Algorithms and Software for Singular Polynomial Systems

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。在这个项目中，PI将开发稳健的符号-数值方法来求解奇异多项式系统和分解奇异变量。这些算法将获得当前常规方法无法提供的关于单一种类的信息：特别是，发现解决方案集的嵌入组件。这种方法将导致数值算法，这些算法将解决通过符号初级分解例程难以解决的问题。这个项目的一个主要部分是软件实现。由Grayson和Stillman创建的免费开源计算机代数系统Macaulay2将为高效实现我们的算法提供一个平台。一个包将用Macaulay2语言编写，在Macaulay2内核中用C语言实现计算密集型例程。由于同伦延拓的基本例程规模很大，因此上述算法的并行实现将在适当的地方进行。多项式系统在科学和工程中的各种数学模型中普遍存在。在现实世界中出现的多项式系统中，即使不是大多数，也有许多包含奇异解分量。制作的软件将帮助广泛的科学家和工程师在他们的工作中面对多项式系统。通过这个项目被任命为研究助理的学生将在结合编程和算法开发的数学研究方面获得宝贵的经验。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In this project the PI will develop robust symbolic-numerical methods for solving singular polynomial systems and decomposing singular varieties. These algorithms would obtain information about a singular variety that the current regular methods can not deliver: in particular, discover embedded components of the solution set. This approach will lead to numerical algorithms that would solve problems that are intractable by symbolic primary decomposition routines. A major part of this project is software implementation. Macaulay2, a free open-source computer algebra system created by Grayson and Stillman, will make a platform for an efficient implementation of our algorithms. A package will be written in the Macaulay2 language with computationally intensive routines implemented in C++ in theMacaulay2 kernel. Since the basic routines for homotopy continuation scale well, parallel implementation of the aforementioned algorithms will be carried out where appropriate.Polynomial systems are ubiquitous in various mathematical models in science and engineering. Many, if not the majority, of the polynomial systems arising in the real world contain singular solution components. The produced software will help a broad range of scientists and engineers confronted with polynomial systems in their work. The students appointed as research assistants through this project will gain invaluable experience in mathematical research combined with programming and development of algorithms.