Some consequences of Schanuel's Conjecture

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dc.creatorCheng, Chuangxun
dc.creatorDietel, Brian
dc.creatorHerblot, Mathilde
dc.creatorHuang, Jingjing
dc.creatorKrieger, Holly
dc.creatorMarques, Diego
dc.creatorMason, Jonathan
dc.creatorMereb, Martin
dc.creatorWilson, S. Robert
dc.date2008-04-22
dc.date2008-05-08
dc.date.accessioned2026-07-07T09:37:29Z
dc.date.available2026-07-07T09:37:29Z
dc.descriptionDuring 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 $\overline{\mathbb{Q}}$.
dc.description8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf
dc.identifierhttps://arxiv.org/abs/0804.3550
dc.identifierhttp://arxiv.org/abs/0804.3550
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/160472
dc.subjectNumber Theory
dc.subject11J81
dc.titleSome consequences of Schanuel's Conjecture
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