Linear and Nonlinear Algebraic Systems of Equations

The monograph is devoted to topical problems of a system of nonlinear algebraic equations with many variables. For algebraic systems containing different degrees and products of variables, analogues of Cramer's determinants are constructed, which contributes to the fulfillment of the criteria for the existence of solutions to nonlinear systems of equations. The elements of the constructed by determinants are numbers composed of the auxiliary coefficients of the system specified in the monograph of the method. The paper proposes two approaches to the study, using those that allow answering the question of the existence of solutions to a nonlinear system of algebraic equations, finding the number of these solutions and from the point of view of their definition. The results of the generalization of the results to the case of nonlinear systems of operator equations in finite Hilbert spaces are also given. The monograph will be useful for senior students, graduate students and mathematicians working in the field of linear algebra and spectral theory of operators.