The approximation of solutions to linear homogeneous differential equations by rational functions

Abstract

Using methods analogous to those in diophantine approximation, the existence of good rational (function) approximations to the general solutions of each of a class of linear homogeneous differential equations is investigated. It is shown that, while many good approximations may exist, no truly extraordinary approximations exist. The proofs are via Lemmas on formal series which remain valid in a much wider context, e. g. even if the field of constants has positive characteristic.