The Fekete-Szego Theorem on Curves, with Splitting Conditions

研究人员和他的学生学习容量理论及其在算术几何中的应用。容量是集合大小的一种度量，它起源于位势理论，在概率论、复分析和数论中都有应用。这个项目的主要目的是证明关于代数曲线的Fekete-Szego定理的一个非常强的版本，该定理断言，如果曲线上的一个adelic集有足够大的容量，那么在adelic意义下，在它附近存在代数点及其所有的共轭。新的定理将是一个基本的存在定理，它产生的代数点受实数和p-进理性条件以及拓扑约束的约束。为了修正想法，对于复杂平面中的集合，圆的容量等于其半径，而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0的集合和正容量的集合：容量0的集合对于全纯函数来说是“看不见的”。在数论中，主要的区别是容量大于1和小于1的集合。经典的Fekete和Szego定理说，对于复共轭稳定的复平面上的一个集合，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Galois轨道。David Cantor将该定理推广到射影直线上的Adelic集，随后研究者又将其推广到代数曲线上的Adelic集。作为应用，作者证明了仿射代数簇上代数整点的一个存在定理，这是Moret-Bailly和Szpiro及其学生大量工作的目标。早些时候，拉斐尔·罗宾逊将Fekete-Szego定理推广到另一个方向，证明了如果实线中包含一组大于1的容量，则每个实邻域包含无限多个全实代数整数的伽罗瓦轨。最近，这位研究人员证明了罗宾逊定理的一个高级版本，证明了在有限个位置上存在全实、全p进的代数数。该项目的目标是将该定理推广到代数曲线。所使用的方法将涉及p-进分析、位势理论和代数函数的逼近理论。丢番图方程的研究(寻找多变量多项式方程的整数解)是一个非常古老且非常困难的课题，可以追溯到希腊人。只是在最近的半个世纪里，运用现代数论的方法才取得了很大的进展。有两个著名的结果是对“希尔伯特第十问题”(1970年由Matiyasevich提出)的负解，该问题询问是否有确定给定方程是否有整数解的算法；以及费马最后定理的负解(由Wiles于1995年提出)，它询问整数的n次方和是否可以是n次方。研究人员的工作追求了一个不同的方向，表明对于比整数大得多的算术域，在适当的条件下确实存在解；此外，还存在判断它们是否存在的算法。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.