Decomposition And Reconstruction Of One-sided Quaternion Polynomial

Although it has taken about two centuries since the birth of the the quaternion, there are still many unsolved problems for the quaternion theory and in the related filed of applied sciences. These problems not only are the important questions for the quaternion theory, but also the focus points in the filed of the other subjects such as the noncommutative algebras, and non-commutative geometry. This dissertation mainly contains our researching works about the method of solving quaternion equations, the decomposition of quaternion polynomials and the relationships between the coefficients and the roots of the quaternion equation. Concretely, the main contents contains:The fundamental notions and results of the quaternion theory, the principle properties of the quaternion polynomials, the related results which are referred in the dissertation such as the relationship between the zero points and the decompositions of the quaternion polynomials, and the background and the current development of the problems about the zero points of quaternion polynomials and the decompositions of the quaternion polynomials are contained in the chapter I and II.The chapter III contains our works about the topological isolation property of the isolated zeros and the computation of the multiplicities of the zeros of one-side polynomials. Firstly, based on the classification of the roots of the quaternion equations and the notion of the multiplicity, using the theory about the extracting roots of the quaternion equations, we prove that the isolated zero points are topologically isolated(Theorem 3.2).Furthermore, we also give a method of computing the multiplicity of the zero points.The content of the chapter IV are our works about the decomposition of quaternion polynomials. We introduce a notion of the fundamental polynomials, and a sufficient and necessary conditions for a polynomial to be a fundamental polynomial(Theorem 4.1).Using fundamental polynomials, we give a general theory about the decomposition of quaternion polynomials: every polynomial can be decomposed in the sense of the fundamental polynomials, and the decomposition is unique in a certain sense(Theorem 4.3,Theorem 4.4). We also give algorithm for the decompositions of quaternion polynomials.By the research about the structure of the polynomials which have the fixed points as zero points, we also do some works about the reconstruction problems of quaternion polynomials. We find that there is only one polynomials with degree n which has the fixed n points as zero points. And this polynomials can be obtained by these n fixed points(Theorem 4.5and Remark 4.4.1).In the chapter V, though the researching about the relationship between the coefficients and the roots of the quaternion equations, we give a necessary condition for a polynomial to be a fundamental polynomial and a sufficient and necessary condition for a quadratic polynomial to be a fundamental polynomial.(Theorem 5.3).