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Schanuel’s conjecture and algebraic powers \(z^w\) and \(w^z\) with \(z\) and \(w\) transcendental. (English) Zbl 1226.11075
The authors prove several conditional results on transcendence assuming Schanuel’s conjecture. The conjecture states that if numbers \(\alpha_1,\dots,\alpha_n\in\mathbb C\) are linearly independent over \(\mathbb Q\) then there are at least \(n\) algebraically independent numbers among \(\alpha_1,\dots,\alpha_n,e^{\alpha_1},\dots,e^{\alpha_n}\).
Under this conjecture the authors prove that if for \(z,w\in\mathbb C\setminus\{0,1\}\) the numbers \(z^w\) and \(w^z\) are algebraic, then \(z\) and \(w\) are either both rational or both transcendental. This, for instance, implies the transcedence of the numbers \(i^{e^\pi}\) and \(i^{i^i}\).
Another conditional result is that if \(\alpha\neq0\) and \(z\) are complex numbers with \(\alpha\) algebraic and \(z\) irrational such that \(\alpha^{\alpha^z}=z\), then \(z\) is transcendental.

11J81 Transcendence (general theory)
Full Text: arXiv