Quadratic Split Quaternion Polynomials: Factorization and Geometry

Factorization of Quaternionic Polynomials of Bi-Degree (n,1)

We consider polynomials of bi-degree $(n,1)$ over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot …

Zeros and factorizations of quaternion polynomials: the algorithmic approach

It is known that polynomials over quaternions may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial. In this paper, we equivalently describe the zero structure of a polynomial in terms of its left and right spherical divisors as well as in terms of left and right indecomposable divisors. Several algorithms are proposed to find left/right zeros and left/right spherical divisors of a quaternion polynomial, to construct a polynomial with prescribed zero structure and more generally, to constr…

Factorization of Dual Quaternion Polynomials Without Study's Condition

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained b…

Quadratic formulas for split quaternions

Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive explicit formulas for computing the roots of $x^{2}+bx+c=0$ in split quaternion algebra.

The column and row immanants of matrices over a split quaternion algebra

The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a non-split quaternion algebra are introduced, and their basic properties are investigated. The key theorem about the column and row immanants of a Hermitian matrix over a split quaternion algebra is proved. Based on this theorem an immanant of a Hermitian matrix over a split quaternion algebra is introduced.

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.

On left spectrum of a split quaternionic matrix

In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations $ax=b$ and $ax^2+bx+c=0$ over split quaternions, we make an attempt to understand the left spectrum of a split quaternion matrix of order 2.

微分四元数代数的分裂

Splitting of differential quaternion algebras

本文研究带导子的四元数代数的微分分裂域问题，指出其未必在代数扩张上分裂，并利用 Riccati 方程的解给出分裂域超越次数的界。

分裂半四元数代数上的Rota-Baxter代数结构

The Rota-Baxter algebra structures on split semi-quaternion algebra

本文研究分裂半四元数代数上的Rota-Baxter算子，给出其矩阵刻画与所有权重下所有算子的矩阵表示，并证明Ma等人关于Sweedler代数的相关结果是本文结论的特例。

OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split

分裂定义在有限域扩张上的四元数代数

Splitting quaternion algebras defined over a finite field extension

该研究通过分析域上二次型系统的2-扩张迷向性，得到有限域扩张上四元数代数的特定分裂域，并证明次数16的中心单代数可被次数至多2^16的2-扩张分裂。

The Moore-Penrose inverses of split quaternions

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.

允许分解为线性因子的运动多项式

Motion Polynomials Admitting a Factorization with Linear Factors

本文研究有界运动多项式的线性因子分解，给出充要条件、计算算法及余因子求法的改进，可用于将有理运动分解为简单旋转或平移并构造机构。

The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

The two-sided quaternionic Fourier transformation (QFT) was introduced in \cite{Ell:1993} for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations \cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions was introduced, then called $\pm$split. In the current \change{chapter} we analyze this split further, interpret it geometrically as \change{an} \emph{orthogonal 2D planes split} (OPS), and generalize it to a freely steerable split of $\H$ into two orthogonal 2D analysis planes. The new general form of the OPS split a…

Splitting quaternion algebras over quadratic number fields

We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.

Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2

We give examples of quaternion and octonion division algebras over a field $F$ of characteristic $2$ that split over a purely inseparable extension $E$ of $F$ of degree $\geq 4$ but that do not split over any subextension of $F$ inside $E$ of lower exponent, or, in the case of octonions, over any simple subextension of $F$ inside $E$. Thus, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms.

Some remarks regarding Quaternions and Octonions

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved. Moreover, we give a set of invertible elements in split quaternion algebras and in split octonion algebras.

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Keywords: Quaternionic adaptive filtering, Hypercom…

一类扭曲偏 Hopf 作用

A class of twisted partial Hopf actions

提出利用分裂四元数、分裂半四元数及 1/4-四元数构造扭曲偏 Hopf 作用的新方法，并由 Sweedler Hopf 代数的扭曲偏作用得到一类新实例。

About some split central simple algebras

In this paper we study certain quaternion algebras and symbol algebras which split.

Hyperbolic Unit Groups and Quaternion Algebras

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R.

Totally decomposable quadratic pairs

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in [3].

From the Fundamental Theorem of Algebra to Kempe's Universality Theorem

This article provides a gentle introduction for a general mathematical audience to the factorization theory of motion polynomials and its application in mechanism science. This theory connects in a rather unexpected way a seemingly abstract mathematical topic, the non-unique factorization of certain polynomials over the ring of dual quaternions, with engineering applications. Four years after its introduction, it is already clear how beneficial it has been to both fields.

双曲 Jacobsthal 旋量序列及其数学性质

Hyperbolic Jacobsthal Spinor Sequences and Their Mathematical Properties

本文通过结合旋量与分裂四元数，首次定义了Jacobsthal、Jacobsthal-Lucas及多项式的双曲旋量序列，推导了其递推关系、生成函数、Binet公式及相关恒等式。