We use the polynomial method of Guth and Katz to establish stronger and {\it more efficient} regularity and density theorems for such $k$-uniform hypergraphs $H=(P,E)$, where $P$ is a finite point set in ${\mathbb R}^d$, and the edge set $E$ is determined by a semi-algebraic relation of bounded description complexity. In particular, for any $0<\epsilon\leq 1$ we show that one can construct in $O\left(n\log (1/\epsilon)\right)$ time, an equitable partition $P=U_1\uplus \ldots\uplus U_K$ into $K=O(1/\epsilon^{d+1+\delta})$ subsets, for any $0<\delta$, so that all but $\epsilon$-fraction of the $k$-tuples $U_{i_1},\ldots,U_{i_k}$ are {\it homogeneous}: we have that either $U_{i_1}\times\ldots\times U_{i_k}\subseteq E$ or $(U_{i_1}\times\ldots\times U_{i_k})\cap E=\emptyset$. If the points of $P$ can be perturbed in a general position, the bound improves to $O(1/\epsilon^{d+1})$, and the partition is attained via a {\it single partitioning polynomial} (albeit, at expense of a possible increase in worst-case running time). In contrast to the previous such regularity lemmas which were established by Fox, Gromov, Lafforgue, Naor, and Pach and, subsequently, Fox, Pach and Suk, our partition of $P$ does not depend on the edge set $E$ provided its semi-algebraic description complexity does not exceed a certain constant. As a by-product, we show that in any $k$-partite $k$-uniform hypergraph $(P_1\uplus\ldots\uplus P_k,E)$ of bounded semi-algebraic description complexity in ${\mathbb R}^d$ and with $|E|\geq \epsilon \prod_{i=1}^k|P_i|$ edges, one can find, in expected time $O\left(\sum_{i=1}^k|P_i|\log (1/\epsilon)+(1/\epsilon)\log(1/\epsilon)\right)$, subsets $Q_i\subseteq P_i$ of cardinality $|Q_i|\geq |P_i|/\epsilon^{d+1+\delta}$, so that $Q_1\times\ldots\times Q_k\subseteq E$.

翻译：我们运用Guth和Katz的多项式方法，为$k$一致超图$H=(P,E)$建立了更强且{\it 更高效}的正则性与密度定理，其中$P$是${\mathbb R}^d$中的有限点集，边集$E$由描述复杂度有界的半代数关系决定。特别地，对于任意$0<\epsilon\leq 1$，我们证明可在$O\left(n\log (1/\epsilon)\right)$时间内构造一个均衡划分$P=U_1\uplus \ldots\uplus U_K$，其中$K=O(1/\epsilon^{d+1+\delta})$（对任意$0<\delta$成立），使得除$\epsilon$比例的$k$元组$U_{i_1},\ldots,U_{i_k}$外，其余元组均为{\it 齐次}的：即要么满足$U_{i_1}\times\ldots\times U_{i_k}\subseteq E$，要么满足$(U_{i_1}\times\ldots\times U_{i_k})\cap E=\emptyset$。若$P$中的点可扰动至一般位置，则界可改进为$O(1/\epsilon^{d+1})$，且划分可通过{\it 单个划分多项式}实现（尽管可能以最坏运行时间的增加为代价）。与先前由Fox、Gromov、Lafforgue、Naor和Pach以及后续由Fox、Pach和Suk建立的此类正则性引理不同，我们对$P$的划分不依赖于边集$E$，只要其半代数描述复杂度不超过特定常数。作为推论，我们证明在${\mathbb R}^d$中任意具有有界半代数描述复杂度的$k$部$k$一致超图$(P_1\uplus\ldots\uplus P_k,E)$中，若其边数满足$|E|\geq \epsilon \prod_{i=1}^k|P_i|$，则可在期望时间$O\left(\sum_{i=1}^k|P_i|\log (1/\epsilon)+(1/\epsilon)\log(1/\epsilon)\right)$内找到子集$Q_i\subseteq P_i$，其基数满足$|Q_i|\geq |P_i|/\epsilon^{d+1+\delta}$，使得$Q_1\times\ldots\times Q_k\subseteq E$。