Collaborative Research: Software for Decomposing Solution Sets of Polynomial Systems

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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.