Symbolic computation is exact -- the result of a computation is guaranteed correct. This comes at a cost -- the computations are sometimes very slow and/or memory intensive; at times, the correct results of a computation are too huge to be of use. Numerical computation, on the other hand, is faster, and, on well conditioned input problem instances, one can expect correct solutions in the sense of backward error analysis. The numerical approach has been very successful in solving a wide spectrum of computational problems in science and engineering. However, there are situations where this approach is not completely satisfactory. In several applications of polynomial system solving arising in areas such as geometric modeling and kinematics, one needs the exact answer to a given problem instance and not to a near-by instance. On the flip side, in many application problems, the input polynomial system is not known exactly but to within small error bounds. This changes the nature of questions that can be asked of a solver. For instance, a question such as ``does a given system of polynomial equations have a root of multiplicity r?'' changes to ``is the given system of polynomial equations near a system that has a root of multiplicity r?'' with respect to some appropriate measure of nearness. Existing symbolic/numeric algorithms are not equipped to handle such questions well. Some of the issues that need to be addressed in this framework are: characterization of the solutions to a system of polynomial equations/inequalities in the presence of small input errors; design of efficient algorithms for solving polynomial systems that work in limited (or adaptive) precision; understanding the factors that affect the stability of these algorithms; investigation of ways to arrange the sometimes independent symbolic phase and numerical phase in polynomial system solving to take maximum advantage of the strengths of each other. This project investigates some specific instance s of polynomial system solving in the above framework with particular focus on approximate polynomial greatest common divisors and several special families of systems of polynomials. A new parametric minimization technique is being studied and applied to the approximate GCD problem and several of its variants. The concept of generic Groebner bases is introduced and in combination with eigenvalue computations, their effectiveness in the approximate solving of certain special families of polynomial systems is being investigated. The investigation of generic Groebner bases throws fresh light on some well-known unresolved issues in Groebner base theory. The new techniques will be applied to polynomial systems arising in geometric modeling and kinematics.
