Diophantine Approximation and Nevanlinna Theory

研究者从类比全纯曲线的值分布（Nevanlinnatheory）的角度研究丢芬图近似，反之亦然。具体来说，这项工作包括：(i)将Faltings 1983年关于Shafarevich猜想的定理与Thue的方法联系起来的工作，（ii）寻找几何对数导数引理的进一步应用，（iii）寻找对数导数引理的数论对应项，以及（iv）使用Thue的方法（戴森引理）证明的技术改进。该项目涉及数论（特别是整数或有理数多项式方程的解，以及相关不等式）和值分布理论（由一个或多个复变量的幂级数函数满足的不等式）。这两个看似不同的数学领域有许多共同的现象。这在一定程度上可以通过两个领域之间的形式类比来解释，这种类比导致了新的猜想、现有定理的简短证明和新的定理。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。

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.