CAMPUS THEORY SEMINAR
               Part of the SCIENCE AT THE EDGE Seminar Series

                          Friday, March 23, 2001
                11:30am, Room 224 Physics-Astronomy Building
                      Refreshments served at 11:15am


                        SOLVING POLYNOMIAL SYSTEMS

                                T. Y. Li
                        Department of Mathematics
                        Michigan State University

 
 Numerically solving isolated zeros of polynomial systems in affine space
has become increasingly important in applications. Many engineering
models, such as formular construction, geometric interection problems,
inverse kinematics, power flow problem with PQ-specified bases, the most 
general six degree of freedom manipulators, computation of equilibrium 
states, etc.

  Elimination theory based methods, most notably the Buchberger algorithm
for constructing Groebner bases, are the classical approach to solving
systems of polynomial equations, but its reliance on symbolic manipulation
makes it seem unsuitable for all but small problems. Moreover, the method
reduces the problem to the ill-conditioned problem of numerically solving a
high-degree polynomial equation in one variable.

  In this talk, a new approach, developed in the last two decades, by using
the homotopy continuation method will be surveyed. The method involves
first solving a trivial system, and then deforming these solutions along
smooth paths to the solutions of the target system. The method has been
successfully implemented in solving many polynomial systems, and the 
amount of computation required to find all solutions can be made roughly 
proportional to the number of solutions.