三维Clifford代数无基底刻画的完全性及其多项式标准型的完全分类

计算机代数在处理有限维代数问题时，通常是通过基底来进行计算. 然而为了简化符号处理的难度，用一些与基底无关的导出的代数性质，也可以对无基底的形式进行符号处理. 研究无基底符号子系统的核心问题是：该子系统在处理与基底无关的问题时与基于基底处理是否等价，既该符号子系统的完全性问题. 在四元数变元生成的多项式环中，一个四元数变元的实部、两个变元的内积和三个变元刻画的体积与环中的任何元素可交换. 这三种交换性就是该环的完全无基底刻画. 三维Clifford 代数是四元数的推广，在由三维向量变元通过Clifford 乘积生成的多项式环中，是否也存在着完全的无基底刻画？同样地，在内积空间出现退化时，是否存在着完全的无基底刻画？本课题将利用Grobner基的方法构造并证明三维Clifford 多项式环无基底刻画的完全性，同时给出多项式标准型的完全分类.

When dealing with finite dimensional algebraic problems, computer algebra systems usually accomplish calculations according to the multiplicative principles of bases. However, this approach involves difficult symbolic computations. We can perform symbolic calculations unrelated to bases by utilizing some algebraic properties in order to reduce the computational complexity. The consistence of dealing with problems independent of bases on a symbolic subsystem and the treatment of problems with bases is a staple point in the research of symbolic subsystems unrelated to bases and it is also the completeness problem of the symbolic subsystem. In the non-commutative polynomial ring generated by quaternion variables, the real part of a quaternion, the inner product of two variables and the volume of three variables are commutative with elements in the ring. Furthermore, the three types of commutativity give a complete description unrelated to bases of this kind of ring. Three dimensional Clifford algebra is a generalization of quaternion. An essential problem is the completeness of description unrelated to bases still available in the non-commutative polynomial ring generated by three dimensional vector variables in Clifford algebra. Moreover, can complete descriptions unrelated to bases still be achieved in degenerative inner product spaces and spaces of Clifford algebra with high dimension? In this project, the completeness of description unrelated to bases of Clifford polynomial ring is proved by the method of Groebner basis and the classification of the normalized polynomial forms is also achieved. This project generalizes the complete description independent of bases from quaternion to three dimensional Clifford algebra and expand the range of establishing symbolic systems unrelated to bases.