Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

TL;DR

Assuming Schanuel's conjecture, the paper explores properties of algebraic powers involving transcendental numbers, proving new results about their algebraic and transcendental nature and deriving implications for specific complex numbers.

Contribution

The paper proves new conditional results on algebraic and transcendental powers assuming Schanuel's conjecture, including properties of z^w and w^z and specific transcendental numbers.

Findings

If (S) holds, z^w and w^z are algebraic only when z,w are both rational or both transcendental.

Under (S), existence of four transcendental numbers with specific exponential properties is established.

Certain complex numbers like sqrt(2)^sqrt(2)^sqrt(2) are shown to be transcendental assuming (S).

Abstract

We give a brief history of transcendental number theory, including Schanuel's conjecture (S). Assuming (S), we prove that if z and w are complex numbers, not 0 or 1, with z^w and w^z algebraic, then z and w are either both rational or both transcendental. A corollary is that if (S) is true, then we can find four distinct transcendental positive real numbers x, y, s, t such that the three numbers x^y=/=y^x and s^t=t^s are all integers. Another application (possibly known) is that (S) implies the transcendence of the numbers sqrt(2)^sqrt(2)^sqrt(2), i^i^i, and i^e^pi. We also prove that if (S) holds and a^a^z=z, where a=/=0 is algebraic and z is irrational, then z is transcendental.