Abstract
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.
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