Related papers: An Algorithm for the Factorization of Split Quater…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
In this paper, we intend to present a new algorithm to factorize large numbers. According to the algorithm proposed here, we prove that there is a common factor between p and q. With this procedure, the time of factorization considerably…
This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
The separability assumption (Donoho & Stodden, 2003; Arora et al., 2012) turns non-negative matrix factorization (NMF) into a tractable problem. Recently, a new class of provably-correct NMF algorithms have emerged under this assumption. In…
Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
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…
We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…
We consider algorithms for the factorization of linear partial differential operators. We introduce several new theoretical notions in order to simplify such considerations. We define an obstacle and a ring of obstacles to factorizations.…
We describe three algorithms for computer-aided symbolic multi-loop calculations that facilitated some recent novel results. First, we discuss an algorithm to derive the canonical form of an arbitrary Feynman integral in order to facilitate…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…
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…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…