A proof of Pisot’s $d^{\mathrm{th}}$ root conjecture


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


Umberto Zannier