Some consequences of Schanuel's Conjecture
Loading...
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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 $π, \log π, \log \log π, \log \log \log π, \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 $\overline{\mathbb{Q}}$.
8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf
8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf