An effective lower bound on the “Diophantine approximation” of algebraic functions by rational functions

Abstract

Publisher Summary This chapter reviews an effective lower bound on the diophantine approximation of algebraic functions by rational functions. In an article in the Proceedings in 1959, Professor Ellis Kolchin showed an analogue of Liouville's theorem, which dealt with the approximation of solutions of algebraic differential equations. The most concrete application of Kolchin's theorem is to the approximation by rational functions of formal power series solutions of a nontrivial algebraic differential equation. Kolchin's theorem says that there exists a natural number n and a real number c such that for every r, s ≠ 0: ord(y1 - r/s)