计算代数方程组孤立奇异解的符号数值方法

Symbolic-numeric methods on computing isolated singular solutions of algebraic systems

求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望.

Solving systems of algebraic equations is one of the most fundamental problems in computational algebraic geometry. It is ubiquitous and widely applied across the engineering and sciences, such as in robotics,computer vision, machine learning, artificial intelligence, cryptography, optimization, control theory and etc. One main challenge is to compute isolated singular solutions, which plays an important rule in geometric modelings.Based on recent research results of the authors and their collaborators, a survey for symbolic-numeric methods on computing isolated singular solutions of algebraic systems is conducted, especially for refining and certifying approximate solutions. Some directions for future studies on the topic are discussed as well.