Power Geometry

All local and asymptotic first approximations of a polynomial, a differential polynomial, and of a system of such polynomials can be selected algorithmically. Here the first approximation of a solution to the system of equations is a solution to the corresponding first approximation of the system of equations. The power transformations induce linear transformations of vector exponents and commute with the operation of selecting first approximations. In the first approximation of a system of equations they allow one to reduce the number of parameters and to reduce the presence of some variables to the form of derivatives of their logarithms. If the first approximation is the linear system, then in many cases the system of equations can be transformed into the normal form by means of the formal change of coordinates. The normal form is reduced to the problem of lower dimension by means of the power transformation. Combining these algorithms, we can resolve a singularity in many problems, find parameters determining the properties of solutions, and obtain the asymptotic expansions of solutions. Some applications from mechanics, celestial mechanics, and hydrodynamics are indicated. The work was partially supported by Russian Foundation for Basic Research, Grant 96-01-01411.