Algebraic and transcendental solutions of some exponential equations

Abstract

We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.

14 pages, 7 figures, corrected top of p. 4

References in corpus (1)

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

Cited by in corpus (3)

• Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental
• Transcendence of values of the iterated exponential function at algebraic points
• The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture