Infinite Galois Theory in the Context of Hilbert's Tenth Problem

在1900年的国际数学家大会上，大卫·希尔伯特列出了他认为下个世纪数学中最重要的23个问题。希尔伯特的第十个问题需要一种算法来决定任何给定的丢番图方程是否有整数解。具体地说，该算法还能够决定任何这样的方程是否有有理数解。希尔伯特的第十个问题一直悬而未决，直到1970年，尤里·马蒂亚塞维奇终于证明了这样的算法不存在。然而，他的论文中介绍的方法只适用于整数解的情况，留下了有理数问题的悬而未决。Matiyasevich在他的原始论文中声称，相信算法存在的希尔伯特可能不会满足于目前只适用于整数的解决方案。他还指出，在1970年撰写本文时，“[理性]案例的进展相当有限。”近年来，柯尼斯曼教授的研究小组在解决希尔伯特关于有理数论的第十个问题方面取得了进展。在我的研究中，我想继续目前取得的进展，并采用新的技术来解决这个问题。具体地说，我想借鉴赋值理论、模型理论和无限伽罗瓦理论等领域的最新结果，这些领域是应用于希尔伯特第十问题的新的和有前途的领域。希尔伯特的第十个问题自然可以被翻译成模型论的语言，相当于有理数的存在一阶理论是否可判定的问题。此外，模型理论中的方法引用了更一般的估值理论。这导致了一些有趣的新结果和猜想，其中包括元素伽罗瓦猜想(EGC)。这一猜想表明，绝对伽罗瓦群与某些亨赛尔估值的存在之间存在着深刻的联系，这将产生深远的后果。一个这样的结果提供了证据，证明了有理数上的绝对伽罗瓦群编码了足够的信息来回答希尔伯特的第十个问题。然而，目前对这一群体的了解还远远不够，所以我的研究将在希尔伯特第十问题的背景下对这一对象进行进一步的调查。初等伽罗瓦猜想也与一个著名的猜想密切相关，尽管这个猜想还没有被很好地理解，Grothendieck的截面猜想在他1983年的程序中被称为“阿贝尔几何”。该项目属于EPSRC逻辑和组合学研究领域。

At the International Congress of Mathematicians in 1900, David Hilbert presented a list of 23 problems he deemed the most important questions in mathematics for the next century. Hilbert's Tenth Problem asks for an algorithm that decides for any given Diophantine equation if it has a solution in the integers or not. In particular, this algorithm would also be able to decide for any such equation if it has a solution in the rationals. Hilbert's Tenth Problem remained open until 1970 when Yuri Matiyasevich finally proved that such an algorithm could not exist. The methods introduced in his paper, however, only apply in the case of integer solutions, leaving the problem over the rationals open. In his original paper, Matiyasevich claims that Hilbert, who believed in the existence of an algorithm, would thus prob-ably not be satisfied with the current solution that only applies to the integers. He also points out that, at the time of writing in 1970, "progress in [the rational] case has been rather meagre." In recent years, progress has been made towards solving Hilbert's Tenth Problem over the rationals, in particular by Professor Koenigsmann's research group. In my research, I want to carry on the progress made so far and employ new techniques towards attacking the problem. In particu-lar, I want to draw from recent results in the areas of valuation theory, model theory, and infinite Galois theory, which are new and promising areas when it comes to applying them to Hilbert's Tenth Problem. Hilbert's Tenth Problem can naturally be translated to the language of model theory, amounting to the question of whether the existential first-order theory of the rational numbers is decidable or not. Moreover, methods from model theory have invoked more general valuation the-ory. This has led to a number of interesting new results and conjectures, among them the Elemen-tary Galois Conjecture (EGC). This conjecture suggests a deep connection between the absolute Galois group and the existence of certain henselian valuations, which would have far-reaching consequences. One such consequence provides evidence that the absolute Galois group over the rational numbers encodes sufficient information to answer Hilbert's Tenth Problem. At the mo-ment, however, this group is far from being fully understood, so my research will comprise a fur-ther investigation of this object in the context of Hilbert's Tenth Problem. The Elementary Galois Conjecture is also closely related to a famous, though not yet well understood conjecture, Grothendieck's Section Conjecture in his program called "anabelian geometry" from 1983. This project falls within the EPSRC Logic and Combinatorics research area.