Polynomial Manipulation using Dixon Resultant Formulation

使用 Dixon 结果公式进行多项式运算

• 批准号: 0203051
• 负责人: Deepak Kapur
• 金额: $ 21万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2006-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203051&HistoricalAwards=false
• 关键词: Polynomial; Manipulation; using; Dixon; Resultant

ABSTRACT0203051Dr. Deepak KapurU of New MexicoThe problems of (i)determining whether a given polynomial equation system has a common solution, (ii)deriving conditions on parameters appearing n polynomial equations,such that they have a common solution,as well as (iii)developing an e .cient representation of common solutions are of fundamental significance. These problems arise in numerous applications:engineering and design, robotics, universe kinematics, manufacturing, design and analysis of nano devices in nanotechnology, image understanding, graphics,solid modeling,implicitization,CAD-CAM design,geometric construction,design,and control theory. Multivariate resultants and related elimination methods have been found useful for addressing these problems. The resultant of a polynomial equation system gives the necessary and su .cient condition on itsparameters for a common solution to exist. When there are parameters in a polynomial system,numerical techniques often do not apply. Investigations of efficient elimination methods are also of considerable importance in algebraic geometry,polynomial ideal theory and other related aspects of computational algebra and symbolic computation.Experimental and theoretical analysis indicates that the generalized Dixon formulation developed by Kapur,Saxena and Yang,based on Bezout-Cayley-Dixon methods and efficient elimination/resultant method. A particularly attractive feature of the generalized Dixon formulation is that t s problem-adaptive since timplicitly exploits the sparse structure of the associated polynomial system as well as its non-genericity.

摘要0203051博士。新墨西哥州的迪帕克·卡普尔大学 (Deepak KapurU) 提出了以下问题：(i) 确定给定的多项式方程组是否具有公共解，(ii) 推导出现 n 个多项式方程组的参数的条件，使得它们具有公共解，以及 (iii) 开发一个共同解决方案的有效表示具有根本意义。 这些问题出现在许多应用中：工程和设计、机器人学、宇宙运动学、纳米技术中纳米器件的制造、设计和分析、图像理解、图形、实体建模、隐式化、CAD-CAM 设计、几何构造、设计和控制理论。 已发现多变量结果和相关消除方法对于解决这些问题很有用。多项式方程组的结果给出了其参数存在共同解的充分必要条件。当多项式系统中有参数时，数值技术通常不适用。 有效消除方法的研究在代数几何、多项式理想理论以及计算代数和符号计算的其他相关方面也具有相当重要的意义。实验和理论分析表明Kapur、Saxena和Yang提出的广义Dixon公式基于Bezout- Cayley-Dixon 方法和高效消除/所得方法。广义狄克逊公式的一个特别有吸引力的特征是它是问题自适应的，因为隐含地利用了相关多项式系统的稀疏结构及其非通用性。