有限域上代数簇的指数和及其应用

In the theory of Diophantine equations it is a central problem to study rational points on varieties. For the number of rational points on a variety over a finite field, it can be expressed in terms of exponential sums. So the study of rational points on varieties is reduced to the study of exponential sums. The Kloosterman sums contain rich information about arithmetic and geometry of the varieties and play a crucial role in the number theory. Especially it has a pivotal position in the development of analytic number theory. The degrees and the Smith normal form of polynomials in the variety play a crucial role in the estimates of exponential sums. However, in the literature, there are a few research works on the exponential sum in the case that the degree changing and the index of the undetermined element is zero. The main research objects of this project are the exponential sum of algebraic varieties over finite fields, the progressive form of generalized Kloosterman sums, and the estimation of multiple linear Kloosterman sums, with focus on the effects of degree reductions and the Smith normal form on estimates of exponential sums. Then we study the number of rational points of algebraic clusters in finite fields by using exponential sums. Meanwhile, we will study the application of these theoretical knowledge in coding and cryptography.

代数簇上的有理点问题是丢番图方程的中心问题。而指数和可以表示有限域上代数簇的有理点个数，从而对有限域上代数簇的有理点研究可归结为对指数和的研究。指数和中的Kloosterman和，蕴含了重要的算术与几何信息，在数论领域历来受到高度的重视，特别是在解析数论的发展中具有举足轻重的地位。代数簇中多项式（组）的次数及未定元指数所构成矩阵的Smith正规型在指数和的估计中起着重要的作用，但已有文献中很少有对于次数变动及未定元的指数为零情形下指数和的研究成果。本项目的主要研究对象是有限域上代数簇的指数和及广义Kloosterman和的渐进式和多重线性Kloosterman和的估计，重点研究降次和未定元指数阵的Smith正规型对指数和的影响，并利用指数和研究有限域上代数簇的有理点个数，同时我们将研究这些理论在编码和密码中的具体应用。

代数簇上的有理点问题是丢番图方程的中心问题。本项目主要研究了有限域上代数簇的有理点问题及在Melas码中的应用。首先，主要利用指数和理论给出了有限域上几类特殊对角方程的有理点的解数。其次，利用代数簇中多项式（组）的次数及未定元指数所构成矩阵的Smith正规型给出了有限域上一类代数簇的有理点的解数公式。最后，探索了有限域上的代数簇的解数在编码和密码中的应用，特别是通过确定方程组的解研究在一定条件下Melas码的覆盖半径问题。