Numerical Algebraic Geometry: Computation of Exceptional Parameter Values

Sommese和Wampler提出构造和实现新的算法来数值描述参数化多项式方程系统解集的异常代数子集；以及解解很少的多项式系统，但有很多定义方程。在机器人和机构设计等应用中，参数表示设备的常量，如连杆的长度，变量表示设备的运动。该提案的重点是机器人和机构的设计技术，其目标是找到特殊的参数值，使它们所代表的设备具有特定的运动特性。这些技术将普遍应用于参数化多项式系统，例如可能出现在计算机图形学、化学、机器人视觉以及其他工程和科学学科中，导致多项式系统比目前正在解决的系统更大。为了处理这些大系统，Sommese和Wampler提出了一种新的逐方程方法，他们称之为再生，来解决多项式系统。这将多项式系统的解简化为顺序地寻找从平凡开始逐渐建立到目标系统的系统的解。方程子系统之间的密切关系可能会导致新的算法效率。Sommese和Wampler建议进一步开发Bertini，他们的多项式系统计算免费软件，包括开发并行版本，以便他们可以处理工程和科学中出现的非平凡系统。他们还提出了计算不变量的算法的发展，这是代数几何学者感兴趣的，虽然符号计算的成本很高，但数值计算很容易。许多技术问题，例如，假肢的设计，工业机器人和其他机器的设计，经济学，以及对关键化学过程的详细了解，例如那些涉及燃烧的化学过程，导致多项式系统非常难以用现有方法解决。Sommese和Wampler正在开发新的数学和计算方法来解决这些目前难以解决的问题。他们还在开发Bertini，这是一个免费的软件包，这样工程师、数学家和科学家就可以在不了解Sommese和Wampler工作背后的广泛理论基础的情况下解决他们工作中出现的多项式系统。

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.