Related papers: On polynomial equations over split-octonions: the …
We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the…
We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.
In this paper, with the help of trinomial coefficients we study some arithmetic properties of certain determiants involving reciprocals of binary quadratic forms over finite fields.
In this paper, we introduce h(x)-Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K (K = R,C), which generalize both h(x)-Fibonacci quaternion polynomials and h(x)-Fibonacci octonion polynomials. For…
In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic…
This paper explores quadratic forms over finite fields with associated Artin-Schreier curves. Specifically, we investigate quadratic forms of $\mathbb F_{q^n}/\mathbb F_q$ represented by polynomials over $\mathbb F_{q^n}$ with $q$ odd,…
Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact…
This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…
The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents…
We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex…
We study solutions of exponential polynomials over the complex field. Assuming Schanuel's conjecture we prove that certain polynomials have generic solutions in the complex field.
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
We integrate nonabelian Toda field equations for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants. In the appendix we review some results concerning…
In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply…
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 $E$…
Fast algorithms for arithmetic on real or complex polynomials are well-known and have proven to be not only asymptotically efficient but also very practical. Based on Fast Fourier Transform (FFT), they for instance multiply two polynomials…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…