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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.

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