Sufficient conditions for a problem of Polya

TL;DR

This paper extends Polya's problem conditions to algebraic numbers and sequences, establishing new criteria for algebraic integers using group ring elements, rational functions, and recurrence sequences, employing advanced number theory techniques.

Contribution

It introduces generalized sufficient conditions for algebraic integers involving group rings, rational functions, and recurrence sequences, broadening Polya's classical results.

Findings

If a group ring element applied to powers of an algebraic number yields algebraic integers infinitely often, then the number is an algebraic integer.

The results are extended to rational functions with algebraic coefficients.

A finite version of Polya's problem is proved for binary recurrence sequences of algebraic numbers.

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.