一般代数集的婴儿步-巨步路线图算法
A baby step-giant step roadmap algorithm for general algebraic sets
摘要：
设$\mathrm{R}$是实闭域，$\mathrm{D} \subset \mathrm{R}$是有序域.我们给出了一个算法，作为输入的多项式$Q \in \mathrm{D}[X_1,\ldots,X_k]$，并计算一个路线图的零，$\mathrm{Zer}(Q,\mathrm{R}^k)$，$Q$在$\mathrm{R}^k$的集合的描述。该算法的复杂性，在有序域$\mathrm{D}$的算术运算的数量来衡量，是由$d^{O(k \sqrt{k})}$，其中$d = \mathrm{deg}(Q)\ge 2$界定。因此，存在用于计算实代数集的半代数连通分量的数目的算法，$\mathrm{Zer}(Q,\mathrm{R}^k)$，其复杂性也由$d^{O(k \sqrt{k})}$限定，其中$d = \mathrm{deg}(Q)\ge 2$。用于构造由次数多项式$d$定义的$\mathrm{R}^k$的实代数子集的路线图的最佳先前已知算法具有复杂度$d^{O(k^2)}$。
英文摘要：
Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We give an algorithm that takes as input a polynomial $Q \in \mathrm{D}[X_1,\ldots,X_k]$, and computes a description of a roadmap of the set of zeros, $\mathrm{Zer}(Q,\mathrm{R}^k)$, of $Q$ in $\mathrm{R}^k$. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $\mathrm{D}$, is bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, $\mathrm{Zer}(Q,\mathrm{R}^k)$, whose complexity is also bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. The best previously known algorithm for constructing a roadmap of a real algebraic subset of $\mathrm{R}^k$ defined by a polynomial of degree $d$ has complexity $d^{O(k^2)}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
