Algebraic Numbers as Product of Powers of Transcendental Numbers

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in $\mathbb{C}\left[x\right]$ , and the problem of finding zeros in $\mathbb{Q}\left[x\right]$ leads to the definition of algebraic and transcendental numbers. Recently, Marques 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 $P_1,\dots ,P_n,Q_1,\dots ,Q_n$ are prescribed, non-constant polynomials in $\mathbb{Q}\left[x\right]$ (under weak conditions). More generally, our result generalizes results on the arithmetic nature of $z^w$ when z and w are transcendental.