Polynomial Manipulation using Dixon Resultant Formulation

ABSTRACT0203051Dr。新墨西哥的Deepak KapurU的问题(i)确定一个给定的多项式方程系统是否有一个共同的解，（ii）推导出现在n个多项式方程的参数的条件，使它们有一个共同的解，以及（iii）发展一个e。共同解的客户表示具有根本意义。这些问题出现在许多应用中：工程和设计、机器人、宇宙运动学、制造、纳米技术中纳米器件的设计和分析、图像理解、图形学、实体建模、隐式化、CAD-CAM设计、几何构造、设计和控制理论。多元结果和相关的消去方法被发现对解决这些问题很有用。一个多项式方程组的结式给出了必要和su。一个通用的解决方案存在的参数的客户条件。当多项式系统中有参数时，数值技术通常不适用。有效消去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基于Bezout-Cayley-Dixon方法和有效消去/合成法提出了广义Dixon公式。广义Dixon公式的一个特别吸引人的特征是它具有自适应问题，因为它隐含地利用了相关多项式系统的稀疏结构以及它的非泛型。

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.