Algorithms and Software for Singular Polynomial Systems

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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 the Macaulay2 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.