Mathematical Sciences: Homotopy Continuation Method for Deficient Polynomial Systems

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8902663 Li Most systems of n polynomial equations in n unknowns arising in applications are "deficient", in the same sense that they have fewer solutions than that predicted by the total degree of the system. A procedure is introduced and will be examined in light of efficiency and applicability for finding all the isolated zeros of deficient polynomial systems which need to be solved repetitively with varying coefficients. The procedure is based on the cheater's homotopy, a continuation method which follows paths to all solutions. All solutions are found with an amount of computational work roughly proportional to the actual number of solutions, while previous general methods normally require an amount of computation roughly proportional to the total degree. The research will further investigate the structure of the deficiency on a case-by-case basis. The main concern is, in a specific case, to find an initial polynomial system for the homotopy that is simple to solve and also produces few extraneous homotopy paths.