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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 α 1 ,,α n are linearly independent over then there are at least n algebraically independent numbers among α 1 ,,α n ,e α 1 ,,e α n .

Under this conjecture the authors prove that if for z,w{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 π and i i i .

Another conditional result is that if α0 and z are complex numbers with α algebraic and z irrational such that α α z =z, then z is transcendental.

11J81Transcendence (general theory)