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.