Computational Methods in Numerical Algebraic Geometry

该项目旨在通过开发和实现用于解决许多应用中出现的多项式系统的新算法，为数值代数几何做出贡献。一个目标是开发一种算法来解决大规模结构化多项式系统，这种系统在计算过约束机制以及计算代数集上的实点和奇点时自然出现。该算法将利用由Hauenstein， Sommese和Wampler开发的再生方法，该方法通过从较小的多项式系统的解中构建来计算多项式系统的解。再生和结构的开发将使人们能够解决许多自然发生的多项式系统，这些系统超出了当前方法的范围。另一个目标是在这一领域培养一名或多名本科生。学生们还将帮助开发一些算法，并测试由本提案开发的软件。此外，作为一个小组，我们将把新开发的算法应用于应用中出现的新问题。多项式系统自然地出现在科学、工程、经济学和生物学的许多领域，它们的解决方案，例如，描述专门机器人的设计，化学反应和经济模型的平衡，以及描述肿瘤的稳定性。这些多项式系统的实解通常引起研究人员的特别兴趣，因为它们通常描述物理上有意义的解，例如，可构造的机器人。开发的新算法和软件将允许广泛的科学家、工程师和经济学家在遇到多项式系统时计算物理上有意义的解决方案，这些解决方案超出了当前解决技术的范围。此外，参与该项目的学生将获得数学科学方面的知识和研究经验。

This project aims to contribute to numerical algebraic geometry by developing and implementing new algorithms used to solve polynomial systems arising in many applications. One goal is the development of an algorithm for solving large-scale structured polynomial systems which naturally arise in computing overconstrained mechanisms as well as computing real and singular points on algebraic sets. This algorithm will utilize the regeneration method, developed by Hauenstein, Sommese, and Wampler, which computes the solutions of a polynomial system by building from the solutions of smaller polynomial systems. Regeneration together with the exploitation of structure will allow one to solve many naturally occurring polynomial systems which are beyond the reach of current methods. Another goal is the training of one or more undergraduate students in this area. The students will also help with the development of some of the algorithms and testing of the software developed by this proposal. Additionally, as a group, we will apply the newly developed algorithms to new problems arising from applications.Polynomial systems naturally arise in many areas of science, engineering, economics, and biology with their solutions, for example, describing the design of specialized robots, equilibria of chemical reactions and economic models, and describing the stability of tumors. The real solutions to these polynomial systems are often of particular interest to researchers as they often describe the physically meaningful solutions, e.g., a constructible robot. The new algorithms and software developed will allow a broad range of scientists, engineers, and economists who encounter polynomial systems to compute physically meaningful solutions to systems which are beyond the reach of current solving techniques. Additionally, the students involved in this project will gain knowledge and research experience in the mathematical sciences.