The Newton Polygon Method for Differential Equations

Abstract

We prove that a first order ordinary differential equation (ODE) with a dicritical singularity at the origin has a one-parameter family of convergent fractional power series solutions. The notion of a dicritical singularity is extended from the class of first order and first degree ODE’s to the class of first order ODE’s. An analogous result for series with real exponents is given. The main tool used in this paper is the Newton polygon method for ODE. We give a description of this method and some elementary applications such as an algorithm for finding polynomial solutions.