Existential Definability over Product Formula Fields

这个项目的主要目标是增加我们对环语言中存在的可判定和可定义的理解。更具体地说，我们专注于存在/丢番图的可定义性，从希尔伯特的第十问题的解决方案已经演变的问题。 该领域的主要开放问题涉及有理数和数域代数整数环的存在判定性。 涉及椭圆曲线的新方法已经显示出使这些问题更接近的希望。 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，我们也看一阶可定义性问题，因为相应的存在可定义性问题似乎遥不可及的时刻。 对丢番图可定义性和可判定性问题的兴趣可以追溯到希尔伯特第十问题（HTP）的解决。 在开始的二十世纪希尔伯特提出了以下问题（除其他外）：是否有一种算法，可以确定是否任意多项式方程在几个变量和整数系数有整数的解决方案？在20世纪70年代早期，尤里·马蒂亚舍维奇在马丁·戴维斯、希拉里·普特南和朱莉娅·罗宾逊的工作基础上证明了丢番图集和可计算的整数集是相同的，从而表明希尔伯特所寻求的算法并不存在。Matijasevich的结果立即提出了另一个问题，这被证明是更令人烦恼：是否有一个算法如上所述，但解决方案的合理数目？ 这个问题至今没有解决。正如数学中常见的难题一样，有理数的HTP及其姊妹问题，数域整数环的HTP，产生了许多新的问题，这些问题本身就很有趣，本提案的作者计划对此进行研究。 其中一些问题原来是数论或代数几何的问题，但它们反过来又在逻辑学中产生了相当有趣的结果。 我们的期望是，起源于HTP的问题将产生数论，代数几何和逻辑之间的相互作用的许多新领域。

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.