论文标题
二次多项式的鲁棒sylvester-gallai型定理
Robust Sylvester-Gallai type theorem for quadratic polynomials
论文作者
论文摘要
在这项工作中，我们将Sylvester-Gallai定理的强大版本扩展到Barak，Dvir，Wigderson和Yehudayoff，并将DVIR，SARAF和WIGDERSON获得的案例扩展到了Quadratic的多项式案例。具体而言，我们证明，如果$ \ Mathcal {q} \ subset \ mathbb {c} [x_1。 有$δ> 0 $，以至于每$ q \ in \ nathcal {q} $至少$Δm$ $ $ $ polyenmials $ p \ in \ nathcal {q} $，以便每当$ q $ p $ and $ q $ p $消失，那么第三个polynomial在$ \ ardcal {q} q} $ setminus \ setminus \ s q p \ $ \ dim（\ text {span}（{\ mathcal {q}}））= \ text {poly}（1/δ）$。 Barak等人的工作。和Dvir等。研究了线性多项式的情况，并在维度上证明了$ O（1/δ）$的上限（在第一件作品中，给出了$ O（1/δ^2）$的上限，在第二次工作中将其提高到$ O（1/δ）$）。
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 $\mathcal{Q}\setminus\{Q,P\}$, then $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/δ)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/δ)$ on the dimension (in the first work an upper bound of $O(1/δ^2)$ was given, which was improved to $O(1/δ)$ in the second work).
