Mathematical Sciences: Diophantine Geometry in Positive Characteristics

该奖项支持沃洛克教授的研究工作， 丢番图方程理论中最重要的一个 丢番图研究 也就是说，寻找整数或 有理数的多项式方程，是一个非常经典的话题 在数论中。 这一课题的最新进展大多源于 从类似问题中获得的见解， 数字被有理函数所取代，即所谓的函数 外勤案件 本项目将研究这最后一个案例。 以前的工作 作者的研究导致了定性性质的一般结果。 希望在这个项目中获得量化的结果， 到有效的界限，让人们真正找到解决方案， 到方程式。 这是数论领域的研究。 数论 从整数和问题开始， 一个整数被另一个整数整除。 它是其中的 最古老的数学领域，它最初是追求 纯粹的美学原因。 然而，在过去的半个世纪里， 它已经成为开发新算法的重要工具， 计算机科学和新的电子纠错码。

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.