On algorithms for solving systems of polynomial equations

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Abstract
Methods for finding numerical solutions of nonlinear algebraic systems of equations have been given considerable attention since the birth of the field of numerical analysis. The fact that these methods find many applications to problems in physics, engineering, economics, and mathematical theory of optimization cannot be overstressed. However, a significant number of these problems contain indeterminants or parameters, which should only be given numerical values at the very end of the computational processes. Sometimes numerical results simply cannot provide enough insight for the analysis of the problem. Furthermore, symbolic solutions via elimination theory provide not only all solutions to a given system of equations but also a classification of solutions into solution surfaces or parametrized solutions. Thus, the symbolic method can provide an infinite number of solutions where this feat is clearly impossible for the numerical methods.

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