Algebraic numbers as product of powers of transcendental numbers

Abstract

The elementary symmetric functions played a crucial role in the study of zeros of non-zero polynomials in $C[x]$, and the problem of finding zeros in $Q[x]$ leads to the definition of algebraic and transcendental numbers. Recently, [Marques, D. Algebraic numbers of the form $P(T)^{Q(T)}$, with $T$ transcendental, \textit{Elem. Math.} {\bf 2010}, {\em 65}, 78--80.] studied the set of algebraic numbers in the form $P(T)^{Q(T)}$. In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form $P_1(T)^{Q_1(T)}\cdots P_n(T)^{Q_n(T)}$ for some transcendental number $T$, where $P_1,\ldots,P_n,Q_1,\ldots,Q_n$ are prescribed, non-constant polynomials in $Q[x]$ (under weak conditions). More generally, our result generalizes results on the arithmetic nature of $z^w$ when $z$ and $w$ are transcendental.