A proof of Pisot's dth root conjecture

Abstract

Let $\{b(n):n\in\N\}$ be the sequence of coefficients in the Taylor expansion of a rational function $R(X)\in\Q(X)$ and suppose that b(n) is a perfect $d^{\rm th}$ power for all large n. A conjecture of Pisot states that one can choose a $d^{\rm th}$ root a(n) of b(n) such that $\sum a(n)X^n$ 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.