Existential Definability over Product Formula Fields

产品公式字段的存在可定义性

• 批准号: 0354907
• 负责人: Alexandra Shlapentokh
• 金额: --
• 依托单位: East Carolina University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354907&HistoricalAwards=false
• 关键词: Existential; Definability; over; Product; Formula

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.

该项目的主要目标是增加我们对环语言中可判定和可定义的存在的理解。更具体地说，我们专注于存在主义/丢番图可定义性问题，这些问题是从希尔伯特第十个问题的解决方案演变而来的。 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判定性。 涉及椭圆曲线的新方法已显示出使这些问题更容易解决的希望。 我们还研究了函数域的存在可定义性。 对于特征为 0 的函数域，我们还研究一阶可定义性问题，因为相应的存在可定义性问题目前似乎遥不可及。 对丢番图可定义性和可判定性问题的兴趣可以追溯到解决希尔伯特第十问题（HTP）的时候。 在二十世纪初，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具有整数系数的多个变量的任意多项式方程是否具有整数解？ 1970 年代初，尤里·马蒂耶维奇 (Yurii Matijasevich) 在马丁·戴维斯 (Martin Davis)、希拉里·普特南 (Hilary Putnam) 和朱莉娅·罗宾逊 (Julia Robinson) 的工作基础上证明了丢番图集和可计算可枚举整数集是相同的，从而表明希尔伯特寻求的算法并不存在。马蒂耶维奇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：除了有理数解之外，是否存在如上所述的算法？ 这个问题至今仍未解决。正如数学中常见的难题一样，有理数的 HTP 及其姐妹问题（数域整数环的 HTP）产生了许多新问题，这些问题本身就非常有趣，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中产生了非常有趣的结果。 人们期望源自 HTP 的问题将在数论、代数几何和逻辑之间产生许多新的相互作用领域。