Solution of a quadratic quaternion equation with mixed coefficients

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space $\mathbb{H}$. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space $\mathbb{H}$.Comment: 19 pages, to appear in the Journal of Symbolic Computatio

Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that the integer-valued polynomials on $S$ form a ring, and, on the other hand, sets $S$ such that the set of polynomials in $R[x]$ that are zero on $S$ is an ideal of $R[x]$.Comment: 9 pages, conference paper for "advances in algebra ..." at Ton Duc Thang University, Vietnam, Dec 18-20, 201

Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic $L^2$-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these $L^2$-spaces, the associated reproducing kernels and the ensuing quaternionic coherent states