Numerical Algebraic Geometry: Computation of Exceptional Parameter Values

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Sommese and Wampler propose constructing and implementing new algorithms to numerically describe exceptional algebraic subsets of the solution sets of systems of parameterized polynomial equations; and to solve polynomial systems with few solutions, but with many defining equations. In applications, such as the design of robots and mechanisms, the parameters represent constants of the device, such as the length of a link, and the variables represent the device's motion. A focus of the proposal is on techniques for the design of robots and mechanisms where the objective is to find the exceptional parameter values so that the device they represent has particular motion characteristics. These techniques, which will apply to parameterized polynomial systems generally, such as may arise in computer graphics, chemistry, robot vision, and other engineering and scientific disciplines, lead to systems of polynomials that are large in comparison to the systems presently being solved. To deal with these large systems, Sommese and Wampler propose a new equation-by-equation approach, that they call regeneration, to solve polynomial systems. This reduces the solution of a polynomial system to sequentially finding the solution of systems that starting trivial are gradually built up to the target system. Close relationship between the subsystems of equations will likely result in new algorithmic efficiencies. Sommese and Wampler propose further development of Bertini, their freely available software for polynomial system computations, including the development of parallel versions so that they can tackle nontrivial systems arising in engineering and science. They propose also the development of algorithms for computing invariants that algebraic geometers are interested in, and which, while expensive to compute symbolically, are easy to compute numerically.Many technological problems, e.g., the design of artificial limbs, the design of industrial robots and other machines, economics, and a detailed understanding of critical chemical processes, such as those involved in combustion, lead to systems of polynomials that are very difficult to impossible to solve by current methods. Sommese and Wampler are developing new mathematical and computational approaches to solve such currently intractable problems. They are also developing Bertini, a freely available software package, so that engineers, mathematicians, and scientists may solve the polynomial systems that come up in their work without knowledge of the extensive theoretical foundations underlying the work of Sommese and Wampler.