arXiv 0804.35502008-04math.NT

Schanuel猜想的一些结果

Some consequences of Schanuel's Conjecture

Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, S. Robert Wilson

备注：: 8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf

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摘要：

在亚利桑那冬季学校2008年（在图森，亚利桑那州举行），我们对以下问题的工作：a）（扩大由S。Lang）。定义$E_0 = \overline{\mathbb{Q}}$归纳地，对于$n \geq 1$，将$E_n$定义为通过数字$\exp(x)=e^x$在$E_{n-1}$上生成的字段的代数闭包，其中$x$的范围在$E_{n-1}$上。令$E$是$E_n$，$n \geq 0$的并集。证明了Schanuel猜想意味着数$π, \log π, \log \log π, \log \log \log π, \ldots $在$E$上代数独立。b）尝试得到涉及如下定义的字段$L$的（结构上的）概括。定义$L_0 = \overline{\mathbb{Q}}$。归纳地，对于$n \geq 1$，将$L_n$定义为由数$y$在$L_{n-1}$上生成的域的代数闭包，其中$y$在复数集合上的范围使得$e^y\in L_{n-1}$。令$L$为$L_n$的并集，即$n \geq 0$。我们能够证明Schanuel猜想意味着$E$和$L$在$\overline{\mathbb{Q}}$上线性不相交。

英文摘要：

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 $\overline{\mathbb{Q}}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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