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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 ${\alpha }_{1},\cdots ,{\alpha }_{n}\in ℂ$ are linearly independent over $ℚ$ then there are at least $n$ algebraically independent numbers among ${\alpha }_{1},\cdots ,{\alpha }_{n},{e}^{{\alpha }_{1}},\cdots ,{e}^{{\alpha }_{n}}$.

Under this conjecture the authors prove that if for $z,w\in ℂ\setminus \left\{0,1\right\}$ 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 $\alpha \ne 0$ and $z$ are complex numbers with $\alpha$ algebraic and $z$ irrational such that ${\alpha }^{{\alpha }^{z}}=z$, then $z$ is transcendental.

##### MSC:
 11J81 Transcendence (general theory)