Diophantine Approximation and Nevanlinna Theory

丢番图近似和奈万林纳理论

• 批准号: 0200892
• 负责人: Paul Vojta
• 金额: $ 20.04万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200892&HistoricalAwards=false
• 关键词: Diophantine; Approximation; Nevanlinna; Theory

The investigator studies diophantine approximation from the point ofview of the analogy with value distribution of holomorphic curves (Nevanlinnatheory), and vice versa. Specifically, this work includes:(i) work on relating Faltings' 1983 theorem on the Shafarevich conjecture to Thue's method,(ii) work on finding further applications of the geometric logarithmic derivative lemma,(iii) a search for a number-theoretic counterpart to the lemma on the logarithmic derivative, and(iv) technical improvements in proofs using Thue's method (Dyson's lemma).This project involves number theory (specifically, solutions of polynomialequations in integers or rational numbers, as well as related inequalities)and value distribution theory (inequalities satisfied by power seriesfunctions of one or several complex variables). These two seeminglydisparate areas of mathematics have many phenomena in common. This hasbeen partly explained by a formal analogy between the two fields, whichhas led to new conjectures, shorter proofs of existing theorems, and newtheorems. The present project uses this analogy to further elucidatethe underlying structure of number theory, value distribution theory,and the analogy between the two fields.

研究人员从与全纯曲线的值分布的类比的角度研究丢番图逼近(nevanlinna理论)，反之亦然。具体地说，这项工作包括：(I)将Faltings 1983年关于Shafarevich猜想的定理与Thue方法联系起来；(Ii)寻找几何对数导数引理的进一步应用；(Iii)寻找与对数导数引理对应的数论；(Iv)使用Thue方法(Dyson引理)对证明进行技术改进。本项目涉及数论(具体地说，整数或有理数中多项式方程的解，以及相关的不等式)和值分布理论(由一个或多个复变量的幂函数所满足的不等式)。这两个看似完全不同的数学领域有许多共同的现象。这部分是由两个领域之间的形式类比来解释的，这导致了新的猜想，现有定理的更短的证明，以及新的定理。本项目使用这个类比来进一步阐明数论、值分布理论的基本结构，以及这两个领域之间的类比。