Explicit methods in arithmetic geometry

数学科学的很大一部分关注于开发各种类型的方程的解的方法。最简单的方程是多项式方程，其中变量仅使用加法和乘法组合。这是研究过的最古老的方程类型。如果仅限于自然数、整数或有理数的解，就称这样的方程为丢番图方程。像这样的方程在许多实际情况下都会出现，例如在共振问题中，将选举结果转化为席位分配，以及关于离散配置的各种问题。一般来说，判定一个多项式方程是否有整值解是一个非常困难的问题。事实上，为了回应希尔伯特在1900年提出的一个挑战，人们已经证明，根本不可能创建一个通用的算法(计算机程序)来确定给定的多项式方程是否存在整数解。相比之下，理性解决方案的类似问题的答案是未知的。对于一类称为曲线的特殊方程，最近的工作导致了一种方法，该方法通常可以确定在实践中是否存在任何有理解。我们甚至有理由相信，这种方法实际上可能总是有效的。这个研究项目旨在提高我们对多项式方程有理解的理解。它将开发新的和改进现有的方法来确定是否有任何解决方案，并确定所有解决方案是否存在。

A large part of the mathematical sciences is concerned with developing methods for finding solutions to various types of equations. The simplest kind of equation is the polynomial equation, where the variables are combined using just addition and multiplication. This is the oldest type of equation that was ever studied. One calls such an equation a Diophantine equation if one restricts to solutions that are natural numbers, integers, or rational numbers. Equations like this come up in many practical situations, such as in resonance problems, translating election results into seat distributions, and various questions about discrete configurations. In general, it is a very hard problem to decide if a polynomial equation has an integer valued solution. In fact, it has been proved, in response to a challenge that Hilbert set in 1900, that it is fundamentally impossible to create a general algorithm (computer program) that decides if there is an integral solution to a given polynomial equation. In contrast, the answer to the similar problem for rational solutions is unknown. For a special class of equations called curves, recent work has resulted in a method that can often decide whether there are any rational solutions in practice. We even have some reason to believe that the method might actually always work. This research program aims to improve our understanding of the rational solutions of polynomial equations. It will develop new and improve existing methods for deciding if there are any solutions at all and to determine all solutions if they exist.