Heights, Mahler Measure and Diophantine Inequalities

高度、马勒测量和丢番图不等式

• 批准号: 0603282
• 负责人: Jeffrey Vaaler
• 金额: $ 23.24万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603282&HistoricalAwards=false
• 关键词: Heights; Mahler; Measure; Diophantine; Inequalities

DMS-0603282Jeffrey VaalerThe research contemplated in this proposal is directed at two related Diophantine problems about polynomials. The first problem is to better understand the distribution of values of the local and global Mahler measure of polynomials with coefficients restricted to either algebraic number fields or their completions. The main technical tool used to accomplish this is to study the Mellin transform of the distribution function associated to Mahler's measure. Then Mellin inversion techniques from analysis are used to obtain estimates or explicitformulas for the distribution functions. A further aspect of the research is to interpret the results in terms of the heights of algebraic numbers. The second problem is to consider the Wronskian of a collection of linearly independent polynomials with coefficients in an algebraic number field or the completion of such afield. We seek an inequality which relates the local and global Mahler measures of the Wronskian with the local and global Mahler measures of the original set of linearly independent polynomials. This is well understood for two polynomials, but only approximate results are known for a collection of three or more polynomials. Againthere is an interpretation in terms of the heights of the roots of the polynomials, and this turns out to be related to the ABC-inequality of Stothers and Mason. A further aspect of this research is to work out applications to certain Diophantine problems.A Diophantine equation is an equation to be solved in integers. The study of such equations is one of the oldest parts of number theory and continues to attract some of the best mathematical researchers. The tools developed in modern mathematics to study Diophantine equations can be extremely technical, and often involve related issues that can be described under the general title of "Diophantine problems". For example, one might study inequalities to be solved in integers, or to be solved in polynomials with integer coefficients. The research considered in this proposal is directed at exactly such Diophantine inequalities for polynomials. In all fields of science there is generally a great intellectual distance between the highly technical efforts of pure researchers and the eventual applications to business, technology or engineering. We do know that the area of Diophantine problems has important applications to computer problems, to the construction of encryption systems, and to other technological questions. A solution to the problems contemplated in this proposal would further our understanding of pure mathematical questions which belong to this important field. Also, this proposal will fund the research of several graduate students at the University of Texas. These students will benefit from the opportunity to concentrate on the research efforts, to travel and talk about their work at scientific meetings, and to consult with other researchers working on related problems. A further important goal of this work is the completion of a monograph written by the PI with the title ``Heights, Mahler measure and Diophantine inequalities". Over a period of many years the PI has written notes for graduate students in classes on this topic that the PI has taught at the University of Texas. These notes will be collected, edited and somewhat expanded into a monograph.

DMS-0603282 Jeffrey Vaaler在这个建议中考虑的研究是针对两个相关的关于多项式的丢番图问题。第一个问题是要更好地了解分布的局部和整体马勒措施的多项式系数限制在代数数域或其完成的值。 用于实现这一目标的主要技术工具是研究与马勒测度相关的分布函数的梅林变换。 然后利用分析中的梅林反演技术得到分布函数的估计或显式公式。 研究的另一个方面是用代数数的高度来解释结果。 第二个问题是考虑代数数域或代数数域的完备化中系数线性无关多项式集合的朗斯基。 我们寻求一个不等式，其中涉及的本地和全球的马勒措施的朗斯基与本地和全球的马勒措施的原始设置的线性无关的多项式。 对于两个多项式，这是很好理解的，但是对于三个或更多个多项式的集合，仅知道近似结果。 再次有一个解释的高度方面的根的多项式，这原来是有关的ABC不等式的Stothers和梅森。 本研究的另一个方面是对某些丢番图问题的应用。丢番图方程是一个需要求解整数的方程。 对这类方程的研究是数论中最古老的部分之一，并继续吸引着一些最好的数学研究人员。 在现代数学中开发的研究丢番图方程的工具可以是非常技术性的，并且经常涉及可以在“丢番图问题”的总标题下描述的相关问题。例如，人们可以研究用整数求解的不等式，或者用整数系数的多项式求解的不等式。 在这项建议中考虑的研究正是针对这样的丢番图不等式多项式。 在所有的科学领域中，在纯研究人员的高度技术性努力与最终应用于商业、技术或工程之间，通常存在着巨大的智力差距。 我们确实知道丢番图问题的领域在计算机问题、加密系统的构造以及其他技术问题上有重要的应用。 一个解决方案中设想的问题，这一建议将进一步我们的理解纯数学问题属于这一重要领域。 此外，这项提案将资助德克萨斯大学几名研究生的研究。 这些学生将有机会集中精力进行研究工作，在科学会议上旅行和谈论他们的工作，并与其他研究人员就相关问题进行咨询。这项工作的另一个重要目标是完成PI撰写的题为"高度、马勒测度和丢番图不等式”的专著。 多年来，PI在德克萨斯大学教授的课程中为研究生写了关于这个主题的笔记。 这些笔记将被收集、编辑并在某种程度上扩展成一本专著。