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 generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], 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. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem.