Effective Diophantine Geometry over Function Fields

函数域上的有效丢番图几何

• 批准号: 9701489
• 负责人: Minhyong Kim
• 金额: $ 6.9万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9701489&HistoricalAwards=false
• 关键词: Effective; Diophantine; Geometry; over; Function

Kim 9701489 This award provides funds for M. Kim to continue his past research on the Diophantine geometry of curves over function fields of arbitrary characteristic. In elementary language, this is the study of equations of type f(x,y,t)=0 (1) viewed as a two-variable polynomial equation whose coefficients are functions of t to which we seek solutions (x,y)=(p(t), q(t)), that is, pairs of polynomials (or rational functions) which satisfy f(p(t),q(t),t)=0 as a function of t. In particular, he will continue his previous work on `effective' Mordell conjectures over function fields, which allows one to find all solutions to (1) (for f's of high genus) by giving a priori bounds on the `height' (which in this case is the degree) of solution pairs (p(t),q(t)). A result of this sort over number fields is the long-term eventual goal of this research This is an outstanding problem in Diophantine geometry whose resolution would provide a crude algorithm for finding all rational solutions to Diophantine equations in two variables over the rational numbers. The specific projects go in three closely related directions: (1) finding more refined geometric height inequalities in arbitrary characteristic to serve as efficient input into algorithmic search for solutions; (2) embedding the PI's previous work into the study of curves on surfaces of general type in the spirit of Bogomolov's theorem on boundedness of curves of bounded genus, and that of his joint work with Shepherd-Barron; (3) approaching geometric height inequalities from a more explicitly algorithmic viewpoint in the hope of finding techniques which would transfer more readily to number fields than the existing ones. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error corre cting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

Kim 9701489该奖项为M. Kim提供了资金，以继续他过去对曲线的二世几何学的研究，而不是任意特征的功能领域。 在基本语言中，这是对F（x，y，t）型方程= 0（1）的研究，被视为一个两种可变性的多项式方程，其系数是t的功能（x，y）=（x，y）=（p（t），q（t），q（t），是（q（t），q（t）），是（或有理由）的功能，该功能是（或有理由）=（t）（t）=（t）（t）（t）（t）（c） t。 特别是，他将继续他先前在功能领域的“有效” Mordell猜想上的工作，这使人们可以通过在“高度”（在这种情况下是溶液对的程度）（p（p（t），q（t））给出先验界限（在这种情况下，这是一个（高属）（高属））。 这类数字领域的结果是这项研究的长期最终目标这是毒液几何学中的一个杰出问题，其分辨率将提供一种粗略的算法，用于在两个变量中在合理数量的两个变量中找到所有理性解决方案。 特定项目沿三个密切相关的方向进行：（1）在任意特征中找到更多精制的几何高度不平等，以作为算法搜索解决方案的有效输入； （2）将PI的先前工作嵌入到Bogomolov定理的一般类型表面上的曲线研究中，这些曲线是关于有限属曲线的界限，以及他与Shepherd-Barron的共同工作； （3）从更明确的算法观点接近几何高度不平等现象，希望找到比现有字段更容易转移到数字字段的技术。 该项目属于算术几何形状的一般领域 - 一个融合了数学最古老的领域的主题：数字理论和几何学。 事实证明，这种组合非常富有成果 - 最近解决了经受住几代人的问题。 它的许多后果包括新的错误corre cing代码。 此类代码对于现代计算机（硬盘）和紧凑磁盘都是必不可少的。