Existential Definability over Product Formula Fields

这个项目的主要目标是增加我们对环的语言中什么是可确定的和可定义的存在的理解。更具体地说，我们专注于从希尔伯特的第十个问题的解决演变而来的存在主义/丢番图可定义性问题。该领域的主要公开问题涉及有理数和数域的代数整数环的存在可判性。涉及椭圆曲线的新方法在使这些问题更容易接近方面显示出了希望。我们还研究了函数域上的存在可定义性。在特征为0的函数域的情况下，我们也考虑一阶可定义性问题，因为相应的存在可定义性问题目前似乎是遥不可及的。丢番图的可定义性和可判定性问题的研究可以追溯到希尔伯特第十问题(HTP)的解决时期。在XX世纪初，希尔伯特提出了以下问题(以及其他问题)：有没有一种算法可以确定一个具有多个变量和整数系数的任意多项式方程是否有整数解？在20世纪70年代初，S、尤里·马蒂亚塞维奇在马丁·戴维斯、希拉里·普特南和朱莉娅·罗宾逊工作的基础上，证明了丢番图集和整数的可计算可枚举集是相同的，从而表明希尔伯特寻求的算法不存在。Matijasevich的结果立即提出了另一个问题，事实证明这个问题更加令人烦恼：是否有如上所述的算法，但对于有理数的解？这个问题至今仍未解决。就像数学中的难题一样，有理数的HTP以及它的姊妹问题--数域整数环的HTP产生了许多新的问题，它们本身就相当有趣，本提案的作者计划对此进行调查。其中一些问题被证明是数论或代数几何的问题，但它们反过来在逻辑中产生了相当有趣的结果。人们的期望是，源自HTP的问题将在数论、代数几何和逻辑之间产生许多新的交互领域。

The main goal of this project is to increase our understanding of what isdecidable and definable existentially in the language of rings. More specifically, we concentrate on issues of existential/Diophantine definability that have evolved from the solution of Hilbert's Tenth Problem. The main open problems in the area concern the existential decidability of rational numbers and rings of algebraic integers of number fields. New methods involving elliptic curves have shown promise in making these problems more approachable. We also investigate existential definability over function fields. In the case of function fields of characteristic 0 we also look at first-order definability problems since the corresponding existential definability problems seem out of reach at the moment. The interest in the questions of Diophantine definability and decidability dates back to the time of the solution of Hilbert's Tenth Problem (HTP). At the beginning of the XX century Hilbert asked the following question (among others): is there an algorithm that can determine whether an arbitrary polynomial equation in several variables and with integer coefficients has integer solutions? In the early 1970's, Yurii Matijasevich, building on the work by Martin Davis, Hilary Putnam and Julia Robinson showed that Diophantine sets and computably enumerable sets of integers were the same and thus showed that an algorithm sought by Hilbert did not exist. Matijasevich's result immediately raised another question which proved to be even more vexing: is there an algorithm as described above but for the solutions in rational numbers? This problem is unsolved to this day. As is often the case with difficult problems in Mathematics, HTP for rational numbers as well as its sister problem, HTP for the rings of integers of number fields, generated many new questions, quite interesting on their own, which the author of this proposal plans to investigate. Some of these questions turned out to be questions of Number Theory or Algebraic Geometry, but they in turn generated quite interesting consequences in Logic. The expectations are that questions originating in HTP will generate many new areas of interaction between Number Theory, Algebraic Geometry and Logic.