The Newton Polyhedron in the Nonlinear Analysis

We give a simple presentation of an algorithm of selecting asymptotical first approximations of equations (algebraic and ordinary differential and partial differential). Here the first approximation of a solution of the initial equation is a solution of the corresponding first approximation of the equation. The algorithm is based on the geometry of power exponents including the Newton polyhedron. We give also a surwey of applications of the algorithm in problems of Celestial Mechanics and Hydrodynamics.