Sufficient conditions for a problem of Polya

Abstract

Let α \alpha be a non-zero algebraic number. Let K K be the Galois closure of Q ( α ) \mathbb {Q}(\alpha ) with Galois group G G and Q ¯ \bar {\mathbb {Q}} be the algebraic closure of Q \mathbb {Q} . In this article, among the other results, we prove the following. If f ∈ Q ¯ [ G ] f\in \bar {\mathbb {Q}}[G] is a non-zero element of the group ring Q ¯ [ G ] \bar {\mathbb {Q}}[G] and α \alpha is a given algebraic number such that f ( α n ) f(\alpha ^n) is a non-zero algebraic integer for infinitely many natural numbers n n , then α \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.