Related papers: Splitting quaternion algebras over quadratic numbe…
We calculate the zeros of an exponential polynomial of three variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the…
In this paper we give new methods to construct zero divisors in A_n =R^(2^n) the Cayley_Dickson algebras over the real numbers, for n larger than 4, and we also relate the set of zero divisors in A_{n+1} with the Stiefel Manifold V_{2^n…
In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations…
The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero…
In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools…
Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically…
We present an algorithm, based on the explicit formula for $L$-functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically…
We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…
In this paper we study certain quaternion algebras and symbol algebras which split.
Let $n=a^2b$, where $b$ is square-free. In this paper we present an algorithm based on class groups of binary quadratic forms that finds the square-free decomposition of $n$, i.e. $a$ and $b$, in heuristic expected time: $$…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n…
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
In this paper we adapt the RBF Solver to work with quaternions by taking advantage of their Lie Algebra and exponential map. This will allow to work with quaternions as if they were normal vectors in R^3 and blend them in a very efficient…
In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots.…
We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We consider X a smooth completion of the complex plane and D an effective divisor supported on the boundary of…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix…
We study algebras $A,$ over a field of characteristic zero, satisfying $(x^p, x^q, x^r)=0$ for $p, q, r$ in ${1, 2}.$ The existence of a unit element in such algebras leads to the third power-associativity. If, in addition, $A$ has degree…
In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic…