Diophantine Problems in Many Variables

9970440本文在Hardy-Littlewood（圆）法的理论和应用的背景下，讨论了多变量的丢番图方程。提议者打算进行一些研究，其目标是提高对齐次方程系统的整数解的存在性和密度的理解，以及在圆法中使用的指数和的相关估计。两个主题在拟议的调查中起着重要作用。一方面，将Hardy-Littlewood方法应用于加性丢番图问题中所熟悉的方法推广到处理更接近一般齐次方程的丢番图方程。在这里，提议者处理二进制形式和光滑数上的指数和的方法是有意义的。在第二个方向上，源于Brauer和Birch工作的更初级的对角化方法将得到推广，以便得到关于丢芬图方程系统有理解密度的更精细的结论。数论研究整数的性质（“整数”）。自古以来，对丢芬图方程（以整数形式求解的方程）的研究已经形成了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).