FINDING ALL SOLUTIONS TO POLYNOMIAL SYSTEMS AND OTHER SYSTEMS OF EQUATIONS

In a previous paper, the authors suggested a procedure for obtaining all solutions to certain systems of n equations in n complex variables. The idea was to start with a trivial system of equations to which all solutions were easily known. The trivial system was then perturbed into the given system. During the perturbation process, one followed the solution paths from each of the trivial solutions into the solutions of the given system. All solutions to the given system were thereby obtained. This paper utilizes a different approach that eliminates the requirement of the previous paper for a leading dominating term in each equation. We add a dominating term artificially and then fade it. Also we rely on mathematically more fundamental concepts from differential topology. These advancements permit the calculation of all solutions to arbitrary polynomials and to various other systems of n equations in n complex variables. In addition, information on the number of solutions can be obtained without calculation. © 1979 The Mathematical Programming Society.