Related papers: Explicit expressions for real roots of a quartic e…

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

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…

It appears that, along with many of my friends and colleagues, I had been brainwashed by the great and tragic lives of Abel and Galois to believe that no general formulas are possible for roots of equations higher than quartic. This seemed…

We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method…

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

Given a real cubic function $f(x)$ with three roots, take an equilateral triangle $ABC$, the projections of which vertices are the roots of $f(x)$. There is a folklore fact that the vertical lines through the extrema of $f(x)$ are tangent…

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…

In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

An expression for four-tangle is obtained by examining the negativity fonts present in a four-way partial transpose under local unitary operations. An alternate derivation of three tangle is also given.

In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results…

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.…

This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.

In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…

This paper derives numerical bounds for and implements the splitting circle method for finding roots of a univariate polynomial in the presence of fixed precision.

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…

In this paper, we will derive the real roots of certain sets of matrices with real entries. We will also demonstrate that real orthogonal matrices can have real root or be involutory. Eventually, we will represent idempotent matrices in a…