AI 中文总结
该研究确定了阿龙-鲁埃达零子空间问题中复齐次多项式及有界次数多项式的精确有限维阈值，两类阈值不同，分别通过特韦列夫定理、德巴尔-马内维尔定理及陈类等方法完成推导与证明。
AI 中文摘要
我们确定了阿龙与鲁埃达提出的零子空间问题中复齐次多项式的精确有限维阈值。更准确地说，对任意的$d$和$k$，我们确定了最小的$m$，使得$\boldsymbol{\text{C}}^m$上的每个$d$次齐次多项式都在一个$k$维线性子空间上消失。我们还确定了次数不超过$d$的任意多项式在$k$维线性子空间上为常数的精确阈值，这两个阈值不同。在齐次情形，精确阈值由特韦列夫关于迷向子空间的定理及关联轨迹的闭性得到；在有界次数情形，我们先通过转到线性齐次分量的核来消除该分量，其余次数为$2,\boldsymbol{\text{...}},d$的分量构成应用德巴尔-马内维尔定理的方程组；对$k=2$，我们利用最高陈类与牛顿不等式给出了单独证明。
英文摘要
We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre--Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.
