Collaborative Research: Software for Decomposing Solution Sets of Polynomial Systems

协作研究：分解多项式系统解集的软件

• 批准号: 0105653
• 负责人: Andrew Sommese
• 金额: $ 18万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0105653&HistoricalAwards=false
• 关键词: Collaborative; Research; Software; Decomposing; Solution

The project is on the development of efficient numerical algorithms and high quality software to solve systems of polynomials. A basic problem is to decompose the positive dimensional subsets of solutions into irreducible components. The numerical approach of Sommese, Verschelde, and Wampler is based on generic slicing with linear spaces, generic projections into lower dimensional linear spaces, and use of classical interpolation techniques to numerically do what elimination theory does in symbolic programs. Given a polynomial system with parameters, a goal of the project is to find equations on the parameters that need to be satisfied for the system to have a positive dimensional component of solutions. Two applications targeted by this project are factoring multivariate polynomials and finding overconstrained mechanisms.A major outcome of this work will be publicly available software to solve polynomial systems that arise in science and engineering. This work is carried out in the research fields of numerical analysis and computer algebra whose mission is to provide the scientific community with software to solve mathematical problems. Since polynomial systems are used as models in application areas as far apart as chemical reaction systems, the design of mechanisms, or economic equilibria to name but a few areas, the focus of the project on such basic models as polynomial systems is appropriate. Besides the technology transfer of advanced mathematical tools into science and engineering, an important aspect of this project is to introduce students in the design and use of the developed software.

该项目旨在开发高效的数值算法和高质量的软件来解决多项式系统。 一个基本问题是把正维解的子集分解成不可约分量。 Sommese，Verschelde和Wampler的数值方法是基于线性空间的通用切片，到低维线性空间的通用投影，以及使用经典插值技术在数值上做消除理论在符号程序中所做的事情。 给定一个带参数的多项式系统，该项目的目标是找到需要满足系统具有正维分量的解的参数方程。 该项目的两个目标应用是分解多元多项式和寻找过约束机制。这项工作的主要成果将是公开可用的软件来解决科学和工程中出现的多项式系统。 这项工作是在数值分析和计算机代数的研究领域进行的，其使命是为科学界提供解决数学问题的软件。 由于多项式系统在化学反应系统、机制设计或经济平衡等应用领域中被用作模型，因此将项目重点放在多项式系统等基本模型上是适当的。 除了将先进的数学工具技术转移到科学和工程领域之外，该项目的一个重要方面是向学生介绍所开发软件的设计和使用。