q_2 \notin \mathcal{K}$ 并且每当 $Q_1$ 和 $Q_2$ 消失时， $\prod_{i\in \mathcal{K}} Q_i$ 也消失，那么 $\mathcal{Q}$ 中多项式的线性跨度具有维度 $O(1)$。这扩展了较早的结果 [Shpilka19]，该结果在 $|\mathcal{K}| 时显示了类似的结论。= 1 美元。我们证明中的一个重要技术步骤是定理对所有可能的情况进行分类，在这些情况下，当其他两个二次多项式消失时，一个二次多项式的乘积可以消失。即，当乘积在理想的根部时，由两个二次方程生成。这一步扩展了 [Shpilka19] 的结果，该结果研究了一个二次多项式在其他两个二次多项式的根中的情况。