Global methods for solving systems of nonlinear algebraic equations

Systems of nonlinear algebraic equations (SNAE) are ubiquitous in the many applications requiring numerical simulation, and more robust and efficient methods for solving SNAE are continuously being sought. In this paper, we present an overview of existing algorithmic approaches for solving SNAE such as reduction to a Groebner basis, the multidimensional resultant method, and the spectral method. A major deficiency in all of these methods is the lack of a theoretical foundation that will allow a priori information about the number of solutions. In the present work, we recognize that SNAE are the principal object of an algebraic geometry and seek to derive qualitative criteria about the solution in an algebraic form. Desirable qualitative criteria include solvability and uniqueness. We show here that the problem of solving SNAE is equivalent to the problem of solving matrices of rank 1 in a given subspace of matrices. Recognizing such equivalencies is an important step to future success in developing improved methods for the solution of SNAE.