Newton polyhedra and power transformations

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. The geometry admits transformations induced by power transformations of coordinates. We give also a survey of applications of the algorithms in problems of Celestial Mechanics and Hydrodynamics. (C) 1998 IMACS/Elsevier Science B.V.