Sufficient conditions for a problem of Polya (Q6583747)

In this paper, the authors study two problems, namely:\N\begin{itemize}\N\item[1.] They consider the generalised power sums of the form \(\lambda_1\alpha_1^l+\ldots+\lambda_k\alpha_k^l\) where \(\lambda_j, \alpha_j\) are algebraic numbers and \(l\geq1\) is any integer. If the power sum is an algebraic integer for infinitely many \(l\)'s, then under some conditions, they conclude that \(\alpha_j\)'s are algebraic integers. As applications of this result (Theorem 2.1), they consider the analogous situation over polynomials, group rings, function fields and a linear combination of trace powers of algebraic numbers.\N\item[2.] Some special cases wherein they restrict \(l\) to a finite (effective) set for a\Ngeneralised power sum.\N\end{itemize}\NThe results of the paper are nice and interesting.