Dr. Deepak Kapur U of New Mexico
The 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 its parameters 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.
