A Proof of Pisot's d th Root Conjecture

Abstract

Let {b(n) : n E N} be the sequence of coefficients in the Taylor expansion of a rational function R(X) E Q(X) and suppose that b(n) is a perfect dth power for all large n. A conjecture of Pisot states that one can choose a dth root a(n) of b(n) such that E a(n)Xn is also a rational function. Actually, this is the fundamental case of an analogous statement formulated for fields more general than Q. A number of papers have been devoted to various special cases. In this note we shall completely settle the general case.