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决定和可定义的理解。更具体地说，我们专注于从希尔伯特第十问题的解决方案演变而来的存在/丢芬图定义性问题。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可决性问题。涉及椭圆曲线的新方法有望使这些问题更容易解决。我们还研究了函数域上的存在可定义性。在特征为0的函数场的情况下，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目前似乎遥不可及。对丢芬图的可定义性和可决性问题的兴趣可以追溯到希尔伯特第十问题（HTP）的解决。20世纪初，希尔伯特提出了这样一个问题（其中之一）：是否存在一种算法，可以确定任意一个多变量整数系数多项式方程是否有整数解？在20世纪70年代早期，yuri Matijasevich在Martin Davis， Hilary Putnam和Julia Robinson的基础上证明了丢芬图集合和可计算枚举的整数集合是相同的，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，这个问题被证明是更加令人烦恼的：是否存在上述算法，但对于有理数的解？这个问题至今没有解决。就像数学中的难题一样，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，它们本身就很有趣，这是本提案的作者计划研究的。其中一些问题后来被证明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。期望源于http的问题将在数论、代数几何和逻辑之间产生许多新的互动领域。

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.