The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture

TL;DR

The paper introduces the Schanuel Subset Conjecture as an alternative to Schanuel's Conjecture and demonstrates that it implies Gelfond's Power Tower Conjecture, providing conditional proofs of these longstanding problems.

Contribution

It proposes the Schanuel Subset Conjecture and shows it implies Gelfond's Power Tower Conjecture, offering a new approach to these conjectures.

Findings

SSC implies Gelfond's Power Tower Conjecture

Conditional proofs of key conjectures assuming SSC

Discussion on the relation between SSC and SC

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.