Algebraic and transcendental solutions of some exponential equations

TL;DR

This paper investigates algebraic and transcendental solutions to specific exponential equations involving positive real numbers, applying classical theorems and Diophantine equations, with implications for iterated exponential functions.

Contribution

It provides a comprehensive analysis of solutions to several exponential equations using classical transcendence theorems and Diophantine methods, extending understanding of exponential function values.

Findings

Characterization of algebraic and transcendental solutions for each equation

Application of Hermite-Lindemann and Gelfond-Schneider theorems to these equations

Insights into the values of iterated exponential functions

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.