On a generalization of Pólya's theorem (Q2460536)

Pólya's theorem: Suppose an entire function \(f\) satisfies \(f(\mathbb{N})\subset\mathbb{Z}\) and \(| f(z)| =O(\exp(\gamma| z| )), \gamma<\log 2\). Then \(f\) is a polynomial. The author generalizes this result as follows: Suppose \(f(U)\subset \mathbb{K}\), where \(U\) is either \(\mathbb{N}\) (Theorem 1) or \(\mathbb{Z}\) (Theorem 2), and \(\mathbb{K}\) is the field of algebraic numbers. If \(f(z)\), the algebraic number \(f(n)\) and all of its conjugates satisfy certain exponential growth conditions, then \(f\) is a polynomial.