整体域上二次 Norm 型方程整点存在性及应用

Norm form equations are classical diophantine equations. It has a long history and wide applications. This project is aimed at the existence of integral points on quadratic norm form equations over global fields and its applications on coding theory and cryptography..Specifically, given a quadratic extension E/F of global fields, we consider equations of the form N_{E/F}(x) = a, where x and a are integral elements in E and F, respectively. .We focus on the cases where F is the rational number field, the rational function field and the maximal real subfield of N-th cyclotomic field. Correspondingly, we obtain binary quadratic equations over the integral ring, binary quadratic equations over the polynomial ring of finite fields, and complex absolute value equations over the ring of cyclotomic integers. Under certain parameter constraints, we will investigate the existence of integral points on these equations, i.e. whether they are integrally solvable. The target is to give explicit description on the existence of integral points. .For complex absolute value equations over the ring of cyclotomic integers, we mainly need their unsolvability, and will apply it on the non-existence of objects in coding theory and cryptography, such as bent functions, different sets, sequences and so on.

Norm型方程是一类经典的丢番图方程,有着悠久历史和广泛应用.本项目主要研究整体域上二次Norm型方程的整点存在性及其在编码和密码学上的应用..具体来说,给定一个整体域的二次扩张E/F,我们考虑形如N_{E/F}(x)=a的方程,其中x和a分别是E和F上的整元..我们重点研究F为有理数域,有理代数函数域,以及N次分圆域的极大实子域这三种情形.相应地,我们分别得到整数环上的二元二次丢番图方程,有限域多项式环上的二元二次方程,以及分圆整数环上的复范数方程.在一定参数约定下,我们将考察这些方程的整点存在性,即是否整可解.目标是给出整点存在性的明显刻画..对于分圆整数环上的复范数方程,我们主要希望证明其不可解性,并将其应用于编码和密码学中的广义Bent函数,差集以及序列等对象的不存在性证明.

给定一个整体域的二次扩张 E/F, 考虑形如 N_{E/F}(x) = a 的方程, 其中 x 和 a 分别是 E 和 F 上的整元, 本项目在 F 为有理数域, 有理代数函数域, 以及 N 次分圆域的极大实子域这三种情形, 拟研究其整点存在性. 到结题为止的相应的主要结果为: 分别对 ZZ 和 FF_q[t] 上的二次型方程 ax^2 + bxy + cy^2 + g = 0, 在少量附加条件下, 得到了整点存在的明显刻画. 将 [A. N. Skorobogatov and Y. G. Zarhin. The Brauer group and the Brauer-Manin set of products of varieties. J. Eur. Math. Soc. (2014)] 的 Brauer 群和 Brauer-Manin 集在乘积下的保持 性质推广到任意代数簇, 去掉原文中的射影条件.