The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture

Abstract

As an alternative to the famous Schanuel's Conjecture (SC), we introduce the Schanuel Subset Conjecture (SSC): Given ${\alpha }_1,...,{\alpha }_n\in \mathbb{C}$ linearly independent over $\mathbb{Q}$, if $\{{\alpha }_1,...,{\alpha }_n,e^{{\alpha }_1},...,e^{{\alpha }_n}\}$ is $\overline{\mathbb{Q}}$-dependent on a subset $\{{\beta }_1,...,{\beta }_n\}$, then ${\beta }_1,...,{\beta }_n$ are algebraically independent}. (A set $X\subset \mathbb{C}$ is called $\overline{\mathbb{Q}}$-dependent on $Y\subset \mathbb{C}$ if $\overline{\mathbb{Q}}(X)\subset \overline{\mathbb{Q}}(Y)$.) We discuss whether SC is equivalent to the a priori weaker SSC. Assuming SSC, we give conditional proofs of Gelfond's Power Tower Conjecture and of two other results.

9 pages, added Proposition 1 proving that SSC implies SC for $n=2$

References in corpus (1)

• Algebraic and transcendental solutions of some exponential equations