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.