Collaborative Research: Software for Decomposing Solution Sets of Polynomial Systems

该项目旨在开发高效的数值算法和高质量的软件来求解多项式系统。一个基本问题是将解的正维子集分解成不可约的分量。Sommese、Verschelde和Wampler的数值方法是基于线性空间的一般切片，到低维线性空间的一般投影，以及使用经典插值技术在数值上完成符号程序中消除理论的工作。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的参数方程，使系统具有解的正维分量。这个项目的两个应用目标是分解多元多项式和发现过度约束机制。这项工作的一个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。这项工作在数值分析和计算机代数的研究领域进行，其任务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，因此该项目将重点放在多项式系统等基本模型上是合适的。除了将先进的数学工具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开发的软件的设计和使用。

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.