Diophantine Definability and Decidability Over the Algebraic Extensions of Global Fields

摘要研究全局域的丢芬图可定义性和可判定性及其代数扩展问题。该项目的长期目标是深入了解Q和数域代数整数环的丢芬图（非）可决性。直接目标是研究数域和函数域的一些子域的丢番图可定义性。特别地，如果把域想象成一个环，允许所有素数出现在元素的“分母”上，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，可以被认为是一个环，分母上不允许有素数或只允许有有限个素数。因此，这个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上的环中会发生什么。这些环构成了首席研究员研究的主要焦点。一个相关的问题是用多项式方程定义质数集合上的完整性问题。对于素数集是有限的情况，这个问题已经在许多领域得到了解决。然而，即使对于有限集，对于剩余域为代数闭域的素数，这个问题也是开放的。研究者还计划研究这个问题，目的是确定这种完整性在丢番图语中无法定义的情况。1900年，在一次国际数学家大会上，伟大的德国数学家大卫·希尔伯特（David Hilbert）提出了一系列对20世纪数学发展产生重大影响的问题，这些问题的影响可能会延续到21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表述，可以这样说。是否有一个计算机程序可以确定任意多项式方程的几个变量是否有整数（整数）的解？答案是“不”。它花了很多年才得到，最终在60年代末，尤里·马蒂亚谢维奇在朱莉娅·罗宾逊，马丁·戴维斯和大卫·普特南的研究成果基础上的工作中出现。有人猜测希尔伯特并没有预料到这个答案。他希望有一种算法可以“解决”所有多项式方程。这样的算法还可以解决所有允许答案为分数的多项式方程。整数解的计算机程序的缺失给我们允许有理数（分数）作为解的情况留下了很大的问题。当我们允许多项式方程的解是分数时会发生什么，这个问题的答案是这个项目的长期目标。然而，目前这个问题似乎很难直接解决。在数学的许多其他情况下，渐进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转移到分数。如果一个人认为有理数是一组允许任何非零数字出现在分母上的数字，而整数是一组不允许任何数字出现在分母上的数字，那么他可以想象一个中间步骤，即研究一组允许某些数字出现在分母上的数字。研究者的直接目标是研究这样的一组数字。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个部分。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.