Related papers: On the Complete Solution to the Most General Fifth…
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…
Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…
A transformation method is applied to the second order ordinary differential equation satisfied by orthogonal polynomials to construct a family of exactly solvable quantum systems in any arbitrary dimensional space. Using the properties of…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…
This study is devoted to the polynomial representation of the matrix $p$th root functions. The Fibonacci-H\"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the…
In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation…
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information…
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of…
We obtain several Cauchy-like and Pellet-like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and the reformulation of the zeros of a scalar polynomial as the…
We derive some rational solutions for the multicomponent and matrix KP hierarchies generalising an approach by Wilson. Connections with the multicomponent version of the KP/CM correspondence are discussed.
For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n…
New modifications of the methods for simultaneous extraction of all roots of polynomials over an arbitrary Chebyshev system are elaborated. A cubic convergence of iterations is proved. The method presented is a generalisation of the…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The…
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
A general formula is presented for any order derivative of Chebyshev polynomials instead of the existing recursive relationship. Hence, the Chebyshev finite difference method is made applicable not only to second order problems but also to…
We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality…
We provide a method for solving the roots of the general polynomial equation a[n]*x^n+a[n-1]*x^(n-1)+..+a1*x+a0=0. To do so, we express x as a powerseries of s, and calculate the first n-2 coefficients. We turn the polynomial equation into…
This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for…