CAREER: Rational Points via Asymptotics and Geometry

职业：通过渐近学和几何学有理点

• 批准号: 1553459
• 负责人: Bianca Viray
• 金额: $ 45万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1553459&HistoricalAwards=false
• 关键词: CAREER; Rational; Points; via; Asymptotics

A wide range of systems of interest in science and engineering can be modeled by systems of polynomial equations. Often, particularly in computer science and cyber security, important quantities can be described by the set of solutions to a system of polynomial equations with rational coefficients, in other words by an algebraic variety. This project is concerned with determining when such a system has a solution where all coordinates are rational numbers. For an arbitrary system of equations, it can be quite difficult to determine whether there exist any such rational solutions. This research project develops a new perspective on this problem using geometry and asymptotic behavior. The research in this proposal is complemented by educational and outreach activities, including workshops for mid-to-late career graduate students focusing on presentation skills and how these skills can be leveraged to develop and broaden one's research program. The research projects fall in two main directions. The first focuses on the change in behavior of rational points and obstructions as the base field varies. In this direction, the project pursues a direct connection between the existence of 0-cycles and rational points on geometrically rational surfaces and explores new asymptotic questions that will shed light on Colliot-Thélène's conjecture on 0-cycles. The second focuses on leveraging dominant rational maps emanating from a variety X to obtain information about the Brauer group, rational points, and 0-cycles of X.

科学和工程领域的各种感兴趣的系统都可以通过多项式方程组来建模，特别是在计算机科学和网络安全中，重要的量可以通过具有有理系数的多项式方程组的一组解来描述。换句话说，该项目涉及确定这样的系统何时具有所有坐标均为有理数的解。对于任意方程组，确定是否存在此类有理解可能相当困难。本研究项目利用几何和渐近行为对这个问题提出了新的视角。该提案中的研究得到了教育和外展活动的补充，包括为中后期职业研究生举办的研讨会，重点关注演示技能以及如何利用这些技能来发展。并扩大研究项目的范围主要有两个方向：第一个重点是随着基础场的变化，理性点和障碍物的行为发生变化。在这个方向上，该项目追求0-存在之间的直接联系。循环和有理点第二个重点是利用来自各种 X 的主导理性图来获取有关布劳尔群、有理点和 0 循环的信息。 X 的。