Analytic Methods For Diophantine Problems

丢番图问题的解析方法

• 批准号: 9622773
• 负责人: Trevor Wooley
• 金额: $ 11.51万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622773&HistoricalAwards=false
• 关键词: Analytic; Methods; Diophantine; Problems

This award provides funding for a project concerned with the theory and application of the Hardy-Littlewood circle method. The investigator is pursuing an improved, detailed understanding of the circle method both from the analytic viewpoint, through estimates for mean values of exponential sums, and from an arithmetic viewpoint, through an understanding of the role played by the trivial solutions of diophantine systems lying on special subvarieties. The analytic viewpoint will be considered in through further investigation and development of the proposer's new method for estimating fractional moments of smooth Weyl sums and through detailed investigations of low moments of exponential sums. The arithmetic viewpoint will be considered by continuing the investigator's research on the number of non-diagonal solutions of systems of symmetric diagonal equations using slicing methods and also perhaps by sieve methods. The latter viewpoint will also be pursued through the testing of the "Quasi- Hardy-Littlewood" model suggested recently by Vaughan and Wooley. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

该奖项为有关Hardy-Littlewood循环方法的理论和应用的项目提供资金。研究者正在从分析的角度（通过对指数和的平均值的估计）和从算术的角度（通过对位于特殊子变种上的丢番图系统的平凡解所起的作用的理解）追求对圆法的改进的、详细的理解。分析的观点将通过进一步的研究和发展提出的估计光滑Weyl和分数阶矩的新方法，并通过对指数和的低阶矩的详细研究来考虑。通过继续研究者对对称对角方程系统的非对角解的数目的研究，研究者将使用切片法和筛分法来考虑算术观点。后一种观点也将通过对Vaughan和Wooley最近提出的“准Hardy-Littlewood”模型的检验来追求。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。