论文标题
强大的激进Sylvester-Gallai定理用于四边形
Robust Radical Sylvester-Gallai Theorem for Quadratics
论文作者
论文摘要
我们证明了对二次多项式的Sylvester-Gallai型定理的鲁棒性概括，从而将结果推广到[S'20]中。更准确地说，给定一个参数$ 0 <δ\ leq 1 $和一个有限的集合$ \ mathcal {f} $是不可修复和成对独立的多项式学位的$ 2，我们说$ \ mathcal {f} $是$（δ，2）$ - 2）$ - 2）$ - radical sylvester-gallai for for任何$ - \ Mathcal {f} $，存在$δ（| \ Mathcal {f} | -1）$ polyenmials $ f_j $，以便$ | | \ mathrm {rad}（f_i，f_i，f_j）\ cap \ cap \ mathcal {f} | \ geq 3 $，即$ f_i，f_j $的激进分子包含第三个多项式。 在这项工作中，我们证明任何$（δ，2）$ - 激进的sylvester-gallai配置$ \ mathcal {f} $都必须具有低维度：那就是$ \ dim \ dim \ dim \ mathrm {span}（\ mathcal {f} f}）
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. In this work, we prove that any $(δ, 2)$-radical Sylvester-Gallai configuration $\mathcal{F}$ must be of low dimension: that is $$\dim \mathrm{span}(\mathcal{F}) = \mathrm{poly}(1/δ).$$
