The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture

2012/12/31 by Marques, Diego, Sondow, Jonathan
#11J72 #11J81 (Primary) 11J85 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1212.6931

Abstract

As an alternative to the famous Schanuel's Conjecture (SC), we introduce the Schanuel Subset Conjecture (SSC): Given α₁,...,α_n∈ ℂ linearly independent over ℚ, if \α₁,...,α_n, e^α₁,...,e^α_n\ is ℚ-dependent on a subset \β₁,...,β_n\, then β₁,...,β_n are algebraically independent. (A set X⊂ ℂ is called ℚ-dependent on Y⊂ ℂ if ℚ(X) ⊂ 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.

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