[Paper Review] Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

Diego Marques, Jonathan Sondow|arXiv (Cornell University)|Oct 29, 2010

History and Theory of Mathematics8 references5 citations

TL;DR

Assuming Schanuel's conjecture, this paper proves that if $ z $ and $ w $ are transcendental complex numbers (not 0 or 1) and both $ z^w $ and $ w^z $ are algebraic, then $ z $ and $ w $ must both be transcendental—ruling out one being algebraic and the other transcendental. Key results include the transcendence of specific iterated powers like $ \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $, $ i^{i^i} $, and $ i^{e^\pi} $, and a proof that solutions to $ \alpha^{\alpha^z} = z $ with algebraic $ \alpha \neq 0,1 $ and irrational $ z $ must have $ z $ transcendental.

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.

Motivation & Objective

• To investigate the arithmetic nature of complex powers $ z^w $ and $ w^z $ when both $ z $ and $ w $ are transcendental.
• To determine whether $ z^w $ and $ w^z $ can both be algebraic when $ z $ and $ w $ are transcendental, under Schanuel's conjecture.
• To establish conditions under which iterated exponential expressions like $ \alpha^{\alpha^z} = z $ imply transcendence of $ z $.
• To extend known results on transcendence of numbers such as $ e^\pi $, $ i^i $, and $ \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $ using Schanuel's conjecture.

Proposed method

• Assumes Schanuel's conjecture (S), which states that if $ \alpha_1, \dots, \alpha_n $ are linearly independent over $ \mathbb{Q} $, then $ \alpha_1, \dots, \alpha_n, e^{\alpha_1}, \dots, e^{\alpha_n} $ contain at least $ n $ algebraically independent numbers.
• Uses the multiplicative independence of algebraic and transcendental numbers to derive algebraic independence of logarithms and iterated logarithms.
• Applies the logarithmic transformation $ w = \log \alpha / \log z $ to relate $ z^w $ and $ w^z $ to logarithmic expressions and analyze their algebraic independence.
• Employs the transcendence degree (trdeg) over $ \mathbb{Q} $ to measure algebraic independence of sets involving $ \alpha, \beta, z, \log \alpha, \log \beta, \log z, \log \log \alpha, \log \log \beta, \log \log z $.
• Derives a contradiction via algebraic independence of logarithmic terms when assuming $ z $ is algebraic while $ w $ is transcendental, under the assumption that $ z^w $ and $ w^z $ are algebraic.
• Uses the identity $ z^{-1} \log \beta = \log \log \alpha - \log \log z $ to expose inconsistency with expected algebraic independence under (S).

Experimental results

Research questions

• RQ1 If $ z $ and $ w $ are transcendental and $ z^w $, $ w^z $ are algebraic, can one of them be algebraic while the other is transcendental?
• RQ2 Does Schanuel's conjecture imply the transcendence of specific iterated exponential constants such as $ \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $, $ i^{i^i} $, and $ i^{e^\pi} $?
• RQ3 Can the equation $ \alpha^{\alpha^z} = z $ with algebraic $ \alpha \neq 0,1 $ and irrational $ z $ have an algebraic solution $ z $?
• RQ4 What constraints does Schanuel's conjecture impose on the multiplicative and algebraic independence of logarithmic expressions involving $ z^w $ and $ w^z $?

Key findings

• If $ z $ and $ w $ are transcendental and $ z^w $, $ w^z $ are algebraic, then $ z $ and $ w $ must both be transcendental—no mixed algebraic-transcendental pairs are possible under Schanuel's conjecture.
• Schanuel's conjecture implies the transcendence of $ \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $, $ i^{i^i} $, and $ i^{e^\pi} $, as these expressions lead to contradictions with algebraic independence under (S).
• There exist transcendental positive real numbers $ x, y, s \neq t $ such that $ x^y \neq y^x $ and $ s^t = t^s $ are both integers, assuming Schanuel's conjecture.
• If $ \alpha \neq 0 $ is algebraic and $ \alpha^{\alpha^z} = z $ with $ z $ irrational, then $ z $ must be transcendental under Schanuel's conjecture.
• The proof relies on showing that the set $ \{ \log \alpha, z \log \alpha, \alpha^z \log \alpha \} $ is linearly independent over $ \mathbb{Q} $, leading to a transcendence degree contradiction under (S).
• The contradiction arises from assuming $ z $ algebraic while $ \alpha^z $ is transcendental, leading to a trdeg of at least 3, but the field extension has degree at most 2, violating Schanuel's conjecture.