Diophantine Problems in Many Variables

9970440这个建议是关于丢番图方程在许多变量的背景下，在理论和应用的Hardy-Littlewood（圆）的方法。 提议者打算进行一些调查，其目标是更好地理解齐次方程组的整数解的存在性和密度，以及圆方法中使用的指数和的相关估计。 两个主题在拟议的调查中发挥着重要作用。 在一个方向上，熟悉的方法从应用的Hardy-Littlewood方法添加剂丢番图问题将扩展到处理丢番图方程更接近一般齐次方程。 在这里，提出者处理二进制形式和光滑数上的指数和的方法是重要的。在第二个方向，更基本的diagonalisation方法，起源于工作的布劳尔和伯奇，将延长，以获得更精细的结论有关的密度合理的解决方案，系统的丢番图equations.Number Theory研究的性质整数（"整数“）。从古代开始，丢番图方程（用整数求解的方程）的研究就已经成为数论的核心组成部分，并且最近影响了代码和密码系统的发展（例如，应用于数据存储系统，如光盘，通信系统和银行安全）。

9970440This proposal is concerned with diophantine equations in many variables, in the context of the theory and application of the Hardy-Littlewood (circle) method. The proposer intends to pursue a number of investigations whose goal is an improved understanding of the existence and density of integer solutions of systems of homogeneous equations, and associated estimates for exponential sums of use in the circle method. Two themes play a large role in proposed investigations. In one direction, methods familiar from the application of the Hardy-Littlewood method to additive diophantine problems will be extended to handle diophantine equations more closely approximating general homogeneous equations. Here the proposers methods for handling exponential sums over binary forms, and over smooth numbers, are significant. In a second direction, the more elementary diagonalisation methods, originating in work of Brauer and Birch, will be extended so as to obtain more refined conclusions concerning the density of rational solutions to systems of diophantine equations.Number Theory studies the properties of integers (``whole numbers''). Since Antiquity, the study of diophantine equations (equations to be solved in integers) has formed a core component of Number Theory, and has recently influenced the development of codes and cryptosystems (applied, for example, in data storage systems such as compact disks, communications systems and banking security).