Diophantine Definability and Decidability Over the Algebraic Extensions of Global Fields

摘要本文主要研究整体域上丢番图的可定义性和可判定性及其代数扩张问题。 该项目的长期目标是深入了解丢番图（联合国）的可判定性Q和环的代数整数数域。 近期目标是研究数域和函数域的某些子环上的丢番图可定义性。 特别是，如果把域看作一个环，其中所有的素数都允许出现在元素的“分母”中，那么整数和整数函数和整数的环，我们对这个问题的理解相对较好，可以被认为是分母中不允许有素数或只允许有200个素数的环。 因此，该项目的一个自然的中间步骤是理解在允许无限多个素数出现在元素的除数的分母中的环中会发生什么。 这些环构成了主要研究者的主要研究重点。 一个相关的问题是使用多项式方程定义素数集合的完整性的问题。 这个问题已经解决了许多领域的情况下，素数集是有限的。 然而，即使是有限集的问题是开放的素数的剩余领域是代数封闭的。 调查人员还计划研究这个问题的目标是确定这种情况下的完整性是不定义的丢番图terms.In 1900年，在国际数学家大会期间，一个伟大的德国数学家大卫希尔伯特提出了一个问题的清单有很大的影响数学的发展在二十世纪和其influence很可能会延伸到二十一世纪。 清单上的第十个问题提出了一个问题，如果用现代术语重新措辞，可以表述如下。 有没有一个计算机程序可以确定一个任意的多元多项式方程是否有整数解？ 答案是否定的。 它花了很多年才获得，它终于出现在六十年代后期的工作尤里Matyasevich建设的结果朱莉娅罗宾逊，马丁戴维斯和大卫普特南。 有人猜测希尔伯特没有预料到这个答案。 他希望有一种算法能“解”所有的多项式方程。 这样的算法也将解决所有多项式方程的答案是允许的分数。 由于没有整数解的计算机程序，当我们允许有理数（分数）作为解时，这个问题就大开了。 当我们允许多项式方程的解是分数时会发生什么的问题的答案是该项目的长期目标。 然而，这个问题目前似乎很难直接解决。 正如数学中的许多其他情况一样，渐进式的攻击可能是必要的。 作为我们程序的第一步，我们计划分阶段从整数到分数。如果一个人认为有理数是一个集合的数字，其中任何非零的数字是允许在分母，和整数作为一个集合的数字，其中没有数字是允许在分母，一个可以想象的中间步骤作为一个研究集合的数字，其中一些但不是所有的数字是允许在分母。 研究人员的直接目标是研究这些数字。 最后，我们应该注意到，有理数上的多项式方程几乎存在于数学及其应用的每一部分。 因此，理解它们的逻辑性质可能会揭示许多其他问题。

ABSTRACTThe investigator will study the issues of Diophantine definability and decidability over global fields and their algebraic extensions. The long term goal of the project is to gain insight into Diophantine (un)decidability of Q and rings of algebraic integers of number fields. The immediate goal is to study Diophantine definability over some subrings of number fields and function fields. In particular, if one thinks of the field as a ring where all the primes are allowed to occur in the "denominators" of the elements, then integers and rings of integral functions and numbers, where we understand the problem relatively well, can be considered as rings where no primes or only finitely many primes are allowed in the denominator. Thus a natural intermediate step for the project is to understand what happens in the rings where infinitely many primes are allowed to appear in the denominator of the divisors of the elements. These rings constitute the main focus of study by the Principal Investigator. A related issue is the problem of defining of integrality at sets of primes using polynomial equations. This problem has been solved for many fields for the case when the prime sets are finite. However, even for the finite sets the question is open for the primes whose residue fields are algebraically closed. The investigator also plans to study this problem with the goal of identifying situations where this kind of integrality is not definable in Diophantine terms.In 1900, during an International Congress of Mathematicians, a great German Mathematician David Hilbert presented a list of problems which had great influence on the development of Mathematics in the XX century and whose influnce is likely to extend to the XXI century. The tenth problem on the list asked a question which, if rephrased in modern terms, can be stated as follows. Is there a computer program which can determine whether an arbitrary polynomial equation in several variables has solutions in integers (whole numbers) ? The answer turned out to be "no". It took many years to obtain and it finally emerged in the late sixties in the work of Yurii Matyasevich building on results of Julia Robinson, Martin Davis and David Putnam. There has been speculation that Hilbert did not expect this answer. He hoped for an algorithm to "solve" all polynomial equations. Such an algorithm would also solve all polynomial equations where the answers are allowed to be fractions. The absence of the computer program for integer solutions left the question wide open for the case when we allow rational numbers (fractions) as solutions. The answer to the question of what happens when we allow solutions to polynomial equations to be fractions is the long term goal of the project. However, this problem currently seems too hard to approach directly. As in many other situations in Mathematics, a gradual assault is probably necessary. As a first step in our program we plan to move from integers to fractions in stages. If one thinks of rational numbers as a collection of numbers where any non-zero number is allowed in the denominator, and integers as a collection of numbers where no number is allowed in the denominator, one can visualize an intermediate step as a the study of collections of numbers where some but not all numbers are allowed in the denominator. The investigator's immediate goal is to study such sets of numbers. Finally we should note that polynomial equations over rational numbers are present in virtually every part of Mathematics and its applications. Thus understanding their logic properties is likely to shed light on many other problems.