本文關鍵詞： 四元數(shù) 多項式 二次 零點 重數(shù) 本質(zhì)數(shù)　出處：《國防科學技術大學》2015年博士論文　論文類型：學位論文

【摘要】：本文研究了四元數(shù)多項式的零點理論,主要內(nèi)容和結論包括以下幾點。對于單邊四元數(shù)多項式,研究了點重數(shù)、擴展重數(shù)、和球面重數(shù)三者之間的聯(lián)系,得到了多種計算它們的方法,證明了次數(shù)重數(shù)關系定理;在反構造方面,刻畫了零點決定多項式的條件,引入了廣義零點的概念,建立了多項式與廣義零點組一一對應的關系定理;在根式可解性方面,給出了根式可解性定義并得到結論:一次或二次的單邊四元數(shù)多項式有基于系數(shù)的根式解,而次數(shù)更高的單邊四元數(shù)多項式一般沒有根式解。對于二次雙邊四元數(shù)一般多項式,得到以下結論:第一,零點只能是孤立的、球面的、或者圓圈的;第二,零點集的連通分支數(shù)至多為八;第三,零點集只能是一個球面并上至多兩個孤立零點、一個圓圈并上至多七個孤立零點、一個元素個數(shù)至多為八的非空離散集這三者之一。對于二次雙邊四元數(shù)標準多項式,提供了零點集的計算公式并且上述第三點加強為:零點集只能是某個四元數(shù)的共軛類、一個圈圈并上至多兩個孤立點、一個元素個數(shù)至多為八的非空離散集這三者之一。最后,考慮了本質(zhì)數(shù)猜想,給出猜想的一個否定回答,并且指出了二次雙邊四元數(shù)標準多項式零點集本質(zhì)數(shù)的上確界。
[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.
【學位授予單位】：國防科學技術大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O174.14

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相關期刊論文 前1條

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本文編號：1482713