奇异点问题是多项式系统求解中最具挑战性的问题之一，在计算机辅助几何设计和多项式优化等许多相关领域中有着广泛的应用。本项目将针对这一问题开展理论与算法研究，所涉及的内容包括：改进的收缩方法、改进的临界点方法、正维系统的欧氏距离次数、算法的复杂度分析和阈值的自动控制等。本项目将致力于分析与解决多项式系统求解中与奇异点相关的若干理论问题，设计并实现高效、鲁棒的零维系统奇异解的精化与验证算法和求解正维系统的符号数值混合算法。科学与工程计算中出现的很多数学问题都可以归结为多项式系统求解问题，奇异点的识别与处理是其中的重点和难点。本项目的研究不仅对多项式系统求解中的奇异点问题，对其他研究领域中的相关问题（如计算隐式曲线曲面的拓扑和实半代数集的凸包等）的发展也有着重要的意义。

A main challenge in polynomial systems solving is to identify and tackle singular points, which naturally occur in CAGD (computer aided geometric design) and polynomial optimization. This project aims to contribute to this problem by developing new theories and new algorithms on singular points in polynomial systems solving. Project subjects include: improvements of the deflation method and the critical points method, the Euclidean distance degree of positive-dimensional systems, complexity analysis and automatic threshold control in algorithms and etc. Project goals are twofold: the first goal is to analyze and solve some theoretic problems related to singular points in polynomial systems solving; the second goal is to develop and implement efficient and robust algorithms used to refine and certify singular solutions of zero-dimensional systems and to solve positive-dimensional systems. Polynomial systems naturally arise in many areas from science and engineering. Furthermore, singular solutions to these systems are often of particular interest to researchers. The new theory established will provide a more explicit description of singular points in polynomial systems solving, and the new algorithms developed will allow a broad range of scientists and engineers, who encounter polynomial systems to compute their singular solutions which are beyond the reach of current solving techniques.