Numerical Algebraic Geometry: Computation of Exceptional Parameter Values

Sommese和Wampler提出了构建和实现新算法来数值描述参数化多项式方程组的解集的例外代数子集;以及解决具有很少解但具有许多定义方程的多项式系统。在机器人和机构设计等应用中，参数表示设备的常数，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制的设计技术，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。这些技术通常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、机器人视觉和其他工程和科学学科中，导致与目前正在解决的系统相比较大的多项式系统。为了处理这些大型系统，Sommese和Wampler提出了一种新的方程-方程方法，他们称之为再生，以解决多项式系统。这减少了多项式系统的解决方案，顺序地找到系统的解决方案，开始平凡逐渐建立到目标系统。方程的子系统之间的密切关系可能会导致新的算法效率。Sommese和Wampler建议进一步发展贝尔蒂尼，他们的免费软件多项式系统计算，包括并行版本的发展，使他们能够解决工程和科学中出现的非平凡系统。他们还提出了算法的发展计算不变量，代数几何学家感兴趣的，而昂贵的计算象征性的，很容易计算数值。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（例如涉及燃烧的那些）的详细理解导致多项式系统，这些多项式系统很难或不可能通过当前方法来求解。 Sommese和Wampler正在开发新的数学和计算方法来解决这些目前棘手的问题。 他们还在开发Bertini，一个免费的软件包，这样工程师、数学家和科学家就可以解决他们工作中出现的多项式系统，而不需要了解Sommese和Wampler工作背后的广泛理论基础。

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.