多簇集的多项式划分
Polynomial partitioning for several sets of varieties
摘要：
我们给出了一个新的，系统的证明，最近的结果拉里古斯，从而也延长了结果的设置与几个家庭的品种：对于任何整数$D\geq 1$和$\mathbb{R}^n$中的低次$k$维变种的集合$Γ_1,\ldots,Γ_j$的任何集合，存在次数至多为$D$的非零多项式$p\in\mathbb{R}[X_1,\ldots,X_n]$，使得$\mathbb{R}^n{\setminus}Z(p)$的每个连通分量与$Γ_i$的变种的$O(jD^{k-n}|Γ_i|)$相交，同时对每个$1\leq i\leq j$进行。对于$j=1$，我们恢复了Guth的原始结果。我们的证明，通过指数计算在等变上同调，显示了如何用于分区的多项式的程度是由拓扑结构，即由欧拉类给出的顶级迪克森多项式。
英文摘要：
We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $Γ_1,\ldots,Γ_j$ of low-degree $k$-dimensional varieties in $\mathbb{R}^n$ there exists a non-zero polynomial $p\in\mathbb{R}[X_1,\ldots,X_n]$ of degree at most $D$ so that each connected component of $\mathbb{R}^n{\setminus}Z(p)$ intersects $O(jD^{k-n}|Γ_i|)$ varieties of $Γ_i$, simultaneously for every $1\leq i\leq j$. For $j=1$ we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely by the Euler class being given in terms of a top Dickson polynomial.
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