Realizing solving for the exact solutions is a too difficult task, we turn to seek approximate solutions. I will present two methods that we have investigated to find real solutions of systems of polynomial equations.
The first method computes the number of solutions in regions of the parameter space. By subdividing the regions into smaller and smaller sizes, we get closer and closer approximations to the solutions. This method uses bilinear mappings induced by polynomials defining the boundaries of the regions, and obtains the solution counts from the positive and negative eigenvalues of the bilinear mappings. These mappings, however, are computed using Groebner bases or resultant matrices. Thus the method has huge computational complexities and so far remains theoretical.
The second method focuses on eliminating regions that contain no solutions in the parameter space instead of obtaining solution counts within the regions. This method also carries a subdivision process. The final small regions of the subdivision process are reported as approximations of the solutions. While most computer algebra systems can not solve systems of six (6) or more equations, this method is able to "solve" up to nine (9) polynomials equations (total degree 14) rising from the docking problems in computer-assisted drug design.