Some consequences of Schanuel's Conjecture

TL;DR

This paper explores the implications of Schanuel's Conjecture on algebraic independence and disjointness of certain fields generated by exponential and logarithmic operations, providing conditional results in transcendence theory.

Contribution

It demonstrates that Schanuel's Conjecture implies algebraic independence of iterated logarithms and disjointness of specific exponential and logarithmic fields.

Findings

Schanuel's Conjecture implies algebraic independence of iterated logs.

Proves linear disjointness of fields generated by exponential and logarithmic processes.

Provides conditional results linking Schanuel's Conjecture to transcendence properties.

Abstract

During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = \overline{\mathbb{Q}}$ Inductively, for $n \geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $\exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n \geq 0$. Show that Schanuel's Conjecture implies that the numbers $\pi, \log \pi, \log \log \pi, \log \log \log \pi, \ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = \overline{\mathbb{Q}}$. Inductively, for $n \geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^y\in L_{n-1}$. Let $L$ be the union of $L_n$, $n \geq 0$. We were able to prove that Schanuel's Conjecture implies $E$ and $L$ are linearly disjoint over $\overline{\mathbb{Q}}$.