Mathematical Sciences: Diophantine Geometry in Positive Characteristics

数学科学：积极特征中的丢番图几何

• 批准号: 9301157
• 负责人: Jose Voloch
• 金额: $ 8.31万
• 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: 1993
• 资助国家: 美国
• 起止时间: 1993-09-01 至 1996-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9301157&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Diophantine; Geometry; Positive

This award supports the research of Professor Voloch to work in the theory of Diophantine equations. The study of Diophantine equations, that is, the search for solutions in integers or rational numbers of polynomial equations, is a very classical topic in number theory. Much of the recent progress in the subject stems from insights obtained from the analogous problem when rational numbers are replaced by rational functions, the so-called function field case. This project will study this last case. Previous work of the author has led to general results of a qualitative nature. It is hoped in this project to obtain quantitative results, leading to effective bounds that allow one to actually find the solutions to the equations. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.

该奖项支持 Voloch 教授在丢番图方程理论方面的研究。 丢番图方程的研究，即寻找多项式方程的整数或有理数解，是数论中非常经典的课题。 该主题的最新进展大部分源于从有理数替换为有理函数时的类似问题（即所谓的函数域情况）中获得的见解。 本项目将研究最后一个案例。 作者之前的工作已经得出了定性的一般结果。 希望在这个项目中获得定量结果，从而产生有效的界限，使人们能够真正找到方程的解。 这是数论领域的研究。 数论从整数和诸如一个整数能否被另一个整数整除之类的问题开始。 它是最古老的数学领域之一，最初纯粹出于美学原因而追求它。 然而，在过去的半个世纪中，它已成为开发计算机科学新算法和电子学新纠错码的重要工具。