Determining All Solutions to Certain Systems of Nonlinear Equations

Abstract
Only during the past decade has it been discovered how, under fairly reasonable conditions, to calculate a solution of a system of n nonlinear equations in n variables. The technique utilized was the so-called “complementarity” or “fixed point” approach to the Brouwer fixed point theorem. In this paper we extend that approach to calculate for certain systems not just one but all possible solutions. All solutions can be calculated when the equations of the system behave like a polynomial in the highest order terms. That is, in each equation the highest order term must be a variable raised to a power while the lower order terms can be more general. Our procedure starts with a trivial system of equations, one to which all solutions are obvious and immedately known. The trivial system is then deformed into the system given with the unknown solutions. The idea is to follow the paths generated from the obvious solutions into the unknown solutions of the given system. This process under rather general conditions calculates all of the solutions.

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