Mathematical Sciences: Diophantine Geometry in Positive Characteristics

该奖项支持沃洛赫教授致力于丢番图方程理论的研究。丢番图方程的研究，即寻找多项式方程的整数或有理数的解，是数论中一个非常经典的课题。这门学科最近的许多进展来自于从有理数被有理函数取代时的类似问题中获得的见解，即所谓的函数域情况。这个项目将研究最后一个案例。作者以前的工作已经导致了定性的一般结果。它希望在这个项目中获得定量的结果，导致有效的界限，使人们能够实际找到方程的解。这是数论领域的研究。数论始于整数和诸如一个整数被另一个整数整除的问题。它是最古老的数学领域之一，最初是出于纯粹的美学原因。然而，在过去的半个世纪里，它已经成为开发计算机科学的新算法和电子产品的新纠错码的必要工具。

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.