Problems of Definability and Decidability over Algebraic Fields

与本建议中讨论的问题相关的一般问题可以表述如下：什么是可判定的和可定义的环的语言？目前的研究路线起源于希尔伯特的第十问题和戴维斯，普特南，罗宾逊和Matiyasevich的解决方案。当尤里Matiyasevich表明，在合理的整数，丢番图集是相同的可计算的集合，他表明，对象起源于两个不同分支的数学是相同的。这一结果引发了对代数和逻辑共同根的搜索，该搜索以两个最重要的对象为中心：整数和有理数。虽然在搜索中采用的方法大多数是代数或几何的，但所问的问题起源于模型论或可计算性理论。马祖尔的progratures，其扩展和其后果存在的丢番图定义和丢番图模型是一个很好的例子，一个问题，在可计算性理论/模型理论产生的问题，代数几何反过来产生的后果，可计算性理论/模型理论。另一个例子之间的相互作用领域所激发的这条线的研究是最近的成果马祖尔和鲁宾关于职级的椭圆曲线超过一些领域的生产，以回答一些问题的丢番图的可定义性。在这个建议中，作者计划继续她的调查的可定义性和可判定性的代数扩展的有理数和合理的功能领域的所有特点。特别是，我们计划探索一阶定义的无限代数扩展的全球领域和存在的不确定性的一大类功能领域的积极特征。 这个提议研究了以下问题：给定一个多元多项式方程，是否有一个算法（或计算机程序）可以告诉我们这个方程是否在一个特定的集合中有解。这类问题可以追溯到德国数学家大卫希尔伯特谁问在二十世纪初的世纪，我们是否可以算法决定时，多项式方程在几个变量的整数系数有整数解。事实证明，这样的算法并不存在和解决方案的戴维斯，普特南，罗宾逊和Matiyasevich花了不少年。不幸的是，我们仍然不知道是否有一种算法来确定有理数（分数）而不是整数的解的存在性。那个问题比原来的那个更难。因为它往往是困难的问题的情况下，为了解决它，甚至只是为了更好地理解困难，数学领域已经创建。该领域认为多项式是一种特殊的数学语言（被许多数学领域和许多科学使用）的一部分，并试图确定人们可以用这种语言说什么，以及人们是否可以通过算法决定这种语言中的句子是否为真。目前的项目在各种环境中考虑这些问题，如“大环”（所有有理数（分数）的子集）和函数域（通过考虑多项式的比率形成的对象）。

The general questions associated with the problems discussed in this proposal can be stated as follows: what is decidable and definable in the language of rings? The current line of research has its origins in Hilbert's Tenth Problem and its solution by Davis, Putnam, Robinson and Matiyasevich. When Yurii Matiyasevich showed that over rational integers, Diophantine sets were the same as computably enumerable sets, he showed that objects originating in two different branches of Mathematics were the same. This result initiated a search for common roots of algebra and logic, a search centered on the two most important objects: integers and rational numbers. While the methods employed in the search are for the most part algebraic or geometric, the questions asked originate in Model Theory or Computability Theory. Mazur's conjectures, their extensions and their consequences for existence of Diophantine definitions and Diophantine models are a fine example of a question in Computability Theory/ Model Theory generating a question in Algebraic Geometry which in turn produces a consequence in Computability Theory/Model Theory. Another example of interaction between the fields spurred by this line of research are the recent results by Mazur and Rubin concerning ranks of elliptic curves over number fields which were produced to answer some questions of Diophantine definability. In this proposal the author plans to continue her investigation of definability and decidability over algebraic extensions of rational numbers and rational function fields of all characteristics. In particular, we plan to explore first-order definability over infinite algebraic extensions of global fields and existential undefinability of a large class of function fields of positive characteristic. This proposal investigates questions of the following sort: given a polynomial equation in several variables, is there an algorithm (or a computer program) that can tell us if this equation has solutions in a particular set. These kinds of questions go back to a German Mathematician David Hilbert who asked at the beginning of the XX century whether we could algorithmically decide when a polynomial equation in several variables with integer coefficients has integer solutions. It turned out that such an algorithm does not exist and the solution by Davis, Putnam, Robinson and Matiyasevich took quite a few years. Unfortunately, we still don't know whether there is an algorithm to determine existence of solutions in rational numbers (fractions) as opposed to integers. That problem turned out to be even harder than the original one. As it is often the case with difficult problems, in order to solve it or even just to understand better the difficulties, an area of Mathematics has been created. This area considers polynomials as a part of a special Mathematical language (used by many areas of Mathematics and many sciences) and tries to determine what one can say in this language and whether one can decide algorithmically whether a sentence in this language is true. The current project considers these questions in various settings such as ``big rings'' (subsets of all rational numbers (fractions)) and function fields (objects formed by considering ratios of polynomials).