An Approach for Solving Systems of Parametric Polynomial Equations

1.1 Abstract An approach for solving nonlinear polynomial equations involving parameters is proposed. A distinction is made between parameters and variables. The objective is to generate from a system of parametric equations, solved forms from which solutions for speciic values of parameters can be obtained without much additional computations. It should be possible to analyze the parametrized solved forms so that it can be determined for diierent parameter values whether there are in-nitely many solutions, nitely many solutions, or no solutions at all. The approach is illustrated for two diierent symbolic methods for solving parametric equations { Grr obner basis computations and characteristic set computations. These methods are illustrated on a number of examples. Many complex phenomena can be modeled using nonlinear polynomial equations. Examples include imaging transformations in computer vision , computing geometric invariants, geometric and solid modeling, constraint-based modeling, reasoning about geometry problems, properties of chemical equilibrium, kinematics, robotics; an interested reader may consult 13, 14, 7, 5] for many examples. Variables in these equations can be classiied into two subsets: independent variables and dependent variables. Independent variables often correspond to input to a phenomenon or the features of a physical subsystem in a phenomenon. Henceforth, independent variables will also be called parameters; dependent variables will be called just variables. For diierent parameter values, dependent variables have diierent behavior. A designer is usually interested in studying the phenomenon on a wide range of parameter