Diophantine Definability and Decidability Over the Algebraic Extensions of Global Fields

本文将研究全局域及其代数扩张上的丢番图可定性和可判断性问题。该项目的长期目标是深入了解数域的代数整数环的q和环的丢番图(非)可判性。近期的目标是研究数域和函数域的某些子环上的丢番图定义性。具体地说，如果把域看作一个环，其中所有的素数都被允许出现在元素的“分母”中，那么我们相对较好地理解这个问题的整数和整数环可以被认为是不允许在分母中出现素数或只允许有限多个素数的环。因此，这个项目的一个自然中间步骤是了解环中发生了什么，在环中，允许无穷多个素数出现在元素的除数的分母中。这些环构成了首席调查员的主要研究重点。一个相关的问题是使用多项式方程定义素数集的完整性的问题。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，对于其剩余域是代数闭域的素数，这个问题也是公开的。研究人员还计划研究这个问题，目的是找出这种完整性不能用丢番图定义的情况。1900年，在一次国际数学家大会上，伟大的德国数学家大卫·希尔伯特提出了一系列问题，这些问题对XX世纪的数学发展产生了重大影响，其影响可能延伸到21世纪。清单上的第十个问题提出了一个问题，如果用现代的话来说，这个问题可以表述如下。有没有计算机程序可以确定任意多项式的多元方程是否有整数(整数)解？结果，答案是否定的。它花了很多年才获得，最终在60年代末尤里·马蒂亚塞维奇在朱莉娅·罗宾逊、马丁·戴维斯和大卫·普特南成果的基础上出现。有猜测称，希尔伯特没有预料到这一答案。他希望有一种算法来“解决”所有的多项式方程。这样的算法还可以求解所有答案允许为分数的多项式方程。当我们允许有理数(分数)作为解决方案时，整数解的计算机程序的缺失使问题变得很大。当我们允许多项式方程的解为分数时会发生什么这个问题的答案是该项目的长期目标。然而，这个问题目前似乎太难直接解决了。就像数学中的许多其他情况一样，循序渐进的攻势可能是必要的。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数字的集合，其中分母允许任何非零数，而整数是一个数字的集合，其中分母不允许有数字，那么可以将中间步骤想象为对数字集合的研究，其中分母允许一些但不是所有的数字。研究人员的直接目标是研究这组数字。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一个部分。因此，了解它们的逻辑属性可能有助于揭示许多其他问题。

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.