Sufficient conditions for a problem of Polya

Abstract

Let $\alpha$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb{Q}(\alpha)$ with Galois group $G$ and $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. In this article, among the other results, we prove the following. If $f\in \bar{\mathbb{Q}}[G]$ is a non-zero element of the group ring $\bar{\mathbb{Q}}[G]$ and $\alpha$ is a given algebraic number such that $f(\alpha^n)$ is a non-zero algebraic integer for infinitely many natural numbers $n$, then $\alpha$ is an algebraic integer. This result generalizes the result of Polya [11], Corvaja and Zannier [2] and Philippon and Rath [9]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [4], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al., [6] which are applications of the Schmidt subspace theorem.