二次多项式的一个广义Sylvester-Gallai型定理
A generalized Sylvester-Gallai type theorem for quadratic polynomials
摘要：
在这项工作中，我们证明了一个版本的Sylvester-Gallai定理的二次多项式，使我们更接近一步获得一个确定性的多项式时间算法测试零的$Σ^{[3]}ΠΣΠ^{[2]}$电路。具体地说，我们证明了，如果一个有限的不可约二次多项式$\mathcal{Q}$满足，每两个多项式$Q_1,Q_2\in \mathcal{Q}$有一个子集$\mathcal{K}\subset \mathcal{Q}$，使$Q_1,Q_2 \notin \mathcal{K}$和每当$Q_1$和$Q_2$消失，然后也$\prod_{i\in \mathcal{K}} Q_i$消失，那么线性跨度的多项式在$\mathcal{Q}$有尺寸$O(1)$。这扩展了早期的结果[Shpilka 19]，该结果在$|\mathcal{K}| = 1$时显示了类似的结论。在我们的证明中，一个重要的技术步骤是一个定理，该定理对所有可能的情况进行了分类，在这些情况下，当另外两个二次多项式消失时，一个二次多项式的乘积可以消失。也就是说，当乘积在理想的根上时，由两个二次曲线生成。这一步扩展了[Shpilka 19]的一个结果，该结果研究了一个二次多项式在两个其他二次多项式的根中的情况。
英文摘要：
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 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.
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