本文關(guān)鍵詞： 四元數(shù) 多項(xiàng)式 二次 零點(diǎn) 重?cái)?shù) 本質(zhì)數(shù)　出處：《國(guó)防科學(xué)技術(shù)大學(xué)》2015年博士論文　論文類(lèi)型：學(xué)位論文

【摘要】：本文研究了四元數(shù)多項(xiàng)式的零點(diǎn)理論,主要內(nèi)容和結(jié)論包括以下幾點(diǎn)。對(duì)于單邊四元數(shù)多項(xiàng)式,研究了點(diǎn)重?cái)?shù)、擴(kuò)展重?cái)?shù)、和球面重?cái)?shù)三者之間的聯(lián)系,得到了多種計(jì)算它們的方法,證明了次數(shù)重?cái)?shù)關(guān)系定理;在反構(gòu)造方面,刻畫(huà)了零點(diǎn)決定多項(xiàng)式的條件,引入了廣義零點(diǎn)的概念,建立了多項(xiàng)式與廣義零點(diǎn)組一一對(duì)應(yīng)的關(guān)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(jié)論:一次或二次的單邊四元數(shù)多項(xiàng)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(xiàng)式一般沒(méi)有根式解。對(duì)于二次雙邊四元數(shù)一般多項(xiàng)式,得到以下結(jié)論:第一,零點(diǎn)只能是孤立的、球面的、或者圓圈的;第二,零點(diǎn)集的連通分支數(shù)至多為八;第三,零點(diǎn)集只能是一個(gè)球面并上至多兩個(gè)孤立零點(diǎn)、一個(gè)圓圈并上至多七個(gè)孤立零點(diǎn)、一個(gè)元素個(gè)數(shù)至多為八的非空離散集這三者之一。對(duì)于二次雙邊四元數(shù)標(biāo)準(zhǔn)多項(xiàng)式,提供了零點(diǎn)集的計(jì)算公式并且上述第三點(diǎn)加強(qiáng)為:零點(diǎn)集只能是某個(gè)四元數(shù)的共軛類(lèi)、一個(gè)圈圈并上至多兩個(gè)孤立點(diǎn)、一個(gè)元素個(gè)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(gè)否定回答,并且指出了二次雙邊四元數(shù)標(biāo)準(zhǔn)多項(xiàng)式零點(diǎn)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(xué)位授予單位】：國(guó)防科學(xué)技術(shù)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O174.14

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 黃禮平;Jordan Canonical Form of a Matrix over the Quaternion Field[J];東北數(shù)學(xué);1994年01期

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本文編號(hào)：1482713