Coefficient-parameter polynomial continuation

摘要

We establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems. We assume that the coefficients for a polynomial system are given by parameters. (In engineering applications, these would be derived from physical parameters.) Since the coefficients themselves can be the parameters, this is completely general. We show that by considering the space of parameters (instead of the space of coefficients), we can exploit any special structure the coefficient-parameter formulas impose on the solution set. In particular, we prove that we can ignore generic positive-dimensional solution sets. We also show that any “side conditions” (additional equations) can be used to reduce the number of paths. These results unify and generalize all current approaches to polynimial continuation. The effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas. For example, we establish a 36-path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory. Also, we show that a 16-path homotopy suffices for the inverse kinematics problem for 6-revolute-joint manipulators, a problem previously requiring 32 paths.