Nonlinear reduction for solving deficient polynomial systems by continuation methods

SummaryNeither continuation methods, nor symbolic elimination methods can be directly applied to compute all finite solutions to polynomial systems, because the amount of computational time is mostly not proportional to the dimension of the system and to the number of finite solutions. The notion of S-polynomials is used to developed a reduction algorithm to lower the total degree of the deficient polynomial system, so that computing the solutions at infinity can be avoided. Applying the reduction algorithm before solving the system with continuation methods, yields a reliable solution method.