多项式映射是仿射代数几何领域的主要研究对象之一，其中雅可比猜想，tame生成子问题都是该领域具有代表性的公开难题。国际上许多著名数学家都对这一领域的发展做出了重要贡献。Bass等人用稳定化方法证明了，要证明雅可比猜想成立，只需证明雅可比猜想对所有三次齐次多项式映射都成立。Drużkowski进一步证明了要证雅可比猜想成立，只需对Drużkowski映射，即三次线性幂映射证明雅可比猜想成立即可。为了研究这类映射，需要研究Drużkowski矩阵，然而Drużkowski矩阵至今还没有一个很好的描述。为了描述一般的Drużkowski矩阵，本项目将利用仿射簇和组合矩阵论的知识工具系统地研究具有幂零性簇的矩阵并给出具有幂零性簇的矩阵的刻画，从而从中找出更一般的Drużkowski矩阵，进而研究雅可比猜想和tame生成子问题。因此，该项目对于雅可比猜想和tame生成子问题的深入理解有重要意义。

Polynomial mapping is one of the main research objects in the field of affine algebraic geometry. Among them, the Jacobian conjecture and the tame generators problem are representative open problems in the field. Many famous mathematicians in the world have made important contributions to the development of this field. Bass et al. used a stabilization method to prove that the Jacobian conjecture holds if it holds for all cubic homogeneous polynomial mappings. Drużkowski further proved that the Jacobian conjecture holds if it holds for all the Drużkowski maps, that is, all the cubic linear mappings.In order to study this kind of maps, we need to study Drużkowski matrices. However, the Drużkowski matrix has not been well described yet. In order to describe general Drużkowski matrices, this project will use the knowledge of affine varieties and combinatorial matrix theory to systematically study matrices with nilpotency varieties and describe the matrices with nilpotency varieties, so as to find more general Drużkowski matrices, and then study the Jacobian conjecture and the tame generators problem. Therefore, the project has important implications for the in-depth understanding of the Jacobian conjecture and the tame generator problem.