Analytic Solving of Equations of Polynomial Type in Variables and Derivatives: A Unified Calculus Based on Power Geometry

Abstract

Equations that are polynomial in variables and their derivatives appear throughout algebraic geometry, dynamical systems, and the theory of differential equations. This article presents a unified analytic calculus for obtaining asymptotic expansions and simplified representations of solutions to such equations. The approach relies on the systematic use of power geometry, including truncated equations, power transformations, logarithmic transformations, and normalizing coordinate changes. The calculus applies uniformly to algebraic equations, ordinary differential equations, autonomous systems, and partial differential equations. This expanded version provides a structured methodological framework, clarifies the stepwise procedure, and illustrates its relevance through conceptual applications. The methods presented here support the systematic construction of asymptotic solutions and enable the analytical treatment of nonlinear problems that often resist classical techniques.