Diophantine Approximation and Nevanlinna Theory

研究者从与全纯曲线值分布相似的角度（Nevanlinna理论）研究丢番图逼近，反之亦然。 具体来说，这项工作包括：（i）工作有关Faltings' 1983年定理的Shafarevich猜想图埃的方法，（ii）工作寻找进一步的应用几何对数导数引理，（iii）搜索数论对应的引理对对数导数，和（iv）技术改进证明使用图埃的方法（戴森引理）。这个项目涉及数论（具体来说，解决多项式方程的整数或有理数，以及相关的不平等）和值分布理论（不等式满足幂级数函数的一个或几个复杂的变量）。 这两个截然不同的数学领域有许多共同的现象。 这已经部分地解释了两个领域之间的形式类比，这导致了新的假设，现有定理的较短证明和新定理。 本项目使用这种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.