代数超曲面实孤立点的计算
Computing the Real Isolated Points of an Algebraic Hypersurface
摘要：
设$\mathbb{R}$为实数域。本文研究了多项式系统的消失集形式的$\mathbb{R}^n$中实代数集的实孤立点的计算问题。该问题对于研究机构的刚度特性在材料设计中具有重要意义。在本文中，我们设计了一个算法来解决这个问题。它是基于临界点的计算，以及路线图回答连接查询在真正的代数集。这导致一个概率算法的复杂性$(nd)^{O(n\log(n))}$计算真正的孤立点的真正代数超曲面的程度$d$。它使我们能够在实践中解决超出最先进水平的情况。
英文摘要：
Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity $(nd)^{O(n\log(n))}$ for computing the real isolated points of real algebraic hypersurfaces of degree $d$. It allows us to solve in practice instances which are out of reach of the state-of-the-art.
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