Advanced Algebraic Frameworks for Solving Systems of Nonlinear Equations

Abstract

Systems of nonlinear equations arise throughout engineering, economics, physics, data science, optimization, and mathematical modelling. In contrast with linear systems, nonlinear systems may possess several isolated roots, singular or multiple roots, positive-dimensional solution sets, disconnected components, or no real solution. Their reliable treatment therefore requires more than a single local numerical iteration. This paper presents a unified algebraic and symbolic--numerical framework for the analysis and solution of nonlinear systems, with particular emphasis on polynomial ideals, Grobner bases, elimination theory, resultants, homotopy continuation, Newton-type refinement, and interval-based certification. Polynomial ideals provide a structural description of the solution set, while Grobner bases and resultants support exact elimination. Homotopy continuation complements symbolic elimination by tracing multiple solution paths and is especially effective for isolated complex roots of polynomial systems. Newton and quasi-Newton methods then provide rapid local refinement when suitable initial approximations are available. Since a small numerical residual alone is not a proof of the existence of a nearby exact root, interval Newton and Krawczyk-type operators are included as certification tools. The resulting workflow separates four tasks: structural analysis, global or algebraic solution generation, local refinement, and mathematical certification. A reproducible two-variable example demonstrates the convergence of Newton's method and illustrates how a Krawczyk inclusion test can certify a unique root in a prescribed interval box. The discussion clarifies the strengths and limitations of the principal methods and shows why hybrid symbolic--numerical strategies provide a more reliable computational approach than exclusive dependence on a single technique.