Number Theory with Emphasis on Algorithms and Algebraic Number Theory

数论，重点是算法和代数数论

• 批准号: 0100485
• 负责人: Hendrik Lenstra
• 金额: $ 22.28万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100485&HistoricalAwards=false
• 关键词: Number; Theory; Emphasis; Algorithms; Algebraic

Lenstra0100485The proposed research belongs to the interface between number theory andalgebra. It is inspired by problems that come up in an algorithmiccontext and in arithmetic algebraic geometry. Altogether, the proposalcontains 19 problem sets: five from Algorithmic Number Theory, threefrom Algebraic Number Theory, five from Commutative and HomologicalAlgebra, three from the Geometry of Numbers, and three from Group Theory.The collection has been composed with a view towards assisting theinvestigator's many current and future graduate students in choosingsuitable thesis subjects. The problems have the appealing features ofappearing to be feasible without being trivial, and of being specificwithout being narrow. They belong to mainstream areas that will alsoserve the students after obtaining their degrees. Of the nineteenproblem sets, the following two are both easy to formulate andattractive. The first is the development of an algorithmic theory ofquadratic forms over rings and fields of arithmetic interest. A typicalquestion is how quickly one can find a representation of a positiveinteger as a sum of four squares. Or: it is known that any odd unimodularindefinite inner product space over the ring of integers isdiagonalizable; given the symmetric matrix that defines the innerproduct, how quickly can one find the change of basis that diagonalizesthe form? A first investigation shows that one may expect a wide spectrumof answers to the algorithmic questions in this area, displaying all theriches of number-theoretic algorithms. The second is giving class numberestimates for orders in number fields. What is a good upper bound for thenumber of equivalence classes of fractional ideals of a giving order,expressed as a function of the degree and the discriminant of the order?And can one find better estimates for orders that have nice properties,such as being Gorenstein? This type of question is of importance in thetheory of abelian varieties, and one will need to apply techniques comingfrom commutative algebra, abelian group theory, combinatorics, andelementary analytic number theory.In order to place the project in perspective one may consider the recentdevelopment of number theory. Present day number theory differs in twoimportant respects from number theory twenty five years ago, namely inthe roles played by algorithms and computers, and by algebraic geometry.It has been found that algorithmic number theory has importantapplications, notably in cryptography, and in addition number theoristshave learned how to use computers for their research. Inventing goodcomputational methods for number-theoretic problems has thus become ofcentral importance. One of the principal investigator's strengths is inthe interaction between theory and practice, on the one hand using recenttheoretical advances for algorithmic purposes and on the other handderiving purely mathematical inspiration from the problems suggested bythe applications. At the other end of the spectrum, knowledge ofalgebraic geometry has become a standard requirement for aspiring numbertheorists. Virtually every breakthrough in number theory over the pastfew decades, including Andrew Wiles's work on Fermat's Last Theorem, hasinvolved arithmetic algebraic geometry. Algebraic geometry depends on abroad spectrum of techniques from algebra and algebraic number theory,and gives rise to an unending array of tantalizing questions in thoseareas, of which the project studies a sample. What is maybe the mostexciting of all, is the way in which arithmetic algebraic geometry andalgorithmic number theory are presently being tied together, both in theapplication of geometric objects to cryptography and in the applicationof algorithmic techniques to investigate geometric objects in numbertheory. The project will be carried out by the investigator's graduatestudents, many of whom will, as experience shows, acquire combinedexpertise in these two areas, which is a very precious but fairly rarecommodity.

所提出的研究属于数论和代数之间的接口。它的灵感来自于算法背景和算术代数几何中的问题。总的来说，该提案包含19个问题集：5个来自算法数论，3个来自代数数论，5个来自交换同调代数，3个来自数的几何，3个来自群论。该集合已组成，以协助研究者的许多当前和未来的研究生在选择合适的论文科目的观点。这些问题具有吸引人的特点，既可行又不琐碎，既具体又不狭隘。它们属于主流领域，在获得学位后也将为学生服务。在19个习题集中，以下两个都很容易表述，也很吸引人。第一个是关于环和算术领域上的二次型的算法理论的发展。一个典型的问题是，一个人多快能找到一个正整数的四个平方和的表示形式。或：已知整数环上的任何奇单模不定内积空间是可对角的；给定定义内积的对称矩阵，一个人能多快找到使形式对角化的基的变换？第一次调查表明，人们可以期待在这个领域的算法问题的广泛的答案，显示所有理论的数论算法。第二种方法是对编号字段中的订单给出类编号估计。一个给定阶的分数理想等价类的数目的上界是什么？用阶的度和阶的判别式的函数来表示？对于具有良好性质的阶数能否找到更好的估计，比如戈伦斯坦阶数？这类问题在阿贝尔变理论中很重要，需要运用交换代数、阿贝尔群论、组合学和初等解析数论的技巧。为了正确地看待这个项目，我们可以考虑一下数论的最新发展。今天的数论在两个重要方面与25年前的数论不同，即算法和计算机所扮演的角色，以及代数几何。人们已经发现算法数论有重要的应用，特别是在密码学中，此外，数论家已经学会了如何使用计算机进行研究。因此，为数论问题发明好的计算方法变得至关重要。首席研究员的优势之一是理论与实践的相互作用，一方面利用最新的理论进展来实现算法目的，另一方面从应用程序提出的问题中获得纯粹的数学灵感。在光谱的另一端，代数几何知识已经成为有抱负的数论学家的标准要求。在过去的几十年里，数论的每一个突破，包括安德鲁·怀尔斯对费马大定理的研究，都涉及到算术代数几何。代数几何依赖于代数和代数数论的广泛技术，并在这些领域产生了无穷无尽的诱人问题，该项目研究了其中的一个样本。也许最令人兴奋的是，算术、代数几何和算法数论目前正在结合在一起，无论是在将几何对象应用于密码学方面，还是在应用算法技术研究数论中的几何对象方面。该项目将由研究者的研究生进行，经验表明，他们中的许多人将获得这两个领域的综合专业知识，这是非常宝贵但相当罕见的商品。