有限域上的置换与勒让德矩阵

有限域上的置换与勒让德矩阵在组合数论中有着广泛的应用。对其的研究需要综合运用数论、组合数学、线性代数、有限域、椭圆曲线等数学分支中的工具。近年来该领域产生了许多有价值的结果与未解决的问题。本项目中我们将聚焦于以下三个方面：.(1) 利用分圆多项式、组合同余式等工具研究一类涉及有限域上平方元与本原元素的置换问题。 .(2) 我们计划以Cohen与Huczynska发展的本原元素特征函数为基本工具，利用有限域上特征和的性质来研究一类与有限域上平方元以及本原元素相关的加法组合问题。.(3) 通过大量数值计算，我们发现一类涉及二元二次型的勒让德矩阵的行列式与有限域上椭圆曲线的有理点个数密切相关。我们计划利用椭圆曲线与Jacobsthal和的性质来研究这类行列式与椭圆曲线上有理点个数之间的关系。

Permutations over finite fields as well as Legendre matrices have many applications in combinatorial number theory. The investigation of this topic involves tools in many branches of mathematics, such as number theory, combinatorics, linear algebra, finite fields, elliptic curves and so on. Recently, many valuable results and unsolved problems have emerged in this field. In this project, we will focus on the following aspects:. (1) By using some tools such as cyclotomic polynomials and combinatorial congruences, we plan to study permutations involving square elements and primitive elements over finite fields.. (2) By using the character functions of primitive elements developed by Cohen and Huczynska as the basic tool, and combining the properties of character sums, we plan to study some problems in additive combinatorics concerning square elements and primitive elements over finite fields. . (3) By computation we find that the determinants of Legendre matrices concerning binary quadratic forms are closely related to the number of rational points on elliptic curves over finite fields. We plan to study the relationship between such determinants and the number of rational points on elliptic curves by using some properties of elliptic curves and Jacobsthal sums.