The Role of Resultants in System of High-order Equations of Two Variables

Recently, a variety of iterative methods have been developed to solve nonlinear equations and systems of nonlinear equations. These methods have been improved through Taylor interpolation polynomial integration formulas, harmonic perturbation methods, and decomposition techniques.Among them, the binary higher-order equation systems, as an important branch of nonlinear equations, is a relatively common and widely applied type of nonlinear equation, and is the research object of this paper. Through the analysis of the structure of the result formula and its application, this study aims to summarize the current methods for solving System of high-order equations of two variables, clarify the conditions for determining the existence of solutions, and emphasize the key role of the result formula in the solution process. The core significance of this research lies in integrating the theory of resultant with modern numerical methods to establish a universal solution framework applicable to various binary polynomial systems, significantly improving the efficiency and reliability of solving binary high-order equation systems in complex engineering and scientific problems, and providing theoretical support and technical references for practical applications.