Related papers: Splitting quaternion algebras over quadratic numbe…
In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra…
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
We give an algorithm for factoring quadratic polynomials over any UFD, Z in particular. We prove the correctness of this algorithm and give examples over Z and Z[i].
We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…
The multiplication of superpositions of numbers is a core operation in many quantum algorithms. The standard method for multiplication (both classical and quantum) has a runtime quadratic in the size of the inputs. Quantum circuits with…
Let A be a modular abelian variety over \Q of arbitrary even dimension. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over \bar\Q. We accomplish this by…
Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an…
We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put…
Quantum computers are known to be qualitatively more powerful than classical computers, but so far only a small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to…
In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$…
A simple algorithm to compute all the zeros of a generic polynomial is proposed.
Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several…
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…
In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more…
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of…
Monotone inclusions involving the sum of three maximally monotone operators or more have received much attention in recent years. In this paper, we propose three splitting algorithms for finding a zero of the sum of four monotone operators,…
This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines. To this end, the division algebra of quaternions is used to derive an effective model for representing computation…
Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form.…
In a companion paper, we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of…