Related papers: On polynomial equations over split octonions

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…

Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other…

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a "companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose…

We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a…

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…

We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…

Octonions are 8-dimensional hypercomplex numbers which form the biggest normed division algebras over the real numbers. Motivated by applications in theoretical physics, continuous octonionic analysis has become an area of active research…

We consider a polynomial version of the Cayley numbers. Namely, we define the ring of Cayley polynomials in terms of generators and relations in the category of alternative algebras. The ring turns out to be an octonion algebra over an…

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…

In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that…

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…

We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for…

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform…

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…

We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical…

For any positive integer n, a new family of periodic functions in power series form and of period n is used to solve in closed form a class of polynomial equation of order n. The n roots are the values of the appropriate function from that…

In this paper, we generalized De Moivre's formula and Euler's formula to octonions and find the roots of generalized octonions using these formulae.

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…

Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the ${\tilde {\partial}}$-equations are studied. More generally functions of several…