A new approach for finding all zeros for systems of nonlinear functions

A new concept in calculation techniques for finding all the zeros for a system of equations of nonlinear functions arising in various applications is presented. The concept is based on the following steps. First, the corresponding system of algebraic equations is created as a homomorphical model for an initial system of nonlinear functions. Second, this system is transformed to a Groebner basis. Third, the algebraic equations are solved by means of the original spectral method using constructing a system of spectral problems for rectangular pencils of matrices. In the paper, the computational symbolic-numerical procedure for this approach is described. The results of calculations based on this technique are presented for an application in theoretical analysis of the properties of the impurity-helium metastable phase under super-low temperatures.