Definability and Decidability over Algebraic Extensions of Product Formula Fields

乘积公式域代数扩展的可定义性和可判定性

• 批准号: 0650927
• 负责人: Alexandra Shlapentokh
• 金额: $ 11.61万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0650927&HistoricalAwards=false
• 关键词: Definability; Decidability; over; Algebraic; Extensions

The main goal of this project is to increase our understanding of what is definable and decidable in the language of rings. The interest in the questions of polynomial definability and decidability dates back to the time of the solution of Hilbert's Tenth Problem (HTP). At the beginning of the XX century a famous German mathematician David 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 Matiyasevich, building on the work by Martin Davis, Hilary Putnam and Julia Robinson showed that the sets of integers which can be defined using polynomial equations and the sets of integers that can be listed by a computer program were the same, and thus showed that an algorithm sought by Hilbert did not exist. Matiyasevich'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 such as Mazur's conjectures and various conjectures for elliptic curves. Many 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 author of this proposal believes that this line of research will continue to reveal new areas of interaction between Number Theory, Algebraic Geometry and Logic, enriching all the fields involved.The language of rings or the language of polynomial equations is widely used in almost all branches of Mathematics, sciences and social sciences, and therefore understanding what can be expressed by this language and whether we can determine algorithmically which sentences in this language are true is of importance to many areas of Mathematics and beyond.

这个项目的主要目标是增加我们对环语言中什么是可定义和可判定的理解。对多项式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）的解决方案。在二十世纪初，一位著名的德国数学家大卫希尔伯特提出了以下问题：是否有一种算法可以确定一个任意的多元多项式方程和整数系数是否有整数解？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可以用多项式方程定义的整数集合和可以用计算机程序列出的整数集合是相同的，从而表明希尔伯特所寻求的算法并不存在。Matiyasevich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？这个问题是未解决的这一天。经常是这样的困难问题的数学，HTP的有理数以及它的姐妹问题，HTP的环整数的号码领域，产生了许多新的问题，如马祖尔的代数和各种代数椭圆曲线。这些问题中有许多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。这一建议的作者认为，这一研究路线将继续揭示数论、代数几何和逻辑之间相互作用的新领域，丰富所涉及的所有领域。环的语言或多项式方程的语言广泛应用于数学、科学和社会科学的几乎所有分支，因此，理解这种语言可以表达什么，以及我们是否可以通过算法确定这种语言中的哪些句子是正确的，这对数学的许多领域以及其他领域都很重要。