Nonlinear homotopies for solving deficient polynomial systems with parameters

By a deficient polynomial system of n polynomial equation in n unknowns a system is meant that has fewer solutions than that predicted by the total degree, or the Bezout number, of the system. In practice, a deficient polynomial system is often associated with a set of parameters $q = (q_1 , \cdots ,q_s )$. In this paper, nonlinear homotopies for numerically determining all isolated solutions of deficient m -homogeneous systems are introduced. The procedure is that when a system $P(q,x) = 0$ with particular parameter $q^0 $ is solved, it can be used to solve $P(q,x) = 0$ for any parameter q. It is assumed that for a generic q, $P(q,x)$ shares the same kind of deficiency at infinity. Thus far fewer paths need to be followed. As an application, it is shown how to compute the 32 solutions of the Tsai–Morgan manipulator problem with 26 parameters by following only 32 solution paths.