Problems in the Geometry of Numbers and Diophantine Analysis

数几何问题和丢番图分析

• 批准号: 2001281
• 负责人: Shabnam Akhtari
• 金额: $ 22.05万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001281&HistoricalAwards=false
• 关键词: Problems; Geometry; Numbers; Diophantine; Analysis

A fundamental and central question in mathematics, going back to antiquity, concerns understanding integer solutions to polynomial equations. This question has been a source of inspiration in modern mathematics, as the techniques used to study this question have helped shape the foundation of several areas such as number theory, algebra, and analysis. This work will bring new perspectives to the field by studying various families of polynomial equations and by finding innovative applications in the rich area of number theory. The project will also enhance the training of graduate students and support mentoring activities and research opportunities for undergraduate students. The principal investigator aims to utilize tools from classical Diophantine analysis and modern applications of analytic number theory and arithmetic geometry to study basic arithmetic structures. This work will combine ideas from the geometry of numbers with both algebraic and analytic tools to study problems such as representation of integers by binary forms and orders in number fields, as well as various problems about heights in number fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

追溯到古代，数学中的一个基本和中心问题是关于理解多项式方程的整数解。 这个问题一直是现代数学的灵感来源，因为用于研究这个问题的技术有助于形成数论，代数和分析等几个领域的基础。这项工作将带来新的视角，通过研究各种家庭的多项式方程，并通过发现创新的应用领域丰富的数论。该项目还将加强对研究生的培训，并为本科生的辅导活动和研究机会提供支助。主要研究者旨在利用经典丢番图分析和解析数论和算术几何的现代应用工具来研究基本的算术结构。这项工作将联合收割机的想法，从几何的数字与代数和分析工具，研究问题，如整数的二进制形式和订单的代表性，以及各种问题的高度在数域。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

项目成果

Representation of integers by sparse binary forms

用稀疏二进制形式表示整数

• DOI: 10.1090/tran/8241
• 发表时间: 2021
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Akhtari, Shabnam;Bengoechea, Paloma
• 通讯作者: Bengoechea, Paloma

A positive proportion of locally soluble quartic Thue equations are globally insoluble

局部可解的四次 Thue 方程的正比例是全局不可解的

• DOI: 10.1017/s0305004121000554
• 发表时间: 2021
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: AKHTARI, SHABNAM

Quartic index form equations and monogenizations of quartic orders

四次指数形式方程和四次阶的单生化

• DOI: 10.2140/ent.2022.1.57
• 发表时间: 2022
• 期刊: Essential Number Theory
• 作者: Akhtari, Shabnam

Independent relative units of low height

低高度独立相对单位

• DOI: 10.4064/aa210714-20-12
• 发表时间: 2022
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: Akhtari, Shabnam;Vaaler, Jeffrey D.
• 通讯作者: Vaaler, Jeffrey D.