四元数多项式的零点

发布时间：2018-02-01 19:42

本文关键词： 四元数多项式二次零点重数本质数　出处：《国防科学技术大学》2015年博士论文　论文类型：学位论文

【摘要】：本文研究了四元数多项式的零点理论,主要内容和结论包括以下几点。对于单边四元数多项式,研究了点重数、扩展重数、和球面重数三者之间的联系,得到了多种计算它们的方法,证明了次数重数关系定理;在反构造方面,刻画了零点决定多项式的条件,引入了广义零点的概念,建立了多项式与广义零点组一一对应的关系定理;在根式可解性方面,给出了根式可解性定义并得到结论:一次或二次的单边四元数多项式有基于系数的根式解,而次数更高的单边四元数多项式一般没有根式解。对于二次双边四元数一般多项式,得到以下结论:第一,零点只能是孤立的、球面的、或者圆圈的;第二,零点集的连通分支数至多为八;第三,零点集只能是一个球面并上至多两个孤立零点、一个圆圈并上至多七个孤立零点、一个元素个数至多为八的非空离散集这三者之一。对于二次双边四元数标准多项式,提供了零点集的计算公式并且上述第三点加强为:零点集只能是某个四元数的共轭类、一个圈圈并上至多两个孤立点、一个元素个数至多为八的非空离散集这三者之一。最后,考虑了本质数猜想,给出猜想的一个否定回答,并且指出了二次双边四元数标准多项式零点集本质数的上确界。
[Abstract]:In this paper, the 00:00 theory of quaternion polynomials is studied. The main contents and conclusions are as follows. For unilateral quaternion polynomials, the relations among point multiplicity, extended multiplicity and spherical multiplicity are studied. Several methods of calculating them are obtained, and the theorem of multiplicity of degrees is proved. In the aspect of anti-construction, the conditions of 00:00 determinant polynomial are described, the concept of generalized 00:00 is introduced, and the relation theorem between polynomial and generalized 00:00 group is established. In the aspect of radical solvability, the definition of radical solvability is given and the conclusion is drawn that the one-sided quaternion polynomial of one or two times has the root solution based on coefficients. For the general polynomial of quadratic bilateral quaternion, the following conclusions are obtained: first, 00:00 can only be isolated, spherical, or circular; Second, the number of connected branches of the 00:00 set is at most eight; Third, the 00:00 set can only be one sphere and up to two isolated 00:00, a circle and up to seven isolated 00:00. One of the three nonempty discrete sets with a number of elements at most eight. For a quadratic bilateral quaternion standard polynomial. The calculation formula of the 00:00 set is provided and the third point above is strengthened as follows: 00:00 set can only be a conjugate class of some quaternion, a circle and up to two isolated points. A nonempty discrete set with at most eight elements is one of the three. Finally, the essential number conjecture is considered and a negative answer to the conjecture is given. The upper bound of the essential number of the 00:00 set of quadratic bilateral quaternion standard polynomial is also pointed out.
【学位授予单位】：国防科学技术大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O174.14

【参考文献】