Explicit and computational approaches to arithmetic geometry

数学科学的很大一部分是关于寻找各种类型方程的解的方法。最简单的一种方程是多项式方程，其中变量仅使用加法和乘法组合。这是有史以来最古老的方程类型。 [2]如果一个方程的解仅限于自然数、整数或有理数，那么我们就称这个方程为丢番图方程。此类方程出现在许多实际情况中，例如共振问题、将选举结果转化为席位分布以及有关离散配置的各种问题。* * 一般来说，判断一个多项式方程是否有整数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，已经证明了创建一个通用算法（计算机程序）来决定给定多项式方程是否存在整数解是根本不可能的。相反，类似问题的合理解的答案是未知的。 对于一类称为曲线的特殊方程，最近的工作已经产生了一种方法，该方法通常可以确定在实践中是否存在任何合理的解决方案。启发式的论点表明，这些方法可能总是有效的，对于一类特殊的曲线，称为超椭圆曲线，实际实验证实了这一点。 本研究计划旨在提高我们对多项式方程有理解的理解。它将制定新的和改进现有的方法，以确定是否有任何解决办法，并确定所有解决办法，如果它们存在的话。它将特别支持确保计算方法扩展到超椭圆曲线的特殊类别之外的努力。

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.*** *It is generally a very hard problem to decide if a polynomial equation has an integer valued solution. In fact, in response to a challenge that Hilbert set in 1900, it has been proved 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. Heuristic arguments suggest that these methods might always work and for a special class of curves, called hyperelliptic curves, practical experiments confirm this.*** 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. It will particularly support efforts to ensure that computational methods get extended beyond the special class of hyperelliptic curves.**