The Fekete-Szego Theorem on Curves, with Splitting Conditions

调查员和他的学生学习能力理论及其应用算术几何。容量是集合大小的度量，它出现在势理论中，并在概率、复分析和数论中有应用。 这个项目的主要目标是证明一个非常强的版本的Fekete-Szego定理代数曲线，它断言，如果一个adelic集的曲线有足够大的容量，那么存在代数点与所有共轭附近，在adelic意义。新的定理将是一个基本的存在定理产生代数点，这是受真实的和p-adic合理性条件，以及拓扑约束。为了确定概念，对于复平面中的集合，圆的容量等于其半径，线段的容量是其长度的四分之一。 在分析中，主要的区别是容量为0的集合和正容量的集合：容量为0的集合对全纯函数是“不可见的”。 在数论中，主要的区别是容量大于1和小于1的集合之间的区别。 Fekete和Szego的经典定理说，对于复平面上的一个在复共轭下稳定的集合，如果该集合的容量大于1，则该集合的每个邻域包含无穷多个代数整数的伽罗瓦轨道。 大卫康托推广定理的adelic集的射影线，随后调查推广到adelic集的代数曲线。 作为一个应用，调查证明了存在定理代数整数点仿射代数簇，这一直是对象的大量工作的Moret-Bailly和Szpiro和他们的学生。 早些时候，拉斐尔·罗宾逊在另一个方向上扩展了费凯特-塞戈定理，证明了如果一个容量大于1的集合包含在真实的直线中，那么每个真实的邻域包含无限多个完全真实的代数整数的伽罗瓦轨道。 最近调查证明了一个adelic版本的罗宾逊定理，证明存在的代数数，这是完全真实的和完全p-adic在有限数量的地方。 该项目的目标是将这个定理推广到代数曲线。 所使用的方法将涉及p-adic分析，潜在的理论和近似理论的代数函数。丢番图方程的研究（寻找多变量多项式方程的整数解）是一个非常古老和非常困难的课题，可以追溯到希腊人。 只是在过去的半个世纪里，使用现代数论的方法才取得了很大的进展。 两个著名的结果是“希尔伯特第十问题”（由Matiyasevich在1970年）的负解，该问题询问是否存在确定给定方程是否有整数解的算法;以及费马大定理的负解（由Wiles在1995年），该问题询问整数的n次幂和是否可以是n次幂。 调查员的工作追求一个不同的方向，表明，为更大的算术域比整数，在适当的条件下确实存在的解决方案;而且存在算法告诉是否存在。 目前的工作将大大减少已知存在解决方案的域的大小。

The investigator and his students study capacity theory and its applications to arithmetic geometry. Capacity is a measure of size for sets, which arises in potential theory and has applications in probability, complex analysis, and number theory. The chief goal of this project is to prove a very strong version of the Fekete-Szego theorem on algebraic curves, which asserts that if an adelic set on a curve has large enough capacity, then there exist algebraic points with all their conjugates near it, in an adelic sense. The new theorem will be a basic existence theorem producing algebraic points which are subject to real and p-adic rationality conditions, as well as topological constraints. To fix ideas, for sets in the complex plane, the capacity of a circle turns out to equal its radius, and the capacity of a line segment is a quarter of its length. In analysis, the primary distinction is between sets of capacity 0 and sets of positive capacity: sets of capacity 0 are 'invisible' to holomorphic functions. In number theory, the main distinction is between sets of capacity greater than 1, and less than 1. The classical theorem of Fekete and Szego says that for a set in the complex plane stable under complex conjugation, if the capacity of the set is greater than 1, then every neighborhood of the set contains infinitely many Galois orbits of algebraic integers. David Cantor generalized the theorem to adelic sets on the projective line, and subsequently the investigator generalized it to adelic sets on algebraic curves. As an application, the investigator proved an existence theorem for algebraic integer points on affine algebraic varieties, which has been the object of considerable work by Moret-Bailly and Szpiro and their students. Earlier, Raphael Robinson had extended the Fekete-Szego theorem in another direction, showing that if a set of capacity greater than 1 were contained in the real line, then every real neighborhood contained infinitely many Galois orbits of totally real algebraic integers. Recently the investigator proved an adelic version of Robinson's theorem, proving the existence of algebraic numbers which were totally real and totally p-adic at a finite number of places. The goal of the project is to generalize this theorem to algebraic curves. The methods used will involve p-adic analysis, potential theory, and approximation theory for algebraic functions. The study of diophantine equations (looking for integer solutions to polynomial equations in several variables) is a very old and very difficult subject, going back to the Greeks. It is only within the last half-century that much progress has been made, using methods of modern number theory. Two famous results were a negative solution to "Hilbert's Tenth problem" (by Matiyasevich in 1970), which asks if there is an algorithm for determining whether or not a given equation has integer solutions; and the negative resolution of Fermat's Last Theorem (by Wiles in 1995), which asks if sums of n-th powers of integers can be n-th powers. The investigator's work pursues a different direction, showing that for much larger arithmetic domains than the integers, under appropriate conditions there do exist solutions; and moreover there exist algorithms for telling whether or not they exist. The current work will considerably reduce the size of the domains where solutions are known to exist.