Definability and Decidability over Algebraic Extensions of Product Formula Fields

这个项目的主要目标是增加我们对戒指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断性问题的兴趣可以追溯到希尔伯特第十问题(HTP)的解的时候。在二十世纪初，著名的德国数学家大卫·希尔伯特提出了以下问题(其中包括)：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，证明了可以使用多项式方程定义的整数集和可以由计算机程序列出的整数集是相同的，从而表明希尔伯特所寻求的算法不存在。Matiyasevich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，而不是针对有理数的解？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，如Mazur猜想和椭圆曲线的各种猜想。这些问题中的许多后来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。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，丰富涉及的所有领域。环的语言或多项式方程的语言在数学、科学和社会科学的几乎所有分支中都被广泛使用，因此，了解这种语言可以表达什么，以及我们是否能够通过算法确定这种语言中的哪些句子是正确的，对于数学和其他领域的许多领域都是重要的。

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.