Greatest Common Divisors, Integral Points, and Diophantine Approximation

最大公约数、积分点和丢番图近似

• 批准号: 2001205
• 负责人: Aaron Levin
• 金额: $ 34.98万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-05-15 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001205&HistoricalAwards=false
• 关键词: Greatest; Common; Divisors; Integral; Points

The project studies topics at the core of arithmetic and number theory. One of the most basic objects in mathematics is the greatest common divisor of two integers. The project will investigate generalizations and analogues of recently developed inequalities for greatest common divisors, and their connections with Vojta’s conjecture, a central and far-reaching conjecture. Another fundamental and important question, going back to antiquity, concerns understanding integer solutions to polynomial equations. The work will bring new ideas and perspectives to this important question, including research toward methods allowing one to algorithmically compute all integer solutions to large classes of equations. A fundamental tool for studying such equations comes from the subject of Diophantine approximation, which in its most basic form studies how well a real number can be approximated by rational numbers. The project will study various generalizations of one of the primary results in this subject, Schmidt’s subspace theorem. This research has close connections to and consequences for diverse areas of mathematics beyond number theory, including complex analysis and geometry. Additionally, the project will support a wide range of mentoring activities and research opportunities, involving the training of undergraduate students, graduate students, and postdoctoral researchers.The research supported by this award will involve the study of several questions revolving around greatest common divisors, integral points, Diophantine approximation, and their interrelations. The first set of projects center on the investigator's recent higher-dimensional generalization of results on greatest common divisors of polynomials evaluated at S-units. The project will study generalizations and extensions related to Vojta's conjecture, analogues in function fields, including applications, and a novel approach to analogous problems for abelian varieties. A second set of projects will focus on integral points on varieties. First, the project will continue work on aspects of Siegel’s theorem for integral points of bounded degree on curves. Second, the investigator will study an approach to effectively proving Siegel’s theorem for genus two curves. Fundamental tools for studying these problems come from the subject of Diophantine approximation, where Schmidt's subspace theorem is a central result.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.

该项目研究算术和数论的核心课题。数学中最基本的对象之一是两个整数的最大公约数。该项目将研究最近发展的最大公约数的不等式的推广和类比，以及它们与Vojta猜想的联系，Vojta猜想是一个中心和深远的猜想。另一个基本而重要的问题，可以追溯到古代，涉及到理解多项式方程的整数解。这项工作将为这一重要问题带来新的想法和视角，包括研究允许人们通过算法计算大类方程的所有整数解的方法。研究这类方程的一个基本工具来自丢番图近似，它的最基本形式是研究实数用有理数逼近的程度。该项目将研究本学科的一个主要结果--施密特子空间定理的各种推广。这项研究与数论以外的不同数学领域有着密切的联系和影响，包括复杂分析和几何。此外，该项目将支持广泛的指导活动和研究机会，包括本科生、研究生和博士后研究人员的培训。该奖项支持的研究将涉及围绕最大公约数、积分点、丢番图近似及其相互关系的几个问题的研究。第一组项目集中于研究人员最近对在S单位求值的多项式的最大公约数的结果的高维推广。该项目将研究与Vojta猜想有关的推广和扩展，函数领域的类似物，包括应用，以及解决阿贝尔变种类似问题的新方法。第二套项目将侧重于品种上的积分点。首先，该项目将继续研究曲线上有界次积分点的Siegel定理。其次，研究人员将研究一种有效证明亏格两条曲线的Siegel定理的方法。研究这些问题的基本工具来自丢番图近似的主题，其中施密特子空间定理是核心结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

On the degeneracy of integral points and entire curves in the complement of nef effective divisors

论nef有效除数的补中积分点和整条曲线的简并性

• DOI: 10.1016/j.jnt.2020.05.013
• 发表时间: 2020
• 期刊: Journal of Number Theory
• 影响因子: 0.7
• 作者: Heier, Gordon;Levin, Aaron
• 通讯作者: Levin, Aaron

Hilbert’s Irreducibility Theorem andideal class groups of quadratic fields

希尔伯特不可约定理和二次域的理想类群

• DOI: 10.4064/aa211224-22-9
• 发表时间: 2022
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: Kulkarni, Kaivalya R.;Levin, Aaron
• 通讯作者: Levin, Aaron

Quadratic fields with a class group of large 3-rank

• DOI: 10.4064/aa191027-22-6
• 发表时间: 2019-10
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: A. Levin;Shengkuan Yan;Luke Wiljanen

A generalized Schmidt subspace theorem for closed subschemes

• DOI: 10.1353/ajm.2021.0008
• 发表时间: 2017-12
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Gordon Heier;A. Levin

Intersections in subvarieties of ${\mathbb {G}}_{\mathrm {m}}^l$ and applications to lacunary polynomials

${mathbb {G}}_{mathrm {m}}^l$ 子类型的交集及其在缺陷多项式中的应用

• DOI: 10.1090/tran/8470
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Corvaja, Pietro;Levin, Aaron;Zannier, Umberto
• 通讯作者: Zannier, Umberto