Sufficient conditions for a problem of Polya

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