Some New Results on Power-Series Solutions of Algebraic Differential Equations

Abstract

This chapter presents some arithmetic properties of coefficients of a power series that satisfies certain equations, such as algebraic equations, algebraic differential equations, or Pfaffian systems of partial differential equations. If φ ɛ Q[[x]] satisfies a linear algebraic differential equation, then φ has a positive radius of convergence with respect to ⌊dm⌋v. The linear part of the differential equation does not have Fuchsian part at x = 0, and the theorem can be proved very easily. The difficulty comes from the Fuchsian part of the linear part of the differential equation. Thus, it naturally led to a Newton iteration procedure that is similar to those methods used in celestial mechanics. Archimedean estimates a power series solution of an algebraic differential equation, then the recurrences imply that the coefficients am in fact belong to a finite extension K of Q, that is, K is a number field.